diff --git "a/JNE1T4oBgHgl3EQfsAWQ/content/tmp_files/load_file.txt" "b/JNE1T4oBgHgl3EQfsAWQ/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/JNE1T4oBgHgl3EQfsAWQ/content/tmp_files/load_file.txt" @@ -0,0 +1,2000 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf,len=1999 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='03361v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='QA] 9 Jan 2023 FINITE-DIMENSIONAL POINTED HOPF ALGEBRAS OVER FINITE SIMPLE GROUPS OF LIE TYPE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' SEMISIMPLE CLASSES IN PSLn(q) AND PSp2n(q) NICOL´AS ANDRUSKIEWITSCH, GIOVANNA CARNOVALE AND GAST´ON ANDR´ES GARC´IA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We show that the Nichols algebra of a simple Yetter-Drin- feld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We prove that orbits of irreducible elements in the projective special linear groups could not be treated with our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We conclude that any finite-dimensional pointed Hopf algebra H with group of group- like elements isomorphic to PSLn(q) (n ≥ 4), PSL3(q) (q > 2), or PSp2n(q) (n ≥ 3), is isomorphic to a group algebra, completing work in arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='06794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Introduction 1 Acknowledgements 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Racks 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Algebraic groups 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Split conjugacy classes 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The special linear groups 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Semisimple conjugacy classes represented in K 29 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The symplectic groups 37 References 41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The conjugacy class of x ∈ G is denoted by OG x or Ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The subgroup of G generated by I ⊂ G is denoted by ⟨I⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider the following properties of a conjugacy class O of G: 2010 Mathematics Subject Classification: 16T05, 20D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Keywords: Nichols algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hopf algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' rack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' finite group of Lie type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 1 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA (C) There are H < G and r, s ∈ H ∩ O such that rs ̸= sr, H = ⟨OH r , OH s ⟩, OH r ̸= OH s and min{|OH r |, |OH s |} > 2 or max{|OH r |, |OH s |} > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (D) There exist r, s ∈ O such that O⟨r,s⟩ r ̸= O⟨r,s⟩ s and (rs)2 ̸= (sr)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (F) There are ra ∈ O, a ∈ I4 = {1, 2, 3, 4}, such that O⟨ra:a∈I4⟩ ra ̸= O⟨ra:a∈I4⟩ rb and rarb ̸= rbra for a ̸= b ∈ I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We say that O is of type C, D, F when the corresponding property holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As explained in the Introduction to [5], the next question has profound im- plications for the classification of finite-dimensional pointed Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Determine which conjugacy classes of a given finite (non- abelian) group G are of type C, D or F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, if a conjugacy class O is type C, D or F, then any Nichols algebra of group type with support isomorphic to O has infinite dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' for brevity we say in this case that O collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For the purposes of this paper further precision on Nichols algebras is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If O is neither of type C, D nor F then we say that it is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It follows at once from the previous definitions that if O ∩ H is either abelian or a single conjugacy class of H for any H ≤ G, then O is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Intuitively, the criteria of types C, D and F are inductive arguments that are more flexible in the language of racks, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Conjugacy classes are the prototypical examples of racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' One may wonder whether there are other inductive arguments that force the collapse of a conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this respect we say that a rack is sober if every subrack is either abelian or indecomposable [1, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' and austere if every subrack generated by two elements is either abelian or indecomposable [3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Clearly, sober implies austere and austere implies kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Subsidiary to Question 1, we propose: Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Determine which conjugacy classes of a given finite (non- abelian) group G are sober or austere, or kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Simple groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It is natural, and convenient, to start addressing Questions 1 and 2 by assuming that G is simple non-abelian, see [9] for the importance of this reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' When G is alternating or sporadic, this was addressed in [7, 8, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The series of papers [1, 2, 3, 4, 5, 12] treat the case when G is simple of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the first five papers, Questions 1 and 2 were answered for non-semisimple conjugacy classes in Chevalley or Steinberg groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The sixth is devoted to Suzuki and Ree groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the present paper we deal with semisimple conjugacy classes in the classical Chevalley groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The main difficulty is due to the deeper influence of arithmetics, as opposed to the unipotent classes and the mixed classes, which can be reduced in most of the cases to a unipotent one in a smaller group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We summarize our main results and then discuss the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 3 Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let O ̸= {e} be a semisimple conjugacy class in a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) If G = PSLn(q), then any O not listed in Table 1 collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If G = PSpn(q), then any O collapses with the possible exception of the orbit of non-trivial involutions if n = 2 and q ∈ {3, 5, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Part (i) is proved in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 for the cases with small q excluded in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Part (ii) is proved in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For other Chevalley groups, there is substantial information in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Roughly the proofs of these results go as follows: pick a simple group G, a surjective morphism of groups π : G → G and conjugacy classes O and O in G and G respectively, such that π(O) = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then look at the subgroups H intersecting O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If H ∩ O splits as more than one conjugacy class of H for one H ≤ G, then work out the details to have that O is of type C or D and that this is preserved by the projection π : O → O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' When G is of Lie type, the subgroup H is usually found by looking in one way or another at the structure of the algebraic group behind G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' But if π(H)∩O is either abelian or one conjugacy class of π(H) for every H ≤ G, then O is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' When this happens, usually G is ‘small’ and has few subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the present paper, we found that it also happens to any conjugacy class O of an irreducible element in PSLn(q) where n is an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' To show this we used the main result of [17] to get the list of the H ≤ G intersect- ing O together with some arithmetic manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This outcome differs significantly with the results of the previous of the series and underlines the connection of semisimple classes with arithmetical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Kthulhu semisimple classes in PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' n q class Remark 2, PSL2(2) ≃ S3 (3) abelian 3, PSL2(3) ≃ A4 (22) abelian 4, PSL2(4) ≃ A5 (5) sober 2 5, PSL2(5) ≃ A5 (1, 22) sober 9, PSL2(9) ≃ A6 (1, 5) kthulhu even and not a square irreducible, order 3 sober all irreducible, order > 3 sober odd prime all irreducible kthulhu 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Applications to Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As in previous papers, we say that a finite group G collapses if every finite-dimensional pointed Hopf algebra H with G(H) ≃ G is necessarily isomorphic to CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As a corollary of our main Theorem and results from previous papers in the series, we obtain new families of groups that collapse, see Theorem III, extending [3, Theorem 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For this, we first draw the complete list of kthulhu classes in the simple groups PSLn(q) and PSp2n(q) for n ≥ 2, any q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This information combines the main result of this paper with a corrected version of [1, Table 1], [3, Table 3] and [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1] and a careful analysis of the cases of the groups PSL2(q) for q = 2, 3, 4, 5, 7, 9, where some exceptions occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We point out that the classes labeled (1r1, 2) in Sp2n(9), occurring in [3, Tables 3,5] are in fact not kthulhu: they are of type C, as each of them includes a non-trivial unipotent class of type (2) in PSL2(9) ≃ A6 which is of type C, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, the classes of involutions in PSL2(7) in [3, Table 1] are not not kthulhu: since PSL3(2) ≃ PSL2(7), they are of type C by [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be either PSLn(q) or PSp2n(q), n ≥ 2 and let O be a non-trivial conjugacy class in G different from the class of a split involution in PSp4(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O is kthulhu if and only if it occurs in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Kthulhu classes, G = PSLn(q) or PSp2n(q), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G n q type of class description/label even or else odd and not a square unipotent (2) 5 semisimple involution PSLn(q) 2 all semisimple irreducible,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' |x| > 3 even and not a square semisimple irreducible |x| = 3 3 2 unipotent (3) odd prime all semisimple irreducible even unipotent W(1)a ⊕ V (2) PSp2n(q) ≥ 2 odd and not a square unipotent (1r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 2) even unipotent W(2) 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 semisimple split involution It remains open to determine whether the conjugacy class of split involu- tions in PSp4(7) is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The next result combines [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4], [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2] and [5, The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2] with Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The groups PSLn(q) with n ≥ 4, PSL3(q) with q > 2, and PSp2n(q), n ≥ 3, collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ In the group PSL3(2), there is just one class that could not be treated, namely the regular unipotent class O, which is sober.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Actually PSL3(2) ≃ PSL2(7) and for this group, O is semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If a ≤ b ∈ N0, then Ia,b denotes {a, a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , b};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' also Ia = I1,a for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The centraliser of X ⊂ G is denoted by CG(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x, y ∈ G, then x ⊲ y := xyx−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We write X ≥ Y , or Y ≤ X, to express that Y is a subrack of X (or a subgroup, or more generally a subobject in a given category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The normality of a subgroup is expressed by N ⊳ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let q = pm, p a prime and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let Fq be the field with q elements and k the algebraic closure of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We denote by Gn(k) the group of n-th roots of unity in a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Acknowledgements We thank Gunter Malle for very helpful email exchanges and Andrea Lucchini for pointing out several references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' was partially supported by CONICET (PIP 11220200102916CO), FONCyT-ANPCyT (PICT-2019-03660) and Secyt (UNC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' was partially supported by CONICET (PIP 11220200100423CO), Secyt (UNLP) and FONCyT-ANPCyT (PICT-2018-00858).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' was partially supported by Projects BIRD179758/17, DOR2207212/22, and BIRD203834 of the Uni- versity of Padova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The results were obtained during visits of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' to the University of Padova, and of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' to the University of C´ordoba, partially supported by the bilateral agreement between these Universities and the INdAM-GNSAGA Visiting Professor program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Racks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As in previous papers we use the language of racks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' see [9] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A rack is a pair (O, ⊲) where O is a non-empty set and ⊲ : O×O → O is a self distributive operation such that ϕx := x⊲ is bijective for any x ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A subset O′ ⊂ O is a subrack if O′ ⊲ O′ ⊂ O′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let InnO be the group generated by the image of the map ϕ : O → SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The main examples of racks considered in this paper are (unions of) conjugacy classes of a finite group with ⊲ being the conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A rack (O, ⊲) is abelian if x ⊲ y = y for any x, y ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, a rack is indecomposable if it can not be presented as the disjoint union of two subracks and decomposable otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The following observation will be useful, especially when dealing with orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be a finite group, N ⊳ G, g ∈ G − N and ON g the orbit of g under the conjugation action of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then ON g is a subrack of OG g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This is a special case of [13, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2] that can be verified directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Notice that if N ≤ G is not normal, then ON g may fail to be a subrack of OG g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For instance, take G = S4, g = (123) and N = ⟨(12)(34)⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' then ON g = {(123), (142)} is not closed under the rack operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Racks of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The following notion was introduced in [3, Defini- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3] motivated by [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A rack X is of type C if there are a decomposable subrack Y = R � S ≤ X, r ∈ R and s ∈ S such that r ⊲ s ̸= s, R = OInn Y r , S = OInn Y s , min{|R|, |S|} > 2 or max{|R|, |S|} > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The group-theoretical reformulation (C) of the definition of type C is [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We need a variation of [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8] in order to encompass the situation in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The proof can be repeated verbatim: we recall it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be a finite group, g ∈ G and N ⊳G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The orbit O = ON g is of type C if and only if there are H ≤ ⟨O⟩, r, s ∈ O ∩ H such that OH r ̸= OH s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) rs ̸= sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) H = ⟨OH r , OH s ⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) min{|OH r |, |OH s |} > 2 or max{|OH r |, |OH s |} > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that r, s and H are as above and set R := OH r and S := OH s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If r′ = h ⊲ r ∈ OH r = R for some h ∈ H, then there exist x1, · · · , xk ∈ O such that h = x1 · · · xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence r′ = x1 ⊲ (x2 ⊲ (· · · (xk ⊲ r)) ∈ O(⊲O(⊲ · · · O ⊲ O))) ⊂ O, so R ⊂ O∩H and similarly S ⊂ O∩H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) the subset Y := R � S ⊂ O is a decomposable subrack, and r ⊲ s ̸= s is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In addition, OInn Y r = O⟨R,S⟩ r = OH r = R, where the second equality follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3), and similarly, OInn Y s = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The estimate on R and S is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence O is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Conversely, let X = O and r, s, R, S, Y be as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Setting H := ⟨R, S⟩, we immediately have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3), H ≤ ⟨O⟩, R = OH r , S = OH s and r, s ∈ O ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G, N and O be as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (a) If there exist r, s ∈ O satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2), O⟨r,s⟩ r ̸= O⟨r,s⟩ s and: min{|O⟨r,s⟩ r |, |O⟨r,s⟩ s |} > 2 or max{|O⟨r,s⟩ r |, |O⟨r,s⟩ s |} > 4, then O is of type C by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 applied to H := ⟨r, s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 7 (b) If |g| is odd and r, s ∈ O, then for any H ≤ G containing r and s the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2), since then |s| = |r| = |g| is odd, whence |OH r | ≥ |O⟨s⟩ r | ≥ 3, and similarly for OH s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This generalizes [3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7 (b)] to the situation of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n ≥ 5 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the conjugacy class O of n-cycles in Sn is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, O splits into two classes O′ and O′′ in An and |O′| = |O′′| = (n−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 2 > n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, if r ∈ O′, there exists s ∈ O′′ such that s ̸∈ CAn(r) = ⟨r⟩ and the result follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The class O corresponding to the partition (12, 22) in A6 is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, H := CA6(56) ≃ S4 and H ∩ O contains all involutions of the form (ab)(cd) for a, b, c, d /∈ {5, 6} and those of the form (ab)(56) for a, b /∈ {5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, |O′ ∩ H| = 12 and O′ contains all involutions in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, the involutions in S4 are parted into two classes of size 6, and S4 contains non-commuting non-conjugate involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, we can find r, s ∈ H ∩ O′ such that r ⊲ s ̸= s and OH r ̸= OH s , with |OH r | = |OH s | = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally, ⟨OH s , OH r ⟩ = H because S4 is generated by its involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We conclude by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The class of 3-cycles in G = A3 or A4 is kthulhu because its intersection with any subgroup of G is either abelian or a conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The class O of 3-cycles in An for n ≥ 5 and the class O′ labeled (32) in A6 are of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, O ∩ A4 splits into the classes O(123) and O(124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since the representatives do not commute, O is of type C by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Any non-inner automorphism of S6 interchanges O and O′ in A6, so O′ is of type C as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here is an easy but useful application of the previous Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G be a finite group, H ≤ G, x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that H is not abelian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5) H = ⟨OH x ⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) there exists s ∈ OG x ∩ H : s /∈ OH x , |OH s | > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7) Then OG x is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There is r ∈ OH x such that rs ̸= sr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' otherwise s ∈ Z(H) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6), hence |OH s | = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) holds and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) are clear by construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally, |OH r | > 2 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5), thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Here is another way to detect racks of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G1 and G2 be finite groups, a1 ̸= b1 ∈ G1, a2, b2 ∈ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Set r = (a1, a2), s = (b1, b2) ∈ G := G1 × G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that a1b1 = b1a1, a2b2 ̸= b2a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8) OG1 a1 = OG1 b1 , OG2 a2 = OG2 b2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9) G2 = ⟨OG2 a2 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10) Then OG r = OG1 a1 × OG2 a2 is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let H = ⟨{a1} × OG2 a2 , {b1} × OG2 a2 ⟩ ∋ r, s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that OH r = {a1} × OG2 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, ⊆ follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8), and ⊇ from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10): y ∈ G2 =⇒ ∃x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xt ∈ OG2 a2 : y = x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' xt =⇒ (a1, y ⊲ a2) = (a1, x1) ⊲ ((a1, x2) ⊲ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (a1, a2)) ∈ OH r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Similarly, OH s = {b1} × OG2 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally, if |OH r | = |OH s | = |OG2 a2 | ≤ 2, then a2 and b2 commute, contradicting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G1 and G2 be finite groups, a1 ̸= b1 ∈ G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9 on G2 hold when G2/Z(G2) is a non-abelian simple group and a2 ∈ G2 is not central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Namely, ⟨OG2 a2 ⟩⊳G2, hence it is all of G2 giving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Furthermore there is b2 ∈ OG2 a2 that does not commute with a2, because G2 is not abelian, as needed in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Racks of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A rack X is of type D if it has a decomposable subrack Y = R � S with elements r ∈ R, s ∈ S such that r ⊲(s ⊲(r ⊲s)) ̸= s [7, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If O is a conjugacy class in a finite group G, then the rack O is of type D if and only if (D) holds, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Algebraic groups Let G be a connected reductive algebraic group defined over k = Fq and let F : G → G be a Frobenius map, that is a Fq-split Steinberg endomorphism [22, Chapter 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus there exists an F-stable torus T such that F(t) = tq for t ∈ T, and GF = G(Fq) is the finite group of Fq-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We make more precise assumptions on G in each Subsection below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The main objectives of the paper are encompassed in the following situations: ⋄ The group G is either SLn(k) or Sp2n(k) (n ≥ 2) and G := GF /Z(GF) = [GF, GF ]/Z([GF , GF ]) is a finite simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ⋄ The group G is either SO2n+1(k) (n ≥ 2, p is odd) or SO2n(k) (n ≥ 4) and G := [GF , GF ]/Z(GF ) is a finite simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 9 In both situations we say that G is a (classical) Chevalley group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We allow SO5(k) whose simply connected cover is Sp6(k), and SL2(3), SL2(4), SL2(5), SL2(9) for flexibility in some recursive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let Φ be the root system of G and fix a subset ∆ of simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let Q, respectively Λ, be the root, respectively weight, lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then Λ = ⊕i∈IZωi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' here θ is the rank of G, I = Iθ and (ωi)i∈I are the fundamental weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let W be the Weyl group of Φ and let sα ∈ W be the reflection corresponding to α ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, α∨ i : k× → T, i ∈ I, are the simple coroots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then ωi(α∨ j (ξ)) = ξδij, ξ ∈ k×, i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If α ∈ Φ, then there is a monomorphism xα : k → G of abelian groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' let Uα = Im xα (a root subgroup) and let U, respectively U−, be the subgroup of G generated by the Uα’s with α ∈ ∆, respectively −α ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The classical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this section we fix notation for the classical groups we will deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For m ≥ 1 we set Jm = � 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We denote by Frq the Frobenius map GLm(k) → GLm(k) given by (aij) �→ (aq ij), and similarly the restriction to any suitable subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will often use the automorphism φ: GLm(k) → GLm(k) given by: φ(A) := Jm tA−1Jm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) The symplectic group Sp2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The symplectic group Sp2n(k) is the subgroup of GL2n(k) leaving invariant the bilinear form � 0 Jn −Jn 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus Sp2n(k) consists of the invertible matrices � A B C D � such that tCJnA = tAJnC, tBJnD = tDJnB, −tCJnB + tAJnD = Jn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) In this case, F = Frq and G = Sp2n(q)/Z(Sp2n(q)) = PSp2n(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The orthogonal group SO2n+1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let p be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The orthogonal group SO2n+1(k) is the subgroup of SL2n+1(k) leaving invariant the bilinear form J2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus SO2n+1(k) consists of the invertible matrices X = � A e B f k g C h D � , A, B, C, D ∈ kn×n, e, tf, tg, h ∈ kn×1, k ∈ k such that det X = 1 and tCJnA + tff + tAJnC = 0, tCJne + tfk + tAJnh = 0, tCJnB + tfg + tAJnD = Jn, thJne + k2 + teJnh = 1, thJnB + kg + teJnD = 0, tDJnB + tgg + tBJnD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) In this case F = Frq and G = [SO2n+1(q), SO2n+1(q)] = PΩ2n+1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA The orthogonal group SO2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The orthogonal group SO2n(k) is the subgroup of matrices in SL2n(k) preserving the quadratic form �n i=1 xix2n−i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If p is odd, such elements automatically preserve the bilinear form with associated matrix J2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus SO2n(k) consists of those matrices � A B C D � ∈ SL2n(k) with A, B, C, D ∈ kn×n, such that tCJnA + tAJnC = 0, tCJnB + tAJnD = Jn, tDJnB + tBJnD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) If p = 2, then SO2n(k) consists of those matrices � A B C D � ∈ SL2n(k) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) and such that the diagonal terms in tCJnA and tBJnD are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case F = Frq and G = [SO2n(q), SO2n(q)]/Z([SO2n(q), SO2n(q)]) = PΩ+ 2n(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' On normalizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 we shall need the finite unitary groups SUn(q) and GUn(q), and the following folklore fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We consider: G = SLn(k), q0 = pm0 with m0|m so that q is a power of q0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' F0 : GLn(k) → GLn(k) is defined either by F0(A) := Frq0(A) or by F0(A) := Frq1/2 0 (φ(A)), for A ∈ GLn(k), φ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1), the latter occurring only for m0 even, in which case, we denote as usual SUn(q1/2 0 ) = GF0 and GUn(q1/2 0 ) = GLn(k)F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NGLn(q)(GF0) = Z(GLn(q))GLn(k)F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We prove that NGLn(q)(GF0) ≤ Z(GLn(q))GLn(k)F0, the other in- clusion being immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let g ∈ NGLn(q)(GF0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For any y ∈ GF0 there holds F0(gyg−1) = gyg−1, that is z := g−1F0(g) ∈ CGLn(q)(GF0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, GF0 contains regular unipotent elements in U and U−, so it follows from [26, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3] that CGLn(q)(GF0) = Z(GLn(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In addition, F0 restricts to a Steinberg endomorphism on the connected group Z(GLn(k)) ≃ k×, hence Lang-Steinberg theorem [22, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7] is in force and there exists ζ id ∈ Z(GLn(k)) such that ζ−1F0(ζ) id = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, ζ−1g ∈ GLn(k)F0 ≤ GLn(q) and so ζ id ∈ Z(GLn(k)) ∩ GLn(q) = Z(GLn(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The subgroup K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We introduce a subgroup K of G that will be useful in Sections 4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this Subsection G is one of the groups Sp2n(k), n ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' SO2n(k), n ≥ 3, or SO2n+1(k), n ≥ 3, where p ̸= 2 when G = SO2n+1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We set n′ = 2n if G = SO2n(k) or G = Sp2n(k) and n′ = 2n + 1 if G = SO2n+1(k), so that G ≤ GLn′(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall φ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then K is the image of the injective group morphism j : GLn(q) → GF, A �→ \uf8f1 \uf8f2 \uf8f3 � A 0 0 φ(A) � , if G = SO2n(k), or Sp2n(k), � A 0 0 0 1 0 0 0 φ(A) � , if G = SO2n+1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Cuspidal classes in the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let S = {sα : α ∈ ∆}, so that (W, S) is a Coxeter group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Given J ⊂ ∆, we set WJ = ⟨sα : α ∈ J⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' PJ = the standard parabolic subgroup of G determined by J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' LJ = the standard (reductive) Levi subgroup of PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [16, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1] A conjugacy class C in W is called cuspidal if C ∩ WJ = ∅ for all proper subsets J of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' an element is cuspidal if its conjugacy class is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A decomposition of w ∈ W is a family Γ = (γj)j∈Il in Φ, such that w = sγ1 · · · sγl, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5) where sγj is the corresponding reflection and l is minimal (with this prop- erty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then l is denoted by ℓa(w) and is called the absolute length of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By a result of Kostant, see [24], Γ is then a linearly independent family and ℓa(w) = rk(id −w) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) in the natural representation of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [16, Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='16], we have w is cuspidal ⇐⇒ ℓa(w) = rk G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7) Notice that ℓa(w) = rk G means that w has no fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Given a decomposition Γ of w, we set ΨΓ = Φ ∩ (Zγ1 ⊕ · · · ⊕ Zγl) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8) GΓ = ⟨T, Uβ : β ∈ ΨΓ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9) Clearly, ΨΓ is a root subsystem of Φ and GΓ is a connected reductive sub- group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If Γ and Γ ′ are different decompositions of the same w, then the subsystems ΨΓ and ΨΓ ′, and the subgroups GΓ and GΓ ′, might be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If w ∈ W is cuspidal, then GΓ is semisimple for any decompo- sition Γ of w, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If w ∈ WJ for some J ⊂ S, then there is a decomposition Γ such that GΓ ≤ LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, any decomposition of w in WJ is necessarily a decomposition in W, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For, w acts trivially in (RJ)⊥, hence rk(id −w) = rk(id −w)|RJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' F-stable tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here we assume that G is connected reductive and F is a Frobenius map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1], there is a bijection from the set of GF-conjugacy classes of F-stable maximal tori to the set of conjugacy classes in W, described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let T′ be an F-stable torus in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then T′ = gTg−1 for some g ∈ G such that g−1F(g) ∈ N(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let w = class of g−1F(g) ∈ N(T)/T ≃ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10) 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA The assignment T′ �→ w gives rise to the mentioned bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We set Tw := gTg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let T′ be an F-stable maximal torus in G such that T′ �→ w in the correspondence above and let G′ be the derived group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then T′ ∩ G′ is an F-stable maximal torus in G′ and T′ ∩ G′ �→ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, T ∩ G′ is a split torus of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The element g ∈ G such that T′ = gTg−1 and g−1F(g) ∈ NG(T) is a representative of w can be written as g = g′z where g′ ∈ G′ and z is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then T′ ∩ G′ = g′(T ∩ G′)(g′)−1 and (g′)−1F(g′) ∈ NG′(T ∩ G′) is a representative of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' An F-stable maximal torus is cuspidal if the corresponding conjugacy class in W as above is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let w ∈ W be a Coxeter element, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' a product w = s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' sθ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12) where (si)i∈Iθ is a numeration of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' this provides a decomposition Γ of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the class of w is cuspidal and GΓ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If W = Sn, the conjugacy class of w is the only cuspidal class in W, [16, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A Coxeter torus is an F-stable maximal torus that corre- sponds to the conjugacy class containing a Coxeter element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By abuse of terminology the intersection of a Coxeter torus of G with GF will be called a Coxeter torus of GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Semisimple classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here G is connected and reductive, unless oth- erwise stated, F is a Frobenius map and T is an F-stable torus such that F(t) = tq for t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x ∈ G = GF/Z(GF ) be semisimple non-trivial and pick x ∈ GF a representative of x, thus x is semisimple but not central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let O = OG x and O = OGF x ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' there is an epimorphism of racks O ։ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y ∈ GF be semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6], there exists an F-stable maximal torus T′ containing y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' however, not all F-stable maximal tori intersecting OGF y are necessarily GF-conjugated to T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consequently we assign to OGF y the set SOGF y of all conjugacy classes C in W corresponding to F-stable maximal tori T′ that intersect OGF y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that G is simple and that we are not in the cases excluded in [22, Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If C ∈ SOGF y , then O[GF ,GF ] y intersects an F-stable maximal torus T′′ in GF corresponding to an element in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 13 Indeed, let g ∈ GF be such that g−1F(g) ∈ NG(T) represents an element in C, and let T′ = gTg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then there exists l ∈ GF such that l ⊲ y ∈ T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, GF = TF[GF , GF], [22, Corollary 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2, Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15, Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='21], so l decomposes as l = t1l1 with t1 ∈ TF and l1 ∈ [GF , GF], and l1 ⊲ y ∈ t−1 1 gTg−1t1 = T′′, where g−1t1F(t−1 1 )F(g) = g−1F(g) represents an element in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For our aim, it is convenient to introduce the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A semisimple conjugacy class OGF y in GF is called cuspidal if the set SOGF y consists of cuspidal conjugacy classes in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In other words, all F-stable maximal tori intersecting OGF y are cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, OGF y is called a Coxeter class if it only intersects Coxeter tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Necessarily, OGF y is then cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since Z(GF) = Z(G)F is contained in every torus of GF, the class OGF y is cuspidal, respectively Coxeter, if and only if its projection O′ in GF/Z(GF ) intersects only cuspidal tori, respectively Coxeter tori in GF/Z(GF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will thus call also O′ cuspidal, respectively Coxeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In particular, if G is simply-connected and O is cuspidal, O will be called cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If G is not simply-connected, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9 guarantees that OGF y is cuspidal, respectively Coxeter, if an only if O[GF ,GF ] y intersects only cuspidal tori, respectively Coxeter tori in GF, and we will call also O[GF ,GF ] y cuspidal, respectively Coxeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume G is simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If T′ is a maximal F- stable torus that intersects O, C is the conjugacy class in W corresponding to T′ and Γ is a decomposition of w ∈ C, then O intersects GF Γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In particular, the following are equivalent: (a) O is not cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (b) O intersects a proper standard Levi subgroup L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since F is a Frobenius automorphism, GΓ is F-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Pick a rep- resentative ˙w of w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' by definition, it belongs to GΓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By the Lang-Steinberg Theorem there is h ∈ GΓ such that h−1F(h) = ˙w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The tori T′ and hTh−1 are GF-conjugate since they both map to w, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' That is, there exists y ∈ GF such that y ⊲ x ∈ hTh−1 ≤ GΓ, hence y decomposes as y = y′ for some y ⊲ x ∈ GF Γ ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 14 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA If O is not cuspidal, then pick C non-cuspidal and apply Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Conversely, if y ∈ O ∩ L, then there is an F-stable maximal torus T′ of L that contains y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence T′ = uTu−1 for some u ∈ L such that σ := u−1F(u) ∈ NL(T) ≤ NG(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' so that the class of σ belongs to the Weyl group of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Recall that a semisimple element y is regular if its centraliser CG(y) con- sists of semisimple elements, or equivalently, if the irreducible component CG(y)◦ of CG(y) containing the identity is a torus, [27, II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This occurs if and only if y lives in a unique maximal torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If our x ∈ GF is regular, then CG(x)◦ is the unique F-stable maximal torus containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x is a cuspidal element, then it is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We thank Gunter Malle for suggesting us the following proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let T0 be an F-stable maximal torus of G containing x, let g ∈ G be such that T0 = gTg−1 and let w be the corresponding Weyl group el- ement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', g−1F(g) ∈ wT as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The F-stable maximal tori in G containing x are also the F-stable maximal tori in the connected reductive group C = CG(x)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Every F-stable maximal torus in C is of the form cT0c−1 for some c ∈ C such that c−1F(c) ∈ NC(T0) ≤ gNG(T)g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let WC = NC(T0)/T0 be the Weyl group of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that WC is triv- ial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume for a contradiction that WC is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let s be a re- flection in WC and let c ∈ C be such that c−1F(c) = ˙s, a representative of s in NC(T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, ˙s′ := g−1 ˙sg would represent a reflection s′ in W and cT0c−1 = cgTg−1c−1 is an F-stable maximal torus of G, containing x and corresponding to g−1c−1F(c)F(g) = (g−1c−1F(c)g)(g−1F(g)) ∈ s′wT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, s′w is cuspidal by hypothesis on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' However, the characteristic polynomial of a cuspidal element is a product of cyclotomic polynomials different from (X − 1), therefore its value at 0 is 1, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' On the other hand, det(s′w) = − det(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, s′w and w can not be both cuspidal elements in W, contradicting our assumption on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore WC has no reflections and C = T0 is the unique maximal torus containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ The following well-known result is instrumental to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume G is simply-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (a) xq ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (b) If xq = x, then O ∩ TF ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' First, OG x is F-stable: if x = hyh−1 for some h ∈ G, then F(y) = F(h)−1xF(h) ∈ OG x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 15 Since x is semisimple, there are t ∈ T and g ∈ G such that x = gtg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus tq = F(t) ∈ OG x and consequently xq ∈ OG x ∩ GF = O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' here the last equality holds because CG(x) is connected, G being simply connected, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [20, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally, if xq = x, then tq = t ∈ TF ∩OG x ⊂ O by the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Split conjugacy classes We keep the notation from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6, namely G is simple and simply connected, but not of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also F is a Frobenius map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' T is an F-stable torus such that F(t) = tq for t ∈ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' e ̸= x ∈ G = GF/Z(GF ) is semisimple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' x ∈ GF a representative of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' O = OG x and O = OGF x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus there is an epimorphism of racks O ։ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We assume additionally that O ∩ TF ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Without loss of generality, we suppose that x ∈ TF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', x is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Adapting the proof of [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9] for type A, but with more work, we deal with such classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will need to consider separately the following particular situation: G is of type Bθ, q is odd, x satisfies sαj(x) = � x if j < θ, α∨ θ (−1)x if j = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) Here θ ≥ 2 (as B2 = C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' When this is the case, then x has the form x = � � i∈Iθ−1 α∨ i ((−1)i) � α∨ θ (η), where η2 = (−1)θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) Notice that if θ is odd, then such a x belongs to GF iff q ≡ 1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here is the main result of this Section: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that q > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G is not of type Aθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' q /∈ {3, 5, 7} if we are in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' and OGF x intersects the split torus TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' When q = 2, TF is trivial and the class of x could not intersect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This follows from Lemmata 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that q > 2 and that we are not in the situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that x ∈ TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will rely on the proof of [5, Lemmata 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It is shown there that, for any simple root α such that sα(x) ̸= x, the subrack Y = xUF α � sα(x)UF α of OGF x is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that we can choose α such that the restriction of the projection π : Y → O is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If G is of type E8, F4, G2, then Z(G) is trivial and G = GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let u ̸= v ∈ Y such that π(u) = π(v), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' there is z ∈ Z(GF ), z ̸= 1, such u = zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence either u = xxα(a) and v = sα(x)xα(a), or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In any case, x ⋆= zsα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 16 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Center of some G, q odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ζ ∈ F× q has order 4 type q Z(G) Bθ ⟨α∨ θ (−1)⟩ Cθ, θ > 2 � � i odd α∨ i (−1) � Dθ, q ≡ 1 mod 4 � � i odd,i≤θ−2 α∨ i (−1)α∨ θ−1(ζ)α∨ θ (ζ3) � θ ∈ 2Z + 1 q ≡ 3 mod 4 � α∨ θ−1(−1)α∨ θ (−1) � Dθ, θ ∈ 2Z � � i odd α∨ i (−1), α∨ θ−1(−1)α∨ θ (−1) � E7 ⟨α∨ 2 (−1)α∨ 5 (−1)α∨ 7 (−1)⟩ Applying sα, we get z2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus G is not of type E6 (here Z(G) ≃ Z/3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' and q should be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By ⋆, we have ωi(x) = ωi(zsα(x)) = ωi(z)sα(ωi)(x), i ∈ Iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) Say α = αj, j ∈ Iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then sα(ωi) = ωi when i ̸= j, hence ωi(z) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now such z exists only in the situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1), see the shape of Z(G) in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If we are in situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) with q = 9, then O is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If we are in situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) with q > 9, then O is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We deal first with θ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' now B2 = C2 and G = Sp4(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let K ≃ GL2(q) be the subgroup of GF which is the image of j as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus we have a monomorphism of groups GL2(q)/{±1} → G = PSp4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y = � 1 0 0 −1 � ∈ GL2(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' by our assumption on x, we know that either x ⋆= j(y) or x = −j(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let ̟ : GL2(q) → PGL2(q) be the canonical projection and y = ̟(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then we have a surjective map of racks O ∩ K/{±1} → OPGL2(q) y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore it is enough to prove that OPGL2(q) y is of type C if q = 9 and of type D for q > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let q = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then PGL2(9) ≃ A6, and through this isomorphism the class OPGL2(q) y corresponds to the class labeled by (12, 22), which is of type C by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 17 Let now q > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q ≡ 3 mod 4, then [6, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b)] applies1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume then that q ≡ 1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let ζ ∈ F× q be a primitive 4-th root of 1 and let u ∈ PGL2(q) be the class of � ζ 0 0 −ζ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then OPGL2(q) y = OPGL2(q) u = OPSL2(q) u which is of type D by [6, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume next that θ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here G = PΩ2θ+1(q) = GF/Z(GF ) ≃ [SO2θ+1(q), SO2θ+1(q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We identify PΩ5(q) with a subgroup of PΩ2θ+1(q) via the inclusion SO5(q) ֒→ SO2θ+1(q), \uf8eb \uf8ec \uf8ed A e B f k g C h D \uf8f6 \uf8f7 \uf8f8 �→ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed A 0 e 0 B 0 idθ−2 0 0 0 f 0 k 0 g 0 0 0 idθ−2 0 C 0 h 0 D \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , k ∈ Fq, A, B, C, D ∈ F2×2 q , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Fix tθ ∈ T of the shape (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) and analogously t2 of the shape (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) but for type B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If π : GF → G is the projection, then π(tθ) = diag (− idθ, 1, − idθ) = π(t2)γ, where γ = diag (id2, − idθ−2, 1, − idθ−2, id2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here diag refers to a diagonal of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O = OG π(tθ) ≥ OPΩ5(q)×⟨γ⟩ π(tθ) ≃ OPΩ5(q) π(t2) which is of type D by the preceding argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence O is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If we are in situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) with n = 2 and q = 3, then O is austere, hence kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, PSp4(3) ≃ PSU4(2) and the semisimple class we are dealing with in the former group corresponds to the unipotent class of type (2, 12) in the latter one, which is austere by [4, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If we are in the situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) with q = 5, then calculations with GAP show that O is austere, hence kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The evidence obtained by performing different computations seems to indicate that in the case q = 7, the class is also kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 1Notice that Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b) in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' refers implicitly to the class of involutions in PGL2(q) not in PSL2(q), as is transparent from the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 18 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Split classes in orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 was proved by as- suming that G is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For recursive arguments on the orthog- onal groups we need an analogous statement for the orbits of split elements in GF = SOn′(q) for the action of [GF, GF ] for n′ = 2n or 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let T ≤ G be the subgroup of diagonal matrices diag(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , tn, t−1 n , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , t−1 1 ), if n′ = 2n, diag(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , tn, 1, t−1 n , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , t−1 1 ), if n′ = 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If y ∈ TF − Z(SOn′(q)), then O[SOn′(q),SOn′(q)] y collapses, except in the situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', when n′ = 2n + 1 and ti = −1 for i ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There is always a simple root α so that the proof of [5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2] carries over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If ti ̸= ti+1 for some i < n, then take α = αi = εi − εi+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' if, instead, ti = ti+1 for all i < n, then our assumptions imply tn ̸= t−1 n and we take α = αn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', εn when n′ = 2n + 1 and εn−1 + εn when n′ = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The argument works also in types B2 and D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The special linear groups In this Section G = SLn(k), that is, we deal with semisimple classes in G = PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6, e ̸= x ∈ G is semisimple, x ∈ GF − Z(GF) is a representative of x, O = OG x and O = OGF x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There is an epimorphism of racks O ։ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For inductive arguments, we will also consider classes of elements in GLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As observed in [1, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1], for any semisimple element y ∈ GLn(q), we have OGLn(q) y = OSLn(q) y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We say that A ∈ GLn(q) is irreducible if its characteristic polynomial pA is irreducible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' necessarily A is regular semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' From our previous work, we know: Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1] If n = 2, and q ̸∈ {2, 3, 4, 5, 9}, then any O not listed in Table 1 collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) [3, Props.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5] If n = 3 and x is irreducible, then O is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii) [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1] If n ≥ 3 and x is not irreducible, then O collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We record information on the semisimple classes with q ∈ {2, 3, 4, 5, 9} for recursive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that PSL2(q) has two conjugacy classes of maximal tori: the split one, of order q − 1/(2, q − 1) and the Coxeter torus, of order q +1/(2, q +1), that contains the irreducible elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 19 If q = 2, then PSL2(2) ≃ S3 and the semisimple elements are the 3- cycles that form an abelian rack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' if q = 3, then PSL2(3) ≃ A4 and the semisimple elements have order 2 and form an abelian rack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q = 4, then PSL2(4) ≃ A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The irreducible elements have order 5 and form two conjugacy classes that are sober by [14, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 (b) and (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The split semisimple elements form the conjugacy class of 3-cycles which is of type C by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q = 5, then PSL2(5) ≃ A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The irreducible elements form the con- jugacy class of 3-cycles which is of type C by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The split semisimple elements are the involutions in the class (1, 22) which is sober because its intersection with any subgroup of A5 is either trivial, abelian or indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q = 9, then PSL2(9) ≃ A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The irreducible elements have order 5 and form two conjugacy classes that are sober by [14, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 (b) and (c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The split semisimple elements are the involutions in the class (12, 22) which is of type C by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Our main result in this Section is: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let O ̸= {e} be a semisimple conjugacy class in PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then any O not listed in Table 1 collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2, we will consider conjugacy classes of irreducible elements assuming n > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will see that if n is prime, then such classes are kthulhu by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15, otherwise, they are of type C by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We start by a classical result whose proof we include for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n ≥ 2 and ǫ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If P(X) = Xn + ǫ ∈ Fp[X] is irreducible over Fq, then n = 2, ǫ = 1 and q ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' First, ǫ = 1 and q is odd, otherwise P(1) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' and n = 2m is even, otherwise P(−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, q ≡ 3 mod 4, otherwise −1 = ξ2 for some ξ ∈ Fq and P(X) = (Xm +ξ)(Xm −ξ) would be reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let now n = 2ha where a is odd, and let Φd(X) be the d-th cyclotomic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We have the factorization over Z, hence over Fp, (Xn − 1)P(X) = X2n − 1 = � d|2n Φd(X) =⇒ P(X) = � d|2n, d∤n Φd(X) Thus Φ2h+1|P(X), hence they are equal and n = 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finally, if X2h + 1 is irreducible over Fq for q ≡ 3 mod 4, then h = 1 by [23, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Coxeter tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We assume in the rest of this Section that n > 3 and that x is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case W = Sn and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12 and Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7 every irreducible class intersects every Coxeter torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We fix the n-cycle w = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By technical reasons, we fix a Coxeter torus Tw in GLn(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' then Tw∩SLn(k) is a Coxeter torus in SLn(k) by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Example 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4], we have |TF w| = qn − 1 = (n)q(q − 1), |TF w ∩ SLn(q)| = (n)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) The group TF w is isomorphic to F× qn, hence it is cyclic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' further, any cyclic subgroup of GLn(q) of order qn − 1 is conjugated to TF w [19, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ([21, Satz II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3] and [25, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 and below]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We have NGLn(q)(Tw) = NGLn(q)(TF w) ≃ TF w ⋊ CW (w), NSLn(q)(TF w ∩ SLn(q)) = NGLn(q)(TF w) ∩ SLn(q), with CW(w) ≃ Z/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let σ be a generator of CW (w) identified as a subgroup of NGLn(q)(TF w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' σ can be chosen so that σ ⊲ y = yq for any y ∈ TF w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y ∈ TF w be irreducible in GLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then OGLn(q) y ∩ TF w = O NGLn(q)(TF w) y = ⟨σ⟩ ⊲ y = {y, yq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , yqj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , yqn−1}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) O NGLn(q)(TF w) y = OSLn(q) y ∩ TF w = O NSLn(q)(TF w) y (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If z ∈ TF w ∩ OGLn(q) y , then there is g ∈ GLn(q) such that gyg−1 = z, so gCGLn(q)(y)g−1 = CGLn(q)(z), that is, g ∈ NGLn(q)(Tw) since clearly z is also irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6 imply (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since OSLn(q) y = OGLn(q) y , the centraliser argument as above gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ We investigate when two elements in an irreducible class have the same image through the natural projection π: SLn(q) → PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that Gn(Fq) is the group of n-th roots of unity in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (Just for this Lemma, n ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y, z ∈ O such that π(y) = π(z), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', y = λz for some 1 ̸= λ ∈ Gn(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (i) There exists j ∈ In−1 such that λz = zqj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) Let j ∈ In−1 be minimal satisfying zqj = λz for some 1 ̸= λ ∈ Gn(Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then j|n and λ is a primitive n j -th root of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii) Let j ∈ In−1 be minimal satisfying zqj = λz for some 1 ̸= λ ∈ Gn(Fq) and let a := n j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the characteristic polynomial pz ∈ Fq[Xa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This observation rectifies [3, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 21 (iv) Let j ∈ In be minimal satisfying π � zqj� = π(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then j ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i): The elements y and z lie in the same (unique) maximal torus, so y = zqj for some j ∈ I1,n by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, λz = zqj and λ ̸= 1 implies j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii): If n = aj + b, with a ≥ 1 and 0 ≤ b < j, then z = zqn = zqaj+b = (zqaj)qb = (λaz)qb = λazqb that is, zqb = λ−ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence b = 0 by minimality and λa = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, if λc = 1 with c ∈ N, then zqcj = λcz = z, hence c ≥ a = n j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii): By assumption pz = pzqj = pλz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If pz(X) = Xn+cn−1Xn−1+· · ·+c0, then pλz(X) = Xn + λcn−1Xn−1 + · · · + λnc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus pz(X) = pλz(X) if and only if ch = 0 for all h ̸∈ n j Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iv): If j = 1 then pz would be Xn+(−1)n by (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5, n = 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Irreducible elements of SLn(q), n not a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this Subsection we assume that n = cd, for some c, d ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Given S ∈ SLd(q) irreducible, we consider y = diag(S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , S) ∈ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then CGLn(q)(y) ≃ GLc(qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that a Coxeter torus �T of CGLn(q)(y) remains a Coxeter torus in GLn(q) hence T := �T ∩ SLn(q) is a Coxeter torus in SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1), we have | �T| = ((qc)d − 1) = (qn − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since �T is cyclic, it is conjugated to TF w as claimed after (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, |T| = (n)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this subsection we will assume that x lies in a Coxeter torus T of GF arising from some y as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x is irreducible, then O = OPSLn(q) x is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n = cd with c prime and d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We set: � M := CGLn(q)(y) ≃ GLc(qd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' M := CSLn(q)(y) = � M ∩ SLn(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' M1 := [� M, � M] ≃ SLc(qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus M1 ≤ M ≤ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7 gives OGLn(q) x ∩ T = OSLn(q) x ∩ T = {x, xq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqn−1} OM x ∩ T = OM1 x ∩ T = O � M x ∩ T = {x, xqd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqd(c−1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence xq ∈ O ∩ T but xq /∈ OM x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that OM xq ̸⊂ N� M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Suppose the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, ⟨OM xq ⟩ would be a non-central, normal subgroup of 22 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA � M ≃ GLc(qd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then SLc(qd) ≃ M1 ≤ ⟨OM xq ⟩ ≤ N� M(T), and so T ∩ M1 would be normal in M1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We pick s ∈ OM xq \\ N� M(T) and set s := π(s) ∈ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' r := π(x) ∈ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' H := ⟨r, s, π(M1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that r, s and H satisfy the assumptions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' First, s ̸∈ N� M(T) implies s ⊲ r ̸= r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, s ⊲ r = r would give s ⊲ x ∈ Z(GF)x that combined with T = CGLn(q)(x) would force s ⊲ T = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In addition, ⟨OM1 x , OM1 s ⟩ = ⟨O � M x , O � M s ⟩ is a non-central, normal subgroup of � M ≃ GLc(qd), hence ⟨M1, x, s⟩ ≤ ⟨OM1 x , OM1 s ⟩ and therefore H = ⟨π(M1), r, s⟩ ≤ ⟨Oπ(M1) r , Oπ(M1) s ⟩ ≤ ⟨OH r , OH s ⟩ ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' That is, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) holds and H ≤ ⟨O⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Observe that π(M) ≃ M/Z(SLn(q)) ∩ M onto PGLc(qd), so the orbits Oπ(M) x and Oπ(M) xq project onto non-trivial orbits in PGLc(qd), and therefore |OH r | ≥ |Oπ(M) r | > 2 and |OH s | ≥ |Oπ(M) s | > 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We finally analyse OH r ∩ OH s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' First of all, π(M1) ≤ H ≤ π(M) and OM1 x = OM x imply that OH r = Oπ(M) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Similarly, OH s = Oπ(M) s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If Oπ(M) r ∩ Oπ(M) s = ∅, then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Otherwise, x ∈ Oπ(M) s ∩ π(T) = Oπ(M) xq ∩ π(T) = {xq, (xq)qd, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , (xq)qd(c−1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore there exists l ∈ I0,c−1 such that xqdl+1 ∈ Gn(Fq)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8 (iv) gives l ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let j ∈ In−1 be minimal satisfying xqj ∈ Gn(Fq)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then j|n by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8 (ii) whose argument shows that j divides also dl + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, (j, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since j > 1 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8 (iv) again, this can occur only if j = c and (c, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case, d has a prime factor c′ different from c and we may repeat the whole construction replacing c by c′ and d by n c′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As j = c ̸= c′, we get that Oπ(M) r ∩ Oπ(M) s = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 were verified, hence O is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Irreducible elements of SLn(q), n > 3 prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here n > 3 is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that e ̸= x ∈ G = PSLn(q), x ∈ GF − Z(GF ) is a representative of x which is irreducible and belongs to the Coxeter torus T := TF w ∩ SLn(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' we set O = OG x and O = OGF x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There is an epimorphism of racks O ։ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will analyse all possible subgroups of GLn(q) intersecting O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We start by a few well-known arithmetic results instrumental for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n be an odd prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 23 (i) If (n, q − 1) = 1, then (n, qn − 1) = (n, (n)q) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If (n, q − 1) = n, then (n2, (n)q) = (n, (n)q) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii) (q − 1, (n)q) = (n, q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iv) (n(q − 1), (n)q) = (n, q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) and (ii) are [25, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1], whilst (iii) follows from the Eu- clidean algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We prove (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Combining (i), (ii) and (iii) we obtain (n, (n)q) = (n, q − 1) = (q − 1, (n)q), hence (n(q − 1), (n)q) = 1 if (n, q − 1) = 1, and n ≤ (n(q − 1), (n)q) ≤ n2 if (n, q − 1) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case, we discard (n(q − 1), (n)q) = n2 using (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Recall σ ∈ NGLn(q)(Tw) from Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let n be a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) Let g ∈ NGLn(q)(TF w) \\ TF w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then |g| divides n(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) O ∩ NGF (Tw) ⊂ TF w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6, there are k ∈ I1,n−1 and t ∈ TF w such that g = σkt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (σkt)n = \uf8eb \uf8ed � τ∈⟨σk⟩ τ ⊲ t \uf8f6 \uf8f8 σnk = � τ∈⟨σ⟩ τ ⊲ t = � n � i=1 tqi � = t(n)q by a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence |g| divides n(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) Let g ∈ O ∩ NGLn(q)(TF w) = O ∩ NGF (TF w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that |x| divides (n)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If g /∈ TF w, then |g| divides (n(q − 1), (n)q) by (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 (iv) |g| divides (n, q − 1), so g is central, contradicting its irreducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ We recall that a primitive prime divisor of qn − 1 is a prime number ℓ such that ℓ|qn − 1 and ℓ ̸ |qe − 1 for every e ∈ In−1 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (Here n is any odd prime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y be an irreducible semisimple element in GLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, (i) Either there exists a primitive prime divisor ℓ of qn − 1 dividing |y| or else |y| divides n(q − 1), it does not divide (q − 1) and n|q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If y ∈ SLn(q), then there always exists a primitive prime divisor ℓ of qn − 1 dividing |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) If for every prime divisor ℓ of |y| there is an e ∈ In−1 such that ℓ divides qe − 1 = (q − 1)(e)q then, any such ℓ divides (q − 1)((n)q, (e)q) for some e < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The latter equals (q − 1)(e, n)q = q − 1 by the Euclidean algorithm, so ℓ|q − 1 for any prime divisor ℓ of |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 24 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Since y is irreducible, |y| cannot divide q − 1, so there is a prime divisor ℓ0 of |y| dividing q − 1 and (n)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 (iii) this is possible only if n|q − 1 and in this case ℓ0 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 (ii) implies that |y| divides n(q − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If y ∈ SLn(q) then |y| divides (n)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume, for a contradiction, that no primitive prime divisors of qn − 1 divides |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, |y| would divide (n(q − 1), (n)q) = (n, q − 1) by (i) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, y cannot be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ In the terminology of [17, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2], Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12 says that if n is an odd prime, then all irreducible elements in SLn(q) are ppd(n, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' n)-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The following result is a consequence of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let ℓ be a primitive prime divisor of qn − 1 dividing |x| and let H ≤ GLn(q) be such that x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then H occurs in the following list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (a) SLn(q0) ≤ H ≤ NGLn(q)(SLn(q0)) where q0 = pm0 with m = m0d, d ∈ N and (d, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (b) SUn(q1/2 0 ) ≤ H ≤ NGLn(q)(SUn(q0)) where q0 = pm0 a square with m = m0d, d ∈ N and (d, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (c) H ≤ NGLn(q)(TF w) = NGLn(q)(T), and ℓ divides |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (d) H/(H ∩ Z(GLn(q)) ≃ M11, n = 5, ℓ = 11, and q5 ≡ 1 mod 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (e) H/(H ∩ Z(GLn(q)) ≃ M23, or M24, n = 11, ℓ = 23, q11 ≡ 1 mod 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (f) PSL2(ℓ) ≤ H/(H ∩ Z(GLn(q))) ≤ PGL2(ℓ), for ℓ ≥ 7, n = 1 2(ℓ − 1) and qn ≡ 1 mod ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The main result in [17] states that the subgroups of GLd(q) contain- ing a ppd(d, q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' e)-element, for some 1 2d < e ≤ d are precisely those occurring in the Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9 listed therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We extract the cases satisfying d = e = n an odd prime > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 (b) and (d) and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 are discarded because they occur for either d or e even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 (a) and (c) are (a) and (b) in our list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 does not occur because it requires an H-stable subspace of the natural representation of GLn(q) and x ∈ H is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 and Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (a) are discarded as they require e ̸= d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b) is the case (c) in our list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6 (a) is discarded because it requires the prime ℓ = n+1, which is impossible because n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6 (b) and (c) are collected in [17, Tables 2,3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In Table 2, d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In Tables 3 and 4 the number e is never odd > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 25 Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7 are listed in [17, Table 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The column with ℓ = e + 1, is immediately discarded, just as all rows for which d is not a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We are left with the three possible choices for H′ ≃ H/H ∩ Z(GLn(q)) listed in (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8 are listed in [17, Table 6] and are discarded because e ̸= d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9 are listed in [17, Tables 7,8] and if d is a prime, then e is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In Table 8, we discard all cases for which ℓ = e + 1 and we are left with the case (f) in our list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G = SLn(k), m = m0d with (d, n) = 1 and q0 = pm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) If (n, q0 − 1) = n, then Z(GLn(q))GLn(q0) ∩ SLn(q) = SLn(q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If m0 is even and (n, q1/2 0 + 1) = n, then Z(GLn(q))GUn(q1/2 0 ) ∩ SLn(q) = SUn(q1/2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) We prove ⊂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let ζ idn ∈ Z(GLn(q)) and g ∈ GLn(q0) be such that ζg ∈ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, |ζ| divides n(q0 − 1) because ζ−n = det(g) ∈ F× q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It also divides q − 1 because ζ ∈ F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence it divides (n(q0 − 1), q − 1) = (q0 − 1) (n, (d)q0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, (d)q0 ≡ d mod n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' since (d, n) = 1, then |ζ| divides q0 − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', ζ ∈ F× q0 and ζg ∈ SLn(q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) We prove ⊂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let ζ idn ∈ Z(GLn(q)) and g ∈ GUn(q1/2 0 ) be such that (ζ idn)g ∈ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, as g = Frq1/2 0 φ(g), we have (det g)q1/2 0 +1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence |ζ| divides n(q1/2 0 +1) and also q −1 because ζ ∈ F× q , and so it divides (n(q1/2 0 + 1), q − 1) = (q1/2 0 + 1) � n, (q1/2 0 − 1)(d)q1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' However, n is an odd prime dividing q1/2 0 + 1 so it does not divide q1/2 0 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' also, (n, (d)q1/2 0 ) = 1 by the argument in (i) applied to q1/2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, |ζ| divides q1/2 0 + 1, that is, ζ idn ∈ Z(GUn(q1/2 0 )), so ζg ∈ SLn(q) ∩ GUn(q1/2 0 ) = SUn(q1/2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ In this Subsection �π: GLn(q) → PGLn(q) is the natural projection, whose restriction to SLn(q) is π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x be an irreducible element in the Coxeter torus T = TF w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We consider all possible intersections O ∩ M for every M ≤ G containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' All such groups have the form M = π(H ∩ SLn(q)) for some H ≤ GLn(q) containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We will show that either O ∩ M = OM x or else O ∩ M is an abelian subrack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This implies that O is kthulhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 26 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA For our analysis, we will make use of the following auxiliary facts: Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CPSLn(x) ∩ O = {x, xq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7 gives CSLn(q)(x) ∩ O = {x, xq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We describe CPSLn(q)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If z ∈ SLn(q) satisfies zxz−1 ∈ Gm(Fq)x∩O, then by primality of n and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8 (ii) and (iv) we conclude that zxz−1 = x, and so CPSLn(q)(x) = π(CSLn(q)(x)) = π(T), whence the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If �π(H) is simple, then �π(H) ≤ PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In particular, we may assume H ≤ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, if �π(H) is simple, then �π(H) = [�π(H), �π(H)] = �π([H, H]) ≤ �π([GLn(q), GLn(q)]) = π(SLn(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We set from now on H1 := H∩SLn(q) and inspect all possible M = π(H1) where H runs through the list of subgroups from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13 containing x, with ℓ a primitive prime divisor of qn − 1 dividing |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The numbering of items is as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here q = pm, q0 = pm0 where m = m0d, d ∈ N and (n, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 gives NGLn(q)(SLn(q0)) = Z(GLn(q))GLn(q0) so SLn(q0) ≤ H1 ≤ Z(GLn(q))GLn(q0) ∩ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) We will first show that O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = OSLn(q0) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5) If (n, q0 − 1) = n, then the inclusions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) are all equalities by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='14 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = O ∩ SLn(q0) = OSLn(k) x ∩ SLn(q0) = OSLn(q0) x where the last two equalities follow from [22, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11] and [20, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume now that (n, q0 −1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since x ∈ H1 ≤ Z(GLn(q))GLn(q0), there are z = ζ idn ∈ Z(GLn(q)) and y ∈ GLn(q0) such that x = zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider x1 ∈ O ∩ Z(GLn(q))GLn(q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let z1 = ζ1 idn ∈ Z(GLn(q)) and y1 ∈ GLn(q0) be such that x1 = z1y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By construction, |ζ| and |ζ1| divide n(q0 − 1) because x, x1 ∈ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since x is irreducible, y and y1 are again irreducible in GLn(q), whence in GLn(q0), because they are regular and lie in a Coxeter torus of GLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let {ηqj : j ∈ I0,n−1} ⊂ Fqn 0 and {ηqj 1 : j ∈ I0,n−1} ⊂ Fqn 0 be the sets of eigenvalues of y and y1, respectively, NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 27 so {ζηqj : j ∈ I0,n−1} and {ζ1ηqj 1 : j ∈ I0,n−1} are the sets of eigenvalues of x and x1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then {ζηqj : j ∈ I0,n−1} = {ζ1ηqj 1 : j ∈ I0,n−1} and so ζη = ζ1ηqj0 1 for some j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore |ζ1ζ−1| = |ηη−qj0 1 | divides (n(q0 − 1), qn 0 − 1) = (q0 − 1)(n, (n)q0) = q0 − 1, where the last equality follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In other words, ζ1 ∈ ζF× q0, and z−1x1 is a regular semisimple matrix in GLn(q0) with the same eigenvalues as y, and it is therefore SLn(q0)-conjugate to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, O ∩ Z(GLn(q))GLn(q0) ⊂ zOSLn(q0) y = OSLn(q0) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let now x′ = π(x′) ∈ O ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, z′x′ ∈ O for some z′ ∈ Z(SLn(q)) and x′ ∈ Z(SLn(q))H1, that is, z′x′ ∈ O ∩ Z(SLn(q))H1 ⊂ O ∩ Z(GLn(q))GLn(q0) ∩ SLn(q) = OH1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' where the equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, x′ ∈ Z(SLn(q))OH1 x and x′ ∈ Oπ(H1) x = OM x , showing that O ∩ M = OM x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here q = pm, q0 = pm0 where m0|m, m0 is even and (n, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We use the same strategy as in case (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 gives NGLn(q)(SUn(q1/2 0 )) = Z(GLn(q))GUn(q1/2 0 ) so SUn(q1/2 0 ) ≤ H1 ≤ Z(GLn(q))GUn(q1/2 0 ) ∩ SLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) We will first show that O ∩ Z(GLn(q))GUn(q1/2 0 ) ∩ SLn(q) = OSUn(q1/2 0 ) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7) If (n, q1/2 0 +1) = n, then the inclusions in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) are all equalities by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='14 (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In this case O ∩ Z(GLn(q))GUn(q1/2 0 ) ∩ SLn(q) = O ∩ SUn(q1/2 0 ) = OSLn(k) x ∩ SUn(q1/2 0 ) = OSUn(q1/2 0 ) x where the last two equalities follow from [22, Theorem 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11] and [20, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume now that (n, q1/2 0 +1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since x ∈ H1 ≤ Z(GLn(q))GUn(q1/2 0 ), there are z = ζ idn ∈ Z(GLn(q)) and y ∈ GUn(q0) ≤ GLn(q0) such that x = zy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider x1 ∈ O∩Z(GLn(q))GUn(q1/2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let z1 = ζ1 idn ∈ Z(GLn(q)) and y1 ∈ GUn(q0) ≤ GLn(q0) be such that x1 = z1y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By construction, |ζ| and |ζ1| divide n(q1/2 0 + 1) because x, x1 ∈ SLn(q) and det(g)q1/2 0 +1 = 1 for any g ∈ GUn(q1/2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since x is irreducible, y and y1 are again irreducible in GLn(q), whence in GLn(q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We show that |y| cannot divide n(q0 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, if this were 28 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA the case, then we would have yq0−1 = ξ idn for some ξ ∈ Gn(F× q ), with ξ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since yq0 ∈ OGLn(q0) y , the characteristic polynomial py would be Xn − det(y) = Xn − ξ−n, which is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12 (i) there is a primitive prime divisor ℓ of |y| dividing qn 0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let F0 : GLn(k) → GLn(k) be given by F0(A) := Frq1/2 0 φ(A), for A ∈ GLn(k), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6] there exists an F0- stable torus T′ in GLn(k) containing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let T = T′ ∩ GUn(q1/2 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 (c)] and an analysis of φ-classes in the symmetric group, there is a partition λ of n such that |T | = � λi even (qλi/2 0 − 1) � λi odd (qλi/2 0 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The latter divides � λi even (qλi/2 0 − 1) � λi odd (qλi 0 − 1) and is divisible by the primitive prime divisor ℓ of qn 0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, λ = (n) and |T | = (qn/2 0 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now we proceed as in case (a): considering the set of eigenvalues for x and x1 and of y and y1, we deduce that |ζ1ζ−1| divides � n(q1/2 0 + 1), qn/2 0 + 1 � = (q1/2 0 + 1)(n, (n)−q1/2 0 ) = (q1/2 0 + 1) where (n, (n)−q1/2 0 ) = 1 because (n)−q1/2 0 divides qn/2 0 + 1 and q1/2 0 is not a root of Xn + 1 = (X + 1)n in Fn by our assumption on q0 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, z1 ∈ zZ(GUn(q1/2 0 )), and z−1x1 is a regular semisimple matrix in GUn(q1/2 0 ) with the same eigenvalues as y, and it is therefore SUn(q0)- conjugate to y by [20, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, O ∩ Z(GLn(q))GUn(q1/2 0 ) ⊂ zOSUn(q1/2 0 ) y = OSUn(q1/2 0 ) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let now x′ = π(x′) ∈ O ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, z′x′ ∈ O for some z′ ∈ Z(SLn(q)) and x′ ∈ Z(SLn(q))H1, that is, z′x′ ∈ O ∩ Z(SLn(q))H1 ⊂ O ∩ Z(GLn(q))GUn(q1/2 0 ) ∩ SLn(q) = OH1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' where the equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, x′ ∈ Z(SLn(q))OH1 x and x′ ∈ Oπ(H1) x = OM x , showing that O ∩ M = OM x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case (c) In this case, M ≤ π(NSLn(q)(T)) = NG(π(T)), where the second equality follows because Z(SLn(q)) ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If y = π(y) ∈ O ∩ M then there is z ∈ Z(SLn(q)) such that y ∈ OSLn(q) zx ∩ NSLn(q)(T), and zx is again irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11 we see that y ∈ T, so O ∩ M ⊂ π(T) is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case (d) In this case n = 5 and ℓ = 11 and �π(H) = M11 = π(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We show that O ∩ M11 = OM11 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The only elements whose order is divisible by ℓ in M11 are of order 11, so |x| = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There are two classes of such elements in M11, say OM11 x and OM11 y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If O ∩ M11 = OM11 x ∪ OM11 y , then ⟨x⟩ − 1 ⊂ O ∩ M11 ∩ CPSLn(q)(x) = {x, xq, xq2, xq3, xq4}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 29 Case (e) In this case n = 11 and ℓ = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The only elements of order divisible by ℓ in M = M23 or M24 have order 23 and there are 2 conjugacy classes of such elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We proceed as in case (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case (f) In this case ℓ|qn − 1 and M = π(H1) ≃ H1/H1 ∩ Z(SLn(q)), and PSL2(ℓ) ≤ H1/H1 ∩ Z(SLn(q)) ≤ PGL2(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As [PGL2(ℓ) : PSL2(ℓ)] ≤ 2, we have M ≃ PGL2(ℓ) or M ≃ PSL2(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In both cases, |x| = ℓ and we claim that O ∩ M = OM x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In PGL2(ℓ) all non-trivial unipotent elements are conjugate and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let y ∈ O ∩ PSL2(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By replacing y with a representative lying in the same Borel subgroup of PSL2(ℓ) as x, we can ensure that y ∈ CPSL2(ℓ)(x) ∩ O ⊂ CPSLn(q)(x) ∩ O = {x, xq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xqn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Without loss of generality we may assume that x is the class of � 1 ξ 0 1 � , for some ξ ∈ F× ℓ so y is the class of � 1 ξ 0 1 �qj = � 1 qjξ 0 1 � for some j ∈ In−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By assumption q ≡ qn+1 mod ℓ hence q is a square modulo ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore x and y are conjugate in PSL2(ℓ), whence the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider either g ∈ M11, |g| = 11, or g ∈ M23 or M24, |g| = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the classes OM11 g , OM23 g or OM24 g are contained either in OPSL5(q) g for some q, or in OPSL11(q′) g for some q′, respectively, according to [17] and Claim 2, see Cases (d) and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15, since g is irreducible in all cases, OG g is kthulhu, where G is either PSL5(q) or PSL11(q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence so are OM11 g , OM23 g and OM24 g , as was previously proved in [10, Teorema 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Semisimple conjugacy classes represented in K In this section we deal with semisimple conjugacy classes intersecting the subgroup K which is the image of the map j : GLn(q) → GF introduced in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We give parallel proofs for two classes of simple groups: G = Sp2n(k) with n ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' here G := GF /Z(GF) and π: GF → G denotes the standard projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G = SOn′(k) with n′ = 2n and n ≥ 4, or n′ = 2n + 1 and n ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' here G := [GF, GF ]/Z(GF ) and π: [GF , GF] → G is the standard projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the symplectic case, GF = [GF , GF ] so for brevity of the exposition we write [GF , GF] in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We also consider such groups with smaller n sometimes for the sake of recursive arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We shall consider a semisimple class O in G, a class O in [GF, GF ] such that π(O) = O and assume that it exists A ∈ GLn(q) such that j(A) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 30 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Here are the main results of this Section: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G = Sp2n(k), n ≥ 2, and let A ∈ GLn(q) − Z(GLn(q)) be a semisimple element, which is not an involution if n = 2 and q ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O = OG π(j(A)) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G = SO2n(k) or SO2n+1(k) with n ≥ 3 in both cases and let A ∈ GLn(q) − Z(GLn(q)) be a semisimple element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume in addition that j(A) does not correspond to situation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) if q ∈ {3, 5, 7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O = OG π(j(A)) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' These theorems are proved in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 after we deal in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 with the case when A is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the orthogonal case, we consider the orbit O[GF ,GF ] j(A) for later applica- tions even if j(A) does not necessarily belong to [GF, GF ], as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' See Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We start by some general considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let A ∈ GLn(q) be a semisimple element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) Oj(SLn(q)) j(A) = O[K,K] j(A) = OK j(A) = Oj(GLn(q)) j(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If A is irreducible, then either OGLn(q) A = OGLn(q) A−1 or else j(A) is regular in GLn′(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii) If A is irreducible, then either O[GF ,GF ] j(A) = O[GF ,GF ] j(A−1) , or else j(A) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (i) is a consequence of the inclusions j(SLn(q)) ≃ [K, K] ≤ K ≃ GLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii): If ζqh, h ∈ I0,n−1, are the (distinct) eigenvalues of A in k, then ζ±qh for h ∈ I0,n−1 (together with 1 when n′ = 2n + 1) are the eigenvalues of j(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that j(A) is not regular in GLn′(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' hence A and A−1 have a common eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the sets of eigenvalues of A and A−1 coincide by irreducibility, that is A and A−1 are conjugate in GLn(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since centralisers in GLn(k) are connected, [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 19], OGLn(q) A = OGLn(q) A−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (iii) follows from (i) and (ii) and the inclusion [K, K] ≤ K ∩ [GF , GF].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A ∈ GLn(q) is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We first analyze this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (Here n ≥ 3 when G = SO2n(k) or G = SO2n+1(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let A ∈ GLn(q) be an irreducible element such that j(A) is not regular in GLn′(q), NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 31 pA(X) ̸= X2 + 1 when G = Sp4(k) and q ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (i) O := O[GF ,GF ] j(A) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If j(A) ∈ O ⊆ [GF , GF], then O collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Observe that (ii) follows directly from (i), that we prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The initial discussion is valid for both orthogonal and symplectic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall φ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The irreduciblity assumption in A ensures that the eigenvalues of A are all distinct, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', A is regular semisimple, so we may assume that A is the companion matrix of its characteristic (and minimal) polynomial pA = Xn +an−1Xn−1 +· · · a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' That is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' φ(A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' tA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' tA−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' and φ(A−1) have the following shape: A = � 0 0 ··· 0 −a0 1 0 ··· 0 −a1 0 1 ··· 0 −a2 ··· ··· ··· ··· ··· 0 0 ··· 1 −an−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' φ(A) = � 0 1 0 ··· 0 0 0 1 ··· 0 ··· ··· ··· ··· ··· 0 ··· ··· 0 1 −1/a0 −an−1/a0 ··· −a2/a0 −a1/a0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' tA = � 0 1 0 ··· 0 0 0 1 ··· 0 ··· ··· ··· ··· ··· 0 0 0 ··· 1 −a0 −a1 −a2 ··· −an−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' tA−1 = � −a1/a0 −a2/a0 ··· −an−1/a0 −1/a0 1 0 ··· 0 0 0 1 ··· 0 0 ··· ··· ··· ··· ··· 0 0 ··· 1 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A−1 = \uf8eb \uf8ed −a1/a0 1 0 ··· 0 −a2/a0 0 1 ··· 0 ··· ··· ··· ··· ··· −an−1/a0 0 0 ··· 1 −1/a0 0 0 ··· 0 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' φ(A−1) = � −an−1 −an−2 ··· ··· −a0 1 0 0 ··· 0 0 1 0 ··· 0 ··· ··· ··· ··· ··· 0 0 0 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also A ̸= A−1, otherwise A would have eigenvalues ±1, contradicting irre- ducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We consider the disjoint subracks: R1 := �� A Y 0 φ(A) � ∈ O � , R2 := �� A−1 Y 0 φ(A−1) � ∈ O � , if n′ = 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' R1 := �� A 0 Y 0 1 0 0 0 φ(A) � ∈ O � , R2 := �� A−1 0 Y 0 1 0 0 0 φ(A−1) � ∈ O � , if n′ = 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now R1 ̸= ∅ by construction, and R2 ̸= ∅ by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It is easy to see that Ri ⊲Rj = Rj, 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We continue with each group separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G = Sp2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let r1 := � idn Jn 0 idn � ⊲ j(A) = � A −AJn+tA−1Jn 0 Jn tA−1Jn � ∈ R1, r2 := j(A−1) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A direct calculation shows that r1r2 := � idn −A tAJn+Jn 0 idn � r2r1 := � idn −Jn+A−1 tA−1Jn 0 idn � so r1r2 = r2r1 if and only if 2 idn = A−1 tA−1 + A tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let us verify that such an equality never holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Comparing the diagonal entries we obtain a2 i = (−1)i+1a2i 0 (1 − a2 0) for i > 0, whereas comparing the entries in the 32 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA first row we obtain a1(a2 + a3 0) = 0 and a1al = −a3 0al−1 for l > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The conditions a2 + a3 0 = 0 and a2 2 = −a4 0(1 − a2 0) lead to a contradiction, hence necessarily a1 = 0 and so al = 0 for any l > 0 and a2 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In other words, pA(X) = Xn±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5, this is possible only if n = 2, q ≡ 3 mod 4 and pA(X) = X2 + 1, which is excluded by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, r1 ⊲r2 ̸= r2 and, for H := ⟨r1, r2⟩ we have OH r1 ∩ OH r2 ⊂ R1 ∩ R2 = ∅ because A2 ̸= id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If p = 2, then |r1| = |r2| is odd and OSp2n(q) j(A) is of type C by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If, instead, p is odd, then r1r2 ̸= r2r1 implies (r1r2)2 ̸= (r2r1)2 as they are p-elements, so OSp2n(q) j(A) is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that the restriction of the projection π: Sp2n(q) → G to R1 � R2 is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, this could fail only if A2 = ±1, but since A is irreducible, we would have A2 = −1 which would give pA(X) = X2 +1, with q ≡ 3 mod 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', the discarded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence OG π(j(A)) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G = SO2n(k) or SO2n+1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For n ≥ 3 we consider the matrices: E := \uf8f1 \uf8f2 \uf8f3 diag(id n 2 , − id n 2 ) if n is even, diag(id[ n 2 ], 0, − id[ n 2 ]) if n is odd, U := \uf8f1 \uf8f2 \uf8f3 � idn E 0 idn � if G = SO2n(k), � idn 0 E 0 1 0 0 0 idn � if G = SO2n+1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then U ∈ [GF, GF ] by [22, Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15, Proposition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='21] and we con- sider the elements ri ∈ Ri, i = 1, 2: r1 := U ⊲ j(A) ∈ R1 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 � A −AE+Eφ(A) 0 φ(A) � if G = SO2n(k), � A 0 −AE+Eφ(A) 0 1 0 0 0 φ(A) � if G = SO2n+1(k), r2 := j(A−1) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A direct calculation shows that r1r2 = r2r1 if and only if 2E = AEφ(A−1) + A−1Eφ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) We verify that this never happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume first that p is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By looking at the (1, 1)-entries we see that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) never holds if n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since r1r2 and r2r1 are p-elements, it follows that π(r1r2)2 ̸= π(r2r1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The restriction of π to R1 � R2 is injective because A2 = − id with A irreducible would imply n = 2, a discarded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence OG π(j(A)) is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 33 Assume that p = 2, so G = SO2n(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) amounts to A2E ⋆= Eφ(A)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If n ≥ 4, by looking at the first row we see that ⋆ never holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If n = 3, then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) holds only when a2 = a−1 0 , a1 = a2 0, but in this case pA(X) = X3 + a−1 0 X2 + a2 0X + a0 = (X + a0)2(X + a−1 0 ) is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since |r1| is odd and π is injective, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b) applies and so OG π(j(A)) is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (Here n ≥ 3 for G = SO2n(k) and n ≥ 2 for G = SO2n+1(k) or Sp2n(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let A ∈ GLn(q) be an irreducible element such that j(A) is regular in GLn′(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (i) O := O[GF ,GF ] j(A) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (ii) If j(A) ∈ O ⊆ [GF , GF], then O collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since A is irreducible, OGLn(q) A = OGLn(q) Aq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We have O = O[GF ,GF ] j(Aq) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3 (i) so we can consider the disjoint subracks: R1 := �� A Y 0 φ(A) � ∈ O � , R2 := �� Aq Y 0 φ(Aq) � ∈ O � , if n′ = 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' R1 := �� A 0 Y 0 1 0 0 0 φ(A) � ∈ O � , R2 := �� Aq 0 Y 0 1 0 0 0 φ(Aq) � ∈ O � , if n′ = 2n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then Ri ⊲ Rj ⊆ Rj for 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let r1 = j(A) ∈ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since j(Aq) is regular, CGLn′(q)(r1) consists of semisimple elements, there exists u ∈ [GF, GF ] unipotent block upper tri- angular, with identity diagonal blocks of size n, n if n′ = 2n and n, 1, n if n′ = 2n + 1, such that r2 := u ⊲ j(Aq) ∈ R2 \\ {j(Aq)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Observe that r2 = j(Aq)v for some non-trivial block upper triangular unipotent element v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now, r1j(Aq) = j(Aq)r1 and v /∈ CGLn′(q)(r1) because the latter consists of semisimple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence, r1r2 ̸= r2r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If p = 2, then |A| is odd and O[GF ,GF ] j(A) is of type C by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let p be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then H := ⟨r1, r2⟩ = ⟨r1, v⟩ = ⟨r2, v⟩, with v a p-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus ��OH ri �� ≥ ���O⟨v⟩ ri ��� ≥ 3 for i = 1, 2, so O[GF ,GF ] j(A) is of type C by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since A is irreducible, A ̸= Aq, hence the restriction of π to R1 � R2 is injective, giving (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let G = Sp4(k), let q ≡ 3 mod 4 and let A = � 0 −1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O = OG π(j(A)) is of type D, hence it collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 34 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By assumption, π(j(A)) is an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let sc,d := � c d d −c � ∈ F2×2 q , where (c, d) ∈ F2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A direct calculation shows that Asc,dA−1 = s−c,−d = −sc,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, if sc,d ∈ GL2(q), then π(j(A)) ⊲ π(j(sc,d)) = π(j(sc,d));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' also φ(sc,d) = −1 c2 + d2 sc,−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We pick (a, b) ∈ F2 q such that a2 + b2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since q ≡ 3 mod 4, we have ab ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As sa,b is semisimple, with same trace and determinant as A, it lies in OSL2(q) A , so π (j(sa,b)) ∈ OG π(j(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider the disjoint, non-empty subracks R1 := � π � A X 0 −A � ∈ OG π(j(A)) � , R2 := � π � sa,b X 0 sa,−b � ∈ OG π(j(A)) � of OG π(j(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then Ri ⊲ Rj = Rj for i, j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We set r := π � id2 id2 0 id2 � ⊲ π(j(A)) = π � A −2A 0 −A � ∈ R1, s := π(j(sa,b)) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now ab ̸= 0 implies that sa,bsa,−b is not diagonal, hence (rs)2 = π � id2 2(id2 +sa,bsa,−b) 0 id2 � ̸= π � id2 −2(id2 +sa,bsa,−b) 0 id2 � = (sr)2, so OG π(j(A)) is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proofs of Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We now drop the irreducibility assumption and proceed to prove the main results of this Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For A irreducible, this is Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If A is not irreducible, then we may assume that A is a block diagonal matrix diag(A1, · · · , Af) where the Ai’s are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If they are all of size 1, then j(A) lies in a Fq-split torus and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If, instead, one of the matrices Ai has size ni ≥ 2, then n > 2 and Ai is non-central in GLni(q) because it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6 imply that O Sp2ni(q) j(Ai) collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The statement follows from injectivity of the composition of rack maps: � i−1 � l=1 {j(Al)} � × O Sp2ni(q) j(Ai) × � f� m=i+1 {j(Am)} � → OSp2n(q) j(A) → OG π(j(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' For A irreducible, this is Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If A is not irreducible, then we may assume that A is a block diagonal matrix diag(A1, · · · , Af) where f > 1 and each Ai is an irreducible ni × ni- matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 35 If q = 2, then A lies in SLn(q) and is not irreducible, so the rack inclusion OSLn(q) A ֒→ O[SOn′(q),SOn′(q)] j(A) combined with [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1] gives the claim because PΩ+ n′(q) = [SOn′(q), SOn′(q)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If ni = 1 for all i, then j(A) lies in a Fq-split torus and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, we assume from now on that q > 2 and ni ≥ 2 for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If ni ≥ 3 for some i, then O [SO2ni(q),SO2ni(q)] j(Ai) collapses by Lemmata 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the claim follows because of the injectivity of the composition of the rack morphisms �i−1 � l=1 {j(Al)} � × O [SO2ni(q),SO2ni(q)] j(Ai) × � f� l=i+1 {j(Al)} � → O[SOn′(q),SOn′(q)] j(A) → OG π(j(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' From now on we assume that n1 = 2, and ni ≤ 2 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If ni = 1 for some i, say i = 2, then A2 = (c) for some c ∈ F× q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since A1 is irreducible, it is regular and has no eigenvalues in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus the block diagonal matrix ˜A1 = diag(A1, c) has 3 distinct eigenvalues in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The ma- trices � A1 v 0 c � , v ∈ F2 q, have the same eigenvalues, hence they lie in OGL3(q) ˜ A1 = OSL3(q) ˜ A1 , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Consider the map j : GL3(q) → SO6(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that j � A−1 1 0 0 c � ∈ O := O[SO6(q),SO6(q)] j( ˜ A1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, there is a representative g of a suitable w ∈ W in the normaliser of the torus of diagonal matrices in [SO6(q), SO6(q)] that satisfies g ⊲ j( ˜ A1) = j � tA−1 1 0 0 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, tA−1 1 ∈ OSL3(q) A−1 , hence j � tA−1 1 0 0 c � ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus R1 = � j � A1 v 0 c � : v ∈ F2 q � , R2 = � j � A−1 1 v 0 c � : v ∈ F2 q � are subracks of O, which are disjoint because A1 is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Clearly, Ri ⊲ Rj ⊂ Rj, 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Pick 0 ̸= v ∈ F2 q and set: r := j � A1 0 0 c � ∈ R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' s := j � A−1 1 v 0 c � ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By a direct calculation, rs = sr implies that c is an eigenvalue of A1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Similarly, (rs)2 = (sr)2 iff c2 = −1, which can occur only if q is even or q ≡ 1 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q is even, then O is of type C by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If q ≡ 3 mod 4 or q ≡ 1 mod 4 and c2 ̸= −1, then O is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that q ≡ 1 mod 4 and c2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that 3 = |{s, rsr−1, r2sr−2}| ≤ ���O⟨r,s⟩ s ��� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) 3 = |{r, srs−1, rsrs−1r−1}| ≤ ���O⟨r,s⟩ r ��� (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) 36 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA By a direct calculation, r2sr−2 = s iff A2 1v = −v = c2v, that is, hence c or −c is an eigenvalue of A1, a contradiction because they both lie in Fq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Similarly, rsrs−1r−1 = srs−1 iff (A1 − c)2v = 0, thus c is an eigenvalue of A1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now srs−1 ̸= r implies rsrs−1r−1 ̸= r and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence O[SO3(q),SO3(q)] j( ˜ A1) is of type C by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since the composition (R1 � R2) × � f� l=3 {j(Al)} � → O[SO3(q),SO3(q)] j( ˜ A1) × � f� l=3 {j(Al)} � → O[SOn′(q),SOn′(q)] j(A) → O = OG π(j(A)) is an injective morphism of racks, the statement is proved in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There remains the case ni = 2 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' It suffices to assume that f = 2, so G = SO8(q), and that A1 and A2 are the companion matrices of their characteristic polynomials pA1 = X2 + aX + b and pA2 = X2 + cX + d, so A1 := � 0 −b 1 −a � , A2 := � 0 −d 1 −c � , A := � A1 0 0 A2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As in the previous step, there is an element in [SO8(q), SO8(q)] mapping j(A) to j � A−1 1 0 0 A2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We consider the subracks of O[SO8(q),SO8(q)] j(A) given by R1 := � j � A1 M 0 A2 � ∈ O[SO8(q),SO8(q)] j(A) � , R2 := � j � A−1 1 M 0 A2 � ∈ O[SO8(q),SO8(q)] j(A) � which are disjoint since A1 is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Clearly, Ri ⊲Rj ⊂ Rj, 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let u := � id2 id2 0 id2 � ∈ SL4(q) and consider r := j(u) ⊲ j(A) = j � A1 A2−A1 0 A2 � ∈ R1, s := j � A−1 1 0 0 A2 � ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A direct calculation in GL4(q) shows that (rs)2 = (sr)2 if and only if (A2 − A1)A2(id2 +A2 2) = A−1 1 (A2 − A1)(id2 +A2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) Now, det(id2 +A2 2) = 0 implies that there exists 0 ̸= v ∈ F2 q such that A2 2v = −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By the irreducibility of A2, we get pA2 = X2 + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', A2 = � 0 −1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that det(id2 +A2 2) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) is equivalent to A2 − A1 = A−1 1 − A−1 2 , which is equivalent to A1 = A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore, if A1 ̸= A2, possibly interchanging the roles of A1 and A2, we can make sure that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) is not satisfied, hence (rs)2 ̸= (sr)2, so OSL4(q) A is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If A1 = A2 ̸= � 0 −1 1 0 � , then we interchange A1 and A−1 1 and argue as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In all cases, the restriction of π to R1 � R2 is injective, so OG π(j(A)) is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 37 Finally, if A1 = A2 = � 0 −1 1 0 � , then A ∈ SL4(q) and OPSL4(q) πSL4(q)(A) is of type D by [3, Lemmata 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='15, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='16, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='17], and Oπ(K) π(j(A)) projects onto it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Whence Oπ(K) π(j(A)) is also of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The symplectic groups In this Section, G = Sp2n(k), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Recall that e ̸= x ∈ G is semisimple, GF ∋ x �→ x, O = OGF x and O = OG x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here is the main result of this Section: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x ̸∈ Z(G) be semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O collapses unless n = 2, q ∈ {3, 5, 7} and x is an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Classes represented in K have been discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We deal in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 with cuspidal classes that are not Coxeter, and then with Coxeter classes in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1 is proved in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Cuspidal classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Here we discuss the semisimple classes that are cuspidal but not Coxeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Below we use without further notice that a cus- pidal class could not meet a standard Levi subgroup by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We start by the following observation: two semisimple symplectic matrices conjugated in GL2n(q) are then conjugated in Sp2n(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In either of the following cases, O is not cuspidal: (a) Some eigenvalue of x lies in Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (b) |x| ∈ {2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, if λ ∈ Fq is an eigenvalue of x, then so is λ−1, hence O contains an element of the form � λ A′ B′ C′ D′ λ−1 � which belongs to a Levi subgroup isomorphic to Sp2(n−1)(k) × k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus O is not cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By (a), we may assume that x has no eigenvalues in Fq, so ±1 are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If |x| ∈ {2, 3, 4}, then x has at most 2 distinct eigenvalues, namely the two primitive roots of 1, so it is not cuspidal by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Cuspidal classes in the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' As is well-known, the Weyl group is W = (Z/2)n⋊Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' let (ej)j∈In be the canonical basis of (Z/2)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We identify W with a subgroup of S2n as in [11]: W ≃ {ς ∈ S2n : ς(2n + 1 − j) = 2n + 1 − ς(j), j ∈ In}, Sn ∋ σ �→ σ′, σ′(j) = � σ(j), if j ∈ In, 2n + 1 − ς(2n + 1 − j), if j ∈ In+1,2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ej �→ τj = (j 2n + 1 − j), j ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Given h ≤ k in In, we consider the 2(k − h + 1)-cycle in S2n defined by ch,k = (h h + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' k 2n + 1 − h 2n − h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 2n + 1 − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1) 38 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Evidently, ch,k ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let now λλλ = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , dt) be a partition of n, denoted λλλ ⊢ n, with d1 ≥ · · · ≥ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Set cλλλ = c1,d1cd1+1,d1+d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' cd1+···+dt−1+1,n ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) By the identification above, we can rephrase [16, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6]: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The conjugacy class of such cλλλ is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The family cλλλ, λλλ ⊢ n, is a complete set of representatives of the cuspidal conjugacy classes of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ For instance, if λλλ = (n), then cλλλ is a Coxeter element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Cuspidal, but not Coxeter, classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let λλλ = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , dt) ⊢ n, with d1 ≥ · · · ≥ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let Gλλλ be the image of the injective morphism of groups Sp2d1(k) × Sp2d2(k) × · · · × Sp2dt(k) −→ Sp2n(k), �� A1 B1 C1 D1 � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , � At Bt Ct Dt �� �−→ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed A1 B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' At Bt Ct Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' C1 D1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' cλλλ has a decomposition Γ such that GΓ = Gλλλ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' First, wj = cd1+d2+···+dj−1+1,d1+d2+···+dj is a Coxeter element of the factor Sp2dj(k) of Gλλλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Up to appropriate identifications, the union of de- compositions Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , Γt of w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , wt is a decomposition of cλλλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If the conjugacy class O in GF is cuspidal but not Coxeter, then it is of type C, hence it collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12, there is partition λλλ ̸= (n) such that O intersects GF λλλ = Sp2d1(q) × Sp2d2(q) × · · · × Sp2dt(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xt) ∈ GF λλλ ∩ O, with xj ∈ Sp2dj(q) for all j, so that O GF λλλ x = O Sp2d1(q) x1 × O Sp2d2(q) x2 × · · · × O Sp2dt(q) xt ≤ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that xj /∈ Z(Sp2dj(q)) for all j ∈ It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, if xj ∈ Z(Sp2dj(q)) for some j, then x belongs to the torus Tci λλλ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='11), where cj λλλ ∈ W is defined as cλλλ in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2) but omitting cd1+···+dj−1+1,d1+···+dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' This is a contradiction because cj λλλ is not cuspidal proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then the lemma follows from NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 39 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9, by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='10 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, xj ̸= xq j, otherwise x would lie in a non-cuspidal torus by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2 □ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If the conjugacy class O in GF is cuspidal but not Coxeter, then the conjugacy class O in G is of type C, hence it collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let π : Sp2n(q) → PSp2n(q) be the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We may assume that q is odd, thus ker π = {± id}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Keep the notation of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The Lemma holds for t > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Indeed, the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='9 provides a subrack of O of type C with the form Y = � {x1} × O Sp2d2(q) x2 × {x3} � � � {xq 1} × O Sp2d2(q) x2 × {x3} � and clearly the restriction of π to Y is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The Lemma holds for t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x1 ̸= −xq 1, then the restriction of π to the subrack of type C Y = � {x1} × O Sp2d2(q) x2 � � � {xq 1} × O Sp2d2(q) x2 � is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' similarly if x2 ̸= −xq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus we may assume that x1 = −xq 1 and x2 = −xq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Now x1 lives in a Coxeter torus TF 1 in Sp2d1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By [22, Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3] and [16, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3], we have |TF 1 | = qd1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence |x1| divides (2(q − 1), qd1 + 1) = \uf8f1 \uf8f2 \uf8f3 2 if qd1 ≡ 1 mod 4, 4 if qd1 ≡ 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By symmetry, we may assume that the same holds for x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence |x| divides 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' this contradicts Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Coxeter classes in Sp2n(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x ∈ T ′ = TF w be a Coxeter element and let O = OGF x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Hence x is regular and its order divides qn + 1, so xqn = x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Arguing as in [5, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5] we see that O ∩ T ′ = {x±qj, j ∈ I0,n−1}, and that the action of w raises x to xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If ξ ∈ Fq is an eigenvalue of x, then all other eigenvalues of x are of the form {ξqj, j ∈ I0,2n−1} = {ξ±qj, j ∈ I0,n−1}, with ξqn = ξ−1 and they are all distinct by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume q is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x is a Coxeter element, then −x ̸∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If −x ∈ O, then with notation as above, −ξ is an eigenvalue of x, so −ξ = ξqj or −ξ = ξ−qj for some j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In the first case ξq2j = (−ξ)qj = ξ, whilst in the second ξq2j = (−ξ−1)qj = −(ξqj)−1 = ξ, with 2j < 2n in both cases, contradicting regularity of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 40 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let x be a Coxeter element in GF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then O is of type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let H be a subgroup of GF isomorphic to SL2(qn), which exists by [21, II Satz 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Any non-split torus of T ′ ≤ H has order qn + 1 by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1), hence it is a Coxeter torus in GF, [16, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Therefore |O ∩ T ′| = |{x±qj, j ∈ I0,n−1}| = 2n, |OH x ∩ T ′| ≤ 2, so the intersection O ∩ H is not a single H-conjugacy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Since H ≃ SL2(qn) with n ≥ 2, the group H/Z(H) is simple, so the non-central normal subgroup ⟨OH x ⟩ coincides with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' In addition, |OH x | = qn(qn − 1) > 4, so O is of type C by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The restriction of the projection π: GF → G to O is injective by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6, so π(O) = O is of type C as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' The general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let L be a split F-stable Levi subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Then, there exist f > 0, m ≥ 0 and ni for i ∈ If satisfying n = e + �f i=1 ni such that L is isomorphic the image of the injective morphism of groups �j : GLn1(k) × · · · × GLnf (k) × Sp2e(k) → Sp2n(k) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) (A1, · · · , Af, A) �→ diag(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , Ar, A, φ(Af), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , φ(A1)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4) Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If n = 2, q = 3 and x is a non-central involution, then O is kthulhu by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume that q /∈ {5, 7} if n = 2 and x is a non-central involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x is cuspidal but not Coxeter, we invoke Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5, whilst if x is Coxeter, then the claim follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If x is not cuspidal, then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='12 we may assume that x ∈ LF for a proper standard Levi subgroup L of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let �j be as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='3) and let x = �j(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' , xf, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Taking ni, for i ∈ If and e to be minimal, and possibly increasing f, we assume that each xi is irreducible in GLni(q) and y is cuspidal in Sp2e(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Under these assumptions xi ∈ Z(GLni(q)) if and only if ni = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If e = 0 the statement follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If e ≥ 2, then we consider the rack embedding {x1} × · · · × {xf} × OSp2e(q) y → OSp2n(q) x → OPSp2n(q) x (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5) and invoke either Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='5 or Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Assume from now on that e = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=', y is irreducible in Sp2(q) ≃ SL2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If there exists and ni such that ni > 1, then we consider the rack embedding {x1} × · · · × O Sp2ni(q) xi × · · · × {xf} × {y} → OSp2n(q) x → OPSp2n(q) x (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='6) and invoke Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' There remains the case in which ni = 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We assume that f = 1, for if f > 1 we can use the rack injection {x1} × · · · × {xf−1} × O Sp2(nf +1)(q) ˜j(xf ,y) → OSp2n(q) x → OPSp2n(q) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 41 Since y is irreducible, it lies in a non-split maximal torus, so its order divides q + 1 and so yq = y−1 ∈ OSL2(q) y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Also, if py = X2 − zX + 1, then z ̸= ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We may assume that y = � 0 1 −1 z � so y−1 = � z −1 1 0 � and x = diag(λ, y, λ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We consider the following subracks of O = OSp4(q) x : R := � x′ = � λ ∗ ∗ 0 y ∗ 0 0 λ−1 � : x′ ∈ O � , S := � x′ = � λ−1 ∗ ∗ 0 y−1 ∗ 0 0 λ � : x′ ∈ O � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' By construction, R ⊲ S ⊂ S and S ⊲ R ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Observe that R ∩ S = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' otherwise y = y−1 and p = 2, but in this case y would not be semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Let M ∈ SL2(q) be such that M ⊲ y = y−1 and let r := � 1 1 0 0 0 1 0 0 0 0 1 −1 0 0 0 1 � ⊲ x = � λ −λ 1 1 0 0 1 1 0 −1 z z−λ−1 0 0 0 λ−1 � ∈ R, s := � 0 0 1 0 M 0 −1 0 0 � ⊲ x = diag(λ−1, y−1, λ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A direct calculation shows that rs = sr only if λ2 = 1 and z = ±2, a discarded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Taking H := ⟨r, s⟩, we see that OH r ∩ OH s ⊂ R ∩ S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Thus, if p = 2, then |x| is odd so OSp4(q) x is of type C by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' If p is odd, then (rs)2 ̸= (sr)2 because rs and sr are upper triangular unipotent matrices by construction, and so OSp4(q) x is of type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' We claim that the restriction of π to R � S is injective: indeed injectivity could fail only for λ2 + 1 = 0 and z = 0 but in this case, λ ∈ Fq would be a root of py which is irreducible, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' □ References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Andruskiewitsch, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Carnovale, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Garc´ıa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Unipotent classes in PSLn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 4, Article ID 1550053, 35 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [3] Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Semisimple classes in PSLn(q), Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Iberoam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 33, 995–1024, (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [4] Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Unipotent classes in Chevalley and Steinberg groups, Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Theory 23, 621–655(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [5] Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Mixed classes in Chevalley and Steinberg groups, Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 166, 605–647 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [6] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Andruskiewitsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Fantino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Garc´ıa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Vendramin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' On Nichols algebras associated to simple racks, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 537 (2011), 31–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 42 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ANDRUSKIEWITSCH, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CARNOVALE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' GARC´IA [7] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Andruskiewitsch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Fantino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Gra˜na, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Vendramin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Finite-dimensional pointed Hopf algebras with alternating groups are trivial, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Pura Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' (4), 190 (2011), 225–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [8] Pointed Hopf algebras over the sporadic simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Algebra 325 (2011), 305–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [9] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Andruskiewitsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Gra˜na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' From racks to pointed Hopf algebras, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 178 (2003), 177–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Beltr´an Cubillos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' ´Algebras de Nichols sobre grupos diedrales y pecios kthulhu en grupos espor´adicos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Tesis doctoral, Universidad Nacional de C´ordoba (2020).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' 25, 49?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='-80 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' NICHOLS ALGEBRAS OVER SEMISIMPLE CLASSES 43 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=': FaMAF-Universidad Nacional de C´ordoba, CIEM (CONICET), Medina Allende s/n, Ciudad Universitaria, 5000 C´ordoba, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Email address: nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='andruskiewitsch@unc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='ar G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=': Dipartimento di Matematica Tullio Levi-Civita, Universit`a degli Studi di Padova, via Trieste 63, 3512,1 Padova, Italia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Email address: carnoval@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='it, +39-049-8271354 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=': Departamento de Matem´atica, Facultad de Ciencias Exactas, Uni- versidad Nacional de La Plata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' CMaLP-CIC-CONICET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Calle 47 y Calle 115, 1900 La Plata, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content=' Email address: ggarcia@mate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='unlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'} +page_content='ar' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE1T4oBgHgl3EQfsAWQ/content/2301.03361v1.pdf'}