diff --git "a/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt" "b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8dE2T4oBgHgl3EQfPwZL/content/tmp_files/load_file.txt" @@ -0,0 +1,975 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf,len=974 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='03762v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='AG] 10 Jan 2023 REGULAR SEMISIMPLE HESSENBERG VARIETIES WITH COHOMOLOGY RINGS GENERATED IN DEGREE TWO MIKIYA MASUDA AND TAKASHI SATO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A regular semisimple Hessenberg variety Hess(S, h) is a smooth subvariety of the flag variety determined by a square matrix S with distinct eigenvalues and a Hessenberg function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The cohomology ring H∗(Hess(S, h)) is independent of the choice of S and is not explicitly described except for a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this paper, we characterize the Hessenberg function h such that H∗(Hess(S, h)) is generated in degree two as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It turns out that such h is what is called a (double) lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Introduction The flag variety Fl(n) consists of nested sequences of linear subspaces in the complex vector space Cn: Fl(n) = {V• = (V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn) | dimC Vi = i (∀i ∈ [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n})}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A Hessenberg function h: [n] → [n] is a monotonically non-decreasing function satisfying h(j) ≥ j for any j ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We often express a Hessenberg function h as a vector (h(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , h(n)) by listing the values of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Given an n × n matrix A and a Hessenberg function h, a Hessenberg variety Hess(A, h) is defined as Hess(A, h) = {V• ∈ Fl(n) | AVi ⊂ Vh(i) (∀i ∈ [n])} where the matrix A is regarded as a linear operator on Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that Hess(A, h) = Fl(n) if h = (n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The family of Hessenberg varieties Hess(A, h) contains important varieties such as Springer fibers (A is nilpotent and h = (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n)), Peterson varieties (A is regular nilpotent and h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n)), and permutohedral varieties (A is regular semisimple and h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n)), which are toric varieties with permutohedra as moment polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Among n × n matrices, regular semisimple ones S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' matrices S having distinct eigenvalues) are generic and Hess(S, h) is called a regular semisimple Hessenberg variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The regular semisimple Hessenberg variety Hess(S, h) has nice properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For instance, it is smooth and its cohomology H∗(Hess(S, h)) becomes a module over the symmetric group Sn on [n] by Tymoczko’s dot action [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remarkably, the solution of Shareshian–Wachs conjecture [18] by Brosnan and Chow [5] (and Guay-Paquet [10]) connected H∗(Hess(S, h)) as an Sn-module and chromatic symmetric functions on certain graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This opened a way to prove the famous Stanley–Stembridge conjecture in graph theory through the geometry or topology of Hessenberg varieties and motivated us to study H∗(Hess(S, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that H∗(Hess(S, h)) (indeed the diffeomorphism type of Hess(S, h)) is independent of the choice of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We write the regular semisimple Hessenberg variety Hess(S, h) as X(h) for brevity since our concern in this paper is its cohomology ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The Sn-module structure on H∗(X(h)) is determined in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, that on H2(X(h)) was explicitly described by Chow [7] combinatorially (through the theorem by Brosnan-Chow mentioned above) and by Cho-Hong-Lee [6] geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Motivated by their works, Ayzenberg and the authors [4] reproved their results by giving explicit additive generators of H2(X(h)) in terms of GKM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The ring structure on H∗(X(h)) is not explicitly described except for a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remember that X(h) for h = (n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) is the flag variety Fl(n) and H∗(Fl(n)) is generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, X(h) for h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n, n) is the permutohedral variety and H∗(X(h)) is also generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, for h = (h(1), n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) with h(1) arbitrary, a result of [2] shows that H∗(X(h)) is generated Date: January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Primary: 57S12, Secondary: 14M15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hessenberg variety, torus action, GKM theory, equivariant cohomology, lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 1 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO in degree 2 as a ring if and only if h(1) = 2 or n, where X(h) = Fl(n) for the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, it is natural to ask when H∗(X(h)) is generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The answer is the following, which is our main result in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Assume that h(j) ≥ j + 1 for j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then H∗(X(h)) is generated in degree 2 as a ring if and only if h is of the following form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) for some 1 ≤ a < b ≤ n, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) h(j) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a + 1 (1 ≤ j ≤ a) j + 1 (a < j < b) n (b ≤ j ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) X(h) is connected if and only if h(j) ≥ j + 1 for any j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When X(h) is not connected, each connected component of X(h) is a product of smaller regular semisimple Hessenberg varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) X(h) is the flag variety Fl(n) when (a, b) = (n − 1, n) and is the permutohedral variety when (a, b) = (1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3) We will give an explicit presentation of the ring structure on H∗(X(h)) for h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) in a forthcoming paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We can visualize a Hessenberg function h by drawing a configuration of the shaded boxes on a square grid of size n × n, which consists of boxes in the i-th row and the j-th column satisfying i ≤ h(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since h(j) ≥ j for any j ∈ [n], the essential part is the shaded boxes below the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For example, Figure 1 below is the configurations of two Hessenberg functions h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) with n = 11: one is h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (1, 7) and the other is h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (3, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We often identify a Hessenberg function h with its configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ ❅❅ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The configurations for h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) and h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) The chromatic symmetric functions and LLT polynomials associated with h of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) are studied from the viewpoint of combinatorics in [8, 13], and when a = 1 or b = n, the corresponding Hessenberg functions h = (2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , b, n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n) or (a + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , a + 1, a + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) are called lollipops in those papers, so the Hessenberg function of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) may be called a double lollipop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In Section 2, we review GKM theory to compute the equivariant cohomology of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the “only if” part in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, we consider a Morse-Bott function fh on X(h), where the inverse image of the minimum or maximum value of fh is a regular semisimple Hessenberg variety X(h′) with h′ of size one less than that of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then a property of the REGULAR SEMISIMPLE HESSENBERG VARIETIES 3 Morse-Bott function fh shows the surjectivity of the restriction map H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q), and this enables us to use an inductive argument to prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In Section 4, we prove the “if” part in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 by applying the method developed in [2, 9] together with the explicit generators of H2(X(h)) obtained in our previous work [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Regular semisimple Hessenberg varieties We first recall some properties of a regular semisimple Hessenberg variety X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 ([14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) X(h) is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) dimC X(h) = �n j=1(h(j) − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3) X(h) is connected if and only if h(j) ≥ j + 1 for ∀j ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (4) Hodd(X(h)) = 0 and the 2k-th Betti number of X(h) is given by #{w ∈ Sn | ℓh(w) = k} where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) ℓh(w) = #{1 ≤ j < i ≤ n | w(j) > w(i), i ≤ h(j)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For calculation of the cohomology ring of X(h), we use equivariant cohomology which we shall explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We assume that the matrix S in X(h) = Hess(S, h) is a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let T be an algebraic torus consisting of diagonal matrices in the general linear group GLn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The linear action of T on Cn induces an action on the flag variety Fl(n) and preserves X(h) since S commutes with T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The fixed point sets of the T -actions on X(h) and Fl(n) consist of all permutation flags, that is, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) X(h)T = Fl(n)T ∼= Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since T can naturally be identified with (C∗)n, the classifying space BT of T is B(C∗)n = (CP ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let pi : T → C∗ be the projection on the i-th diagonal component of T and ti = p∗ i (t) ∈ H2(BT ) where p∗ i : H∗(BC∗) → H∗(BT ) and t ∈ H2(BC∗) is the first Chern class of the tautological line bundle over BC∗ = CP ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) H∗(BT ) = Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The equivariant cohomology of the T -variety X(h) is defined as H∗ T (X(h)) := H∗(ET ×T X(h)) where ET is the total space of the universal principal T -bundle ET → BT and ET ×T X(h) is the orbit space of the product ET × X(h) by the diagonal T -action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The projection ET × X(h) → ET on the first factor induces a fibration X(h) ρ−→ ET ×T X(h) π−→ BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since Hodd(X(h)) = 0 as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 and Hodd(BT ) = 0, the Serre spectral sequence of the fibration above collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It implies that ρ∗ : H∗ T (X(h)) → H∗(X(h)) is surjective and induces a graded ring isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) H∗(X(h)) ∼= H∗ T (X(h))/(π∗(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , π∗(tn)) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, one can find the ring structure on H∗(X(h)) through H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since Hodd(X(h)) = 0, it follows from the localization theorem that the homomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) H∗ T (X(h)) → H∗ T (X(h)T ) = � w∈Sn H∗ T (w) = � w∈Sn Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn] = Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) induced from the inclusion map X(h)T → X(h) is injective, where X(h)T is identified with Sn by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) and Map(P, Q) denotes the set of all maps from P to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The T -variety X(h) is what is called a GKM manifold and the image of the homomorphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) is described in [20] as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) {f ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) | f(w) − f(w(i, j)) ∈ (tw(i) − tw(j)), for ∀w ∈ Sn, j < i ≤ h(j)}, 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO where (i, j) denotes the transposition interchanging i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that the image of π∗(ti) ∈ π∗(H∗(BT )) ⊂ H∗ T (X(h)) by the homomorphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) is the constant function ti ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Guillemin and Zara [11] assigned a labeled graph to a GKM manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The labeled graph of X(h) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The vertex set is the fixed point set X(h)T = Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' There is an edge between vertices w and v if and only if v = w(i, j) for some j ≤ i ≤ h(j), and the edge between w and w(i, j) is labeled by tw(i) − tw(j) up to sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For h = (2, 3, 3) and h′ = (3, 3, 3), the labeled graphs of X(h) and X(h′) are drawn in Figure 2, where we use the one-line notation for each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ❞ ❞ ❞ ❞ ❞ ❞ ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟ ✟ 123 321 132 312 213 231 X(h) ❞ ❞ ❞ ❞ ❞ ❞ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ✟✟✟ ✟✟✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ 123 321 132 312 213 231 X(h′) = Fl(3) labels : t1 − t2 : t2 − t3 : t1 − t3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The labeled graphs of X(h) and X(h′) In general, labeled graphs and their graph cohomologies are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let R be a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' A labeled graph (Γ, α) consists of a graph Γ = (V, E) and a labeling α: E → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The graph cohomology of a labeled graph (Γ, α) is defined as H∗(Γ, α) = {f ∈ Map(V, R) | f(w) − f(v) ∈ (α(e)) for ∀e = wv ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The graph cohomology H∗(Γ, α) is a subring of Map(V, R) with the coordinate-wise sum and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that we may ignore the signs of the labels α(e) since (α(e)) = (−α(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The observation above shows that the graph cohomology of the labeled graph of X(h) coincides with H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Sending ti to tσ(i) for σ ∈ Sn and i ∈ [n] induces an action of Sn on Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then, the module Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) becomes an Sn-module under what is called the dot action defined by (σ · f)(w) := σ(f(σ−1w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As easily checked, the graph cohomology of X(h) is invariant under the dot action and H∗ T (X(h)) becomes a module over Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, since the action of Sn preserves the ideal (π∗(t1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , π∗(tn)), the action descends to H∗(X(h)) and H∗(X(h)) also becomes an module over Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Obviously, constant functions in Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) satisfy the condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' They are ele- ments corresponding to π∗(H∗(BT )) ⊂ H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Below are three types of elements xi, yj,k, and τA in Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) which satisfy the condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6), so they are in H∗ T (X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let ⊥(h) : = {j ∈ [n − 1] | h(j − 1) = h(j) = j + 1} L(h) : = {j ∈ [n − 1] | h(j − 1) = j and h(j) = j + 1} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7) where we understand h(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (1) For i ∈ [n], xi(w) := tw(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (2) For j ∈ [n − 1] with j ∈ ⊥(h) and k ∈ [n], yj,k(w) := � tk − tw(j+1) (if k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(j)}) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' REGULAR SEMISIMPLE HESSENBERG VARIETIES 5 (3) For A ⊂ [n] with |A| ∈ L(h) τA(w) := � tw(|A|) − tw(|A|+1) (if {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(|A|)} = A) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The cohomological degrees of the elements xk, yj,k, τA are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' One can easily check that the dot actions of σ ∈ Sn on these elements are given as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) σ · xk = xk, σ · yj,k = yj,σ(k), σ · τA = τσ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here is a geometrical meaning of xk’s (regarded as elements in H2(X(h)) through the isomor- phism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' There is a nested sequence of tautological vector bundles over the flag variety Fl(n): F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn = Fl(n) × Cn where Fk := {(V•, v) ∈ Fl(n) × Cn | v ∈ Vk} and V• = ({0} = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then xk (k ∈ [n]) is the image of the first Chern class of the quotient line bundle Fk/Fk−1 over Fl(n) by the homomorphism ι∗ : H∗(Fl(n)) → H∗(X(h)) induced from the inclusion map ι: X(h) → Fl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The dot action on H∗(Fl(n)) is trivial, so the image of ι∗ must be contained in the ring of invariants H∗(X(h))Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In fact, it follows from [1, Theorems A and B] that the image of ι∗ agrees with H∗(X(h))Sn when tensoring with Q and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) H∗(X(h))Sn ⊗ Q = Q[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , xn]/(fh(1),1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , fh(n),n) where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) fh(j),j = j � k=1 \uf8eb \uf8edxk h(j) � ℓ=j+1 (xk − xℓ) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, the Hilbert series of H∗(X(h))Sn is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) Hilb(H∗(X(h))Sn, √q) = n−1 � j=1 [h(j) − j]q where the Hilbert series of a graded algebra A = �∞ r=0 Ar over Z is defined as Hilb(A, q) := ∞ � r=0 (rankZAr)qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Through the isomorphism (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4), the elements xk, yj,k, τA determine elements in H2(X(h)), denoted by the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 ([4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The elements {xk, yj,k, τA | k ∈ [n], j ∈ ⊥(h)\\{n − 1}, A ⊂ [n] with |A| ∈ L(h)\\{n − 1}} generate H2(X(h)) with relations (1) �n k=1 xk = 0, (2) �n k=1 yj,k = (x1 + · · · + xj) − jxj+1 for j ∈ ⊥(h)\\{n − 1}, (3) � |A|=j τA = xj − xj+1 for j ∈ L(h)\\{n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 (see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 in [4] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The element yj,k is defined by looking at the j-th column of the configuration associated to the Hessenberg function h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly, one can define an element y∗ i,k of H∗ T (Hess(S, h)) by looking at the i-th row of the configuration as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For i ∈ [n], we define h∗(i) := min{j ∈ [n] | h(j) ≥ i}, 6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO so that the shaded boxes in the i-th row and under the diagonal in the configuration associated to h are at positions (i, ℓ) (h∗(i) ≤ ℓ < i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h∗(i) = i − 1, we define (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) y∗ i,k(w) := � tk − tw(i−1) (k ∈ {w(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' One can see that y∗ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='k is in H2 T (Hess(S, h)) and we may replace yj,k’s for j ∈ ⊥(h)\\{n − 1} in the generating set in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 by y∗ i,k’s for i ≥ 3 such that h∗(i) = h∗(i + 1) = i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) in Figure 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (a, b) = (3, 7)), we have ⊥(h) = {3, 10}, L(h) = {4, 5, 6}, so Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 says that H2(X(h)) is generated by xk (k ∈ [11]), y3,k (k ∈ [11]), τA for A ⊂ [11] with |A| = 4, 5 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, it follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 that y3,k above may be replaced by y∗ 8,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Necessity In this section, we study a necessary condition on h for H∗(X(h)) to be generated in degree 2 as a ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moment maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let µ: Fl(n) → Rn be the standard moment map on the flag variety Fl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Its image is the permutohedron Πn obtained as the convex hull of the orbits of (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n) by permuting its coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, if ew (w ∈ Sn) denotes the permutation flag associated with w, then we have µ(ew) = (w−1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w−1(n)) ∈ Rn (see [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) Sr n := {w ∈ Sn | w(r) = n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then µ(Sr n) is the set of all vertices of Πn whose n-th coordinate is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore the projection πn : Πn → R, πn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , xn) = xn on the n-th coordinate takes minimum on S1 n and maximum on Sn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The composition of µ and πn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) f := πn ◦ µ: Fl(n) → R is the moment map induced from the following S1-action on Cn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , zn) → (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , zn−1, gzn) (g ∈ S1 ⊂ C), and it is a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let hj be the Hessenberg function obtained by removing all the boxes in the j-th row and all the boxes in the j-th column from its configuration (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To be precise, hj is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' hj(i) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 h(i) (i < j, h(i) < j) h(i) − 1 (i < j, h(i) ≥ j) h(i + 1) − 1 (i ≥ j) REGULAR SEMISIMPLE HESSENBERG VARIETIES 7 j-th row → ↓ j-th column h ❀ remove ← տ ↑ ❀ hj Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The configuration corresponding to hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following is a key lemma in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The restriction maps H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q), H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(hn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) are surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let fh be the map f in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) restricted to X(h), which is also a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The inverse image of the minimum value under fh is X(h1), so it follows from [19, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1] that the restriction map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) H∗ S1(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is surjective, where the S1-action on X(h) is the induced one from the S1-action defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since the S1-action on X(h1) is trivial, we have H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) = H∗(BS1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q)⊗ H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) and hence the forgetful map H∗ S1(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, the surjectivity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) implies the surjectivity of the restriction map H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) → H∗(X(h1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) in ordinary cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The same argument applied to −fh proves the statement for X(hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The surjectivity of the above restriction maps (even with Z coefficients) can also be verified by GKM theory as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that the inclusion of the fixed point set induces an injective homomorphism H∗ T (X(h)) → H∗ T (X(h)T ) ∼= Map(Sn, H∗(BT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The equivariant cohomology H∗ T (X(h)) has an H∗(BT )- module basis {σw,h | w ∈ Sn} (see [6, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It corresponds to a natural paving and then it is a ‘flow-up basis.’ Note that any element of Sn n = Sn−1 is not greater than any element of Sn \\ Sn n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The restriction of {σw,h | w ∈ Sn n} onto X(hn), that is, its restriction onto the fixed point set Sn n = X(hn)T as elements of Map(Sn, H∗(BT )), is a flow-up basis of H∗ T (X(hn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence H∗ T (X(h)) → H∗ T (X(hn)) is surjective, and then H∗(X(h)) → H∗(X(hn)) is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The surjectivity of H∗(X(h)) → H∗(X(h1)) can be verified by a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Given a Hessenberg function h, we obtain a smaller Hessenberg function by removing the first column and row or the last column and row repeatedly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' by taking h1 or hn repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We call it a minor of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following corollary follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let h′ be a minor of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' If H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is generated in degree 2 as a ring, then so is H∗(X(h′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' An easy argument shows that h being of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) can be rephrased as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The Hessenberg function h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) if and only if h has neither (α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β), (β − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β − 1, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β � �� � α ) for 3 ≤ α < β, nor (2, γ − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , γ − 1, γ, γ) for γ ≥ 5 as its minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO Recall that if h† denotes the Hessenberg function obtained by flipping the configuration of h along the anti-diagonal, then X(h†) ∼= X(h) as varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore X((α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β)) ∼= X((β − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β − 1, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β � �� � α )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here, we know that H∗(X((α, β, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , β));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is not generated in degree 2 for 3 ≤ α < β by [2, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus, it suffices to treat the last case in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3, which we shall discuss in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this subsection we prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q) is not generated in degree 2 when h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Some computation is involved in the proof of this proposition but the idea of the proof is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We compute the Poincar´e polynomial of X(h) using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, using explicit generators of H2(X(h)) by [4], we compute an upper bound of the Hilbert series of the subring of H∗(X(h)) generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then it turns out that the latter is strictly smaller than the former at a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Poincar´e polynomial of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The following proposition, which easily follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(4), enables us to compute the Poincar´e polynomial of X(h) inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5 ([4, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) Poin(X(h), √q) = n � j=1 qh(j)−j Poin(X(hj), √q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Using the proposition above, the Poincar´e polynomial of X(h) is explicitly computed as follows when h = (h(1), n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6 ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (h(1), n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) Poin(X(h), √q) = [h(1)]q[n − 1]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)qh(1)−1[n − h(1)]q[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', where [m]q = 1 − qm 1 − q , [m]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' = [1]q[2]q · · · [m]q = m � j=1 1 − qj 1 − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Now, let h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) and set Pn(q) := Poin(X(h), √q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For n ≥ 5, the following recurrence formula holds Pn(q) = (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 2)(q + q2)[n − 3]q[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(q + qn−3) {(1 + q)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 3)q[n − 4]q[n − 4]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='} + (q + q2 + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let Fn(q) denote the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) with h(1) = 2, that is, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7) Fn(q) := (1 + q)[n − 1]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)q[n − 2]q[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Then we have Poin(X(h1), √q) = Poin(X(hn), √q) = Fn−1(q) Poin(X(h2), √q) = Poin(X(hn−1), √q) = (n − 1)Fn−2(q) Poin(X(hj), √q) = Pn−1(q) (3 ≤ j ≤ n − 2), REGULAR SEMISIMPLE HESSENBERG VARIETIES 9 where we note that X(h2) consists of n − 1 copies of Fl(n − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5), we have Pn(q) = qFn−1(q) + (n − 1)qn−3Fn−2(q) + (qn−4 + · · · + q)Pn−1(q) + (n − 1)qFn−2(q) + Fn−1(q) = (1 + q)Fn−1(q) + (n − 1)(q + qn−3)Fn−2(q) + (q + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Combining this equation with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7), we obtain the desired equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For n ≥ 4, let Qn(q) = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then we have Pn(q) ≡ Qn(q) mod (qn−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In other words, Pn(q) and Qn(q) coincide up to degree n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the lemma by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When n = 4, we have P4(q) = 1 + 11q + 11q2 + q3, Q4(q) = 1 + 11q + 20q2 + 12q3, and the lemma is true for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let n be given and suppose that the lemma is true for n − 1, that is, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) Pn−1(q) ≡ Qn−1(q) mod (qn−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hereafter, in this proof, all congruences will be taken modulo qn−2 unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since we have (q + q2)[n − 3]q[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ≡ (q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' q2[n − 4]q[n − 4]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ≡ q2[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', the recurrence formula in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7 reduces to the following congruence relation: Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(q + (n − 2)q2)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8) and the definition of Qn that the sum of the last two terms above becomes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q) ≡ (n − 1)qn−3 + � 1 + (2n − 2)q + (n − 1)(n − 2)q2� (q + · · · + qn−4)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 4) 2 qn−3 = � 1 − q + (n − 1)q(1 − q) + nq + (n − 1)2q2� (q + · · · + qn−4)[n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 2) 2 qn−3 ≡ � q − qn−3 + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n − 1)(n − 2) 2 qn−3 ≡ � q + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 By substituting it to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9), we obtain Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + � (nq + (n − 1)2q2) + (nq + (n − 1)2q2)(q + · · · + qn−4) � [n − 3]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 ≡ (1 + nq + (n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (nq + (n − 1)2q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + n(n − 3) 2 qn−3 = Qn(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO This completes the induction step and the lemma has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of the subring generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n − 1, n, n) for n ≥ 5, we first observe H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='7), we have ⊥(h) = {n − 2}, L(h) = {1, n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, it follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 that H2(X(h)) is generated by the following elements (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) xk, yk := yn−2,k, τk := τ{k} (k ∈ [n]), where xk(w) = tw(k), yk(w) = yn−2,k(w) = � tk − tw(n−1) (if k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n − 2)}) 0 (otherwise), τk(w) = τ{k}(w) = � tw(1) − tw(2) (if k = w(1)) 0 (otherwise) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) for w ∈ Sn by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) n � k=1 yk = x1 + · · · + xn−2 − (n − 2)xn−1, n � k=1 τk = x1 − x2 by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We also have σ · xk = xk, σ · yk = yσ(k), σ · τk = τσ(k) for σ ∈ Sn by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To make the following argument clearer, we introduce elements ρk for k ∈ [n] defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) ρk(w) := � tw(n−1) − tw(n) (if k = w(n)) 0 (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to τk, the ρk satisfies the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) so that it defines an element of H2 T (X(h)) and H2(X(h)) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='14) n � k=1 ρk = xn−1 − xn, σ · ρk = ρσ(k) for σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' An elementary check shows that (yk − yℓ)(w) − (ρk − ρℓ)(w) = tk − tℓ (k, ℓ ∈ [n], w ∈ Sn) and hence yk − yℓ = ρk − ρℓ in H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, �n k=1 yk and �n k=1 ρk are both linear polynomials in xi’s by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='14), so we may replace yk’s in the generating set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10) by ρk’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Namely H2(X(h)) is generated by xk, τk, ρk (k ∈ [n]) with relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='15) n � k=1 xk = 0, n � k=1 τk = x1 − x2, n � k=1 ρk = xn−1 − xn, and the actions of σ ∈ Sn on those generators are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='16) σ · xk = xk, σ · τk = τσ(k), σ · ρk = ρσ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' REGULAR SEMISIMPLE HESSENBERG VARIETIES 11 Our purpose is to find a sharp upper bound of the Hilbert series of the subring R(h) of H∗(X(h)) generated by H2(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let A(h) be the subring of H∗(X(h)) generated by xk’s and we regard R(h) as a module over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) that τkτℓ = � (x1 − x2)τk (k = ℓ) 0 (k ̸= ℓ), ρkρℓ = � (xn−1 − xn)ρk (k = ℓ) 0 (k ̸= ℓ), τkρk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, R(h) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The subring A(h) itself is a submodule of R(h) over A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We consider three other submodules of R(h) over A(h): B(h) :={ n � k=1 bkτk | bk ∈ A(h), n � k=1 bk = 0}, C(h) :={ n � k=1 ckρk | ck ∈ A(h), n � k=1 ck = 0}, D(h) :={ � 1≤i,j≤n dijτiρj | dij ∈ A(h), n � j=1 dij = 0 for i ∈ [n], n � i=1 dij = 0 for j ∈ [n]} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='17) where dkk = 0 for k ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that A(h)⊗Q agrees with the ring of invariants H∗(X(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Q)Sn as mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' R(h) is additively generated by A(h), B(h), C(h), and D(h) when tensoring with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since H∗(X(h)) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h), it suffices to show that any element of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='18) n � k=1 bkτk + n � k=1 ckρk + � 1≤i,j≤n dijτiρj (bk, ck, dij ∈ A(h), dkk = 0) can be expressed as a sum of elements in A(h), B(h), C(h), and D(h) when tensoring with Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Set b := �n k=1 bk and c := �n k=1 ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since �n k=1 τk = x1 −x2 and �n k=1 ρk = xn−1 −xn by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='15), we have n � k=1 bkτk + n � k=1 ckρk = n � k=1 � bk − b n � τk + b n(x1 − x2) + n � k=1 � ck − c n � ρk + c n(xn−1 − xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here the two sums at the right hand side above respectively belong to B(h) ⊗ Q and C(h) ⊗ Q, and the remaining two terms belong to A(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As for the last term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='18), since �n i=1 τi = x1 − x2, we have � 1≤i,j≤n dijτiρj = n � j=1 � n � i=1 � dij − dj n � τi � ρj + n � j=1 dj n (x1 − x2)ρj = � 1≤i,j≤n ˜dijτiρj + n � j=1 dj n (x1 − x2)ρj (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) where dj := n � i=1 dij and ˜dij := dij − dj n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The last sum in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) is a sum of elements in A(h) ⊗ Q and C(h) ⊗ Q by Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We shall show that the sum � 1≤i,j≤n ˜dijτiρj in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='19) is a sum of elements in A(h) ⊗ Q, B(h) ⊗ Q, and D(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='20) n � i=1 ˜dij = n � i=1 � dij − dj n � = n � i=1 dij − dj = 0 12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO and set (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='21) ˜di := n � j=1 ˜dij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since �n j=1 ρj = xn−1 − xn, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) � 1≤i,j≤n ˜dijτiρj = n � i=1 \uf8eb \uf8ed n � j=1 � ˜dij − ˜di n � ρj \uf8f6 \uf8f8 τi + n � i=1 ˜di n (xn−1 − xn)τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here the second sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) is a sum of elements in A(h) ⊗ Q and B(h) ⊗ Q by Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As for the coefficients ˜dij − ˜di n of τiρj in the first sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22), it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='21) that we have n � i=1 � ˜dij − ˜di n � = n � i=1 ˜dij − 1 n n � i=1 ˜di = − 1 n n � i=1 n � j=1 ˜dij = − n � j=1 � n � i=1 ˜dij � = 0, n � j=1 � ˜dij − ˜di n � = n � j=1 ˜dij − ˜di = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus, the first sum at the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='22) belongs to D(h) ⊗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ We shall calculate upper bounds of the Hilbert series of A(h), B(h), C(h), and D(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of A(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and h = (2, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 1, n, n) in our case, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='23) Hilb(A(h), √q) = n−1 � j=1 [h(j) − j]q = (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Hilbert series of B(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that (x1 − tk)τk vanishes at every w ∈ Sn, so we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='24) (x1 − tk)τk = 0 in H∗ T (X(h)) and hence x1τk = 0 in H∗(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, B(h) is indeed a module over A(h)/(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Here A(h)/(x1) ⊗ Q = A(h1) ⊗ Q by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since h1 = (n − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , n − 2, n − 1, n − 1), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) that Hilb(A(h)/(x1), √q) = n−2 � j=1 [h1(j) − j]q = (1 + q)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Since B(h) is a module over A(h)/(x1) generated by τi − τi+1 (i ∈ [n − 1]) and the cohomological degrees of τk’s are two, we obtain an upper bound of Hilb(B(h), q) as follows: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25) Hilb(B(h), √q) ≤ (n − 1)q Hilb(A(h)/(x1), √q) = (n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Here �∞ i=0 aiqi ≤ �∞ i=0 biqi (ai, bi ∈ Z) means that ai ≤ bi for all i’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hilbert series of C(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To f ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) we associate f ∨ ∈ Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) defined by f ∨(w) := f(ww0) for w ∈ Sn, where w0 denotes the longest element in Sn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' w0 = n n − 1 · · · 2 1 in one-line notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This defines an involution on Map(Sn, Z[t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , tn]) and one can easily check that x∨ k = xn−k+1, τ ∨ k = −ρk, ρ∨ k = −τk REGULAR SEMISIMPLE HESSENBERG VARIETIES 13 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence the involution gives an isomorphism between B(h) and C(h), and the same inequality as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25) holds for C(h), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='26) Hilb(C(h), √q) ≤ (n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Hilbert series of D(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We have x1τk = 0 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly we have xnρk = 0 since (x1τk)∨ = −xnρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' (The fact xnρk = 0 also follows from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='13) of xk and ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=') Therefore, D(h) is indeed a module over A(h)/(x1, xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' As mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1, A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and it is the image of the restriction map ι∗ : H∗(Fl(n)) → H∗(X(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, A(h)/(x1, xn) is the image of the restriction map from H∗(Fl(n−2)) and hence Hilb(A(h)/(x1, xn), √q) ≤ [n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. (In fact, the equality holds above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=') There are 2n relations among dij (i ̸= j) in the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='17) of D(h), but one relation can be obtained from the other 2n−1 relations because �n i=1 ��n j=1 dij � = �n j=1 (�n i=1 dij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, there are n(n − 1) number of dij’s and the cohomological degree of τiρj is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='27) Hilb(D(h), √q) ≤ Hilb(A(h)/(x1, xn), √q) {n(n − 1) − (2n − 1)} q2 ≤ (n2 − 3n + 1)q2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='9, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='23), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='25), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='26), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='27) that Hilb(R(h), √q) ≤ (1 + q)2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + 2(n − 1)(q + q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' + (n2 − 3n + 1)q2[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' = (1 + 2nq + n(n − 1)q2)[n − 2]q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='. The coefficient of qn−3 in the last term above is less than that of Pn(q) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='8 by n(n − 3)/2, proving the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Sufficiency The purpose of this section is to prove the following proposition, which implies the sufficiency of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the equivariant cohomology H∗ T (X(h)) is generated in degree 2 as an algebra over H∗(BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(3), X(h) is not connected when h(k) = k for some 1 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, a flag V• = (V0 ⊂ V1 ⊂ · · · ⊂ Vn) ∈ X(h) is of the form Vk = ⟨ei1, ei2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , eik⟩ for some {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ik} ⊂ [n], where ei is the i-th standard basis vector of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, decomposing V• into two flags (V0 ⊂ V1 ⊂ · · · ⊂ Vk) and (V ′ 0 ⊂ V ′ 1 ⊂ · · · ⊂ V ′ n−k), where V ′ i = Vk+i/Vk, one can see that X(h) is the disjoint union of �n k � copies of X(h1) × X(h2), where h1 and h2 are the Hessenberg function obtained by restricting h onto intervals [k] and [k + 1, n], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Each copy corresponds to the choice of a k-subset {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ik} ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' To be precise, h2 : [n − k] → [n − k] is given by shift−1 k h ◦ shiftk, where shiftk : [n − k] → [k + 1, n] shifts integers by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Suppose h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) and 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then X(hr) is not connected ⇐⇒ a + 1 ≤ r ≤ b by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1(3) and that hr is also of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) when r < a + 1 or r > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When a + 1 ≤ r ≤ b, each connected component of X(hr) is isomorphic to X(h1) × X(h2) and both h1 and h2 are of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let Γ(Sn, h) denote the labeled graph of X(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that H∗ T (X(h)) ∼= H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For the subset Sr n ⊂ Sn in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), let Γ(Sr n, h) be the induced labeled subgraph of Γ(Sn, h) on the subset Sr n of vertices, and let Γ0(Sr n, h) denote a connected component of Γ(Sr n, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the restriction map H2(Γ(Sn, h)) → H2(Γ0(Sr n, h)) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We admit the lemma and complete the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Before that, we shall observe that Γ0(Sr n, h) is essentially a connected component of a labeled graph of X(hr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, for 1 ≤ r ≤ n, let cr be the cyclic permutation (r r + 1 r + 2 · · · n) and ϕr : Γ0(Sr n, h) → Γ0(Sn−1, hr) 14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO a graph isomorphism defined by ϕr(w) = wcr for w ∈ Sr n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When i, j ̸= r, the (i, j)-th box in the configuration for h corresponds to the (c−1 r (i), c−1 r (j))-th box in the configuration for hr (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In particular, v = w(i, j) corresponds to vcr = wcr(c−1 r (i), c−1 r (j)) and the edges between these vertices have the same label tw(i) − tw(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, ϕr induces an isomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1) ϕ∗ r : H∗(Γ0(Sn−1, hr)) ∼ = −→ H∗(Γ0(Sr n, h)) of graded algebras over H∗(BT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Recall that H∗ T (X(h)) ∼= H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We prove the proposition by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Let 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For any z ∈ H∗(Γ(Sn, h)) that vanishes on �r−1 j=1 Sj n, it is sufficient to show the existence of a polynomial f in elements of H2(Γ(Sn, h)) such that z − f vanishes on �r j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then the induction on r proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We shall show the existence of f by division into cases according to the value of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case 1 ≤ r ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, Γ(Sr n, h) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We note that z vanishes on �r−1 j=1 Sj n and this implies that z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) z(w) = \uf8eb \uf8ed r−1 � j=1 (tw(j) − tn) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Indeed, for w ∈ Sr n, we have w(r) = n and w(j, r) ∈ Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' If j ≤ r − 1, then there is an edge in the graph Γ(Sn, h) between the vertices w and w(j, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The label on the edge is tw(j) −tw(r) = tw(j) −tn and z vanishes at w(j, r) ∈ Sj n (j ≤ r − 1) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore z(w) is divisible by the product in the big parenthesis in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) and g ∈ Map(Sr n, H∗(BT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Furthermore, one can easily check that the g is indeed in H∗(Γ(Sr n, h)) since z is in H∗(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since H∗(Γ(Sr n, h)) ∼= H∗(Γ(Sn−1, hr)) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), g is a polynomial in elements of H2(Γ(Sr n, h)) by induc- tion on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2, there is a polynomial ˜g in H2(Γ(Sn, h)) which coincides with g on Sr n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, �r−1 j=1(xj −tn) coincides with the product in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2) on Sr n since xj(w) = tw(j) by definition of xj, and vanishes on �r−1 j=1 Sj n since xj(w) = tw(j) = tn for w ∈ Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, � �r−1 j=1(xj − tn) � ˜g coincides with the element z on �r j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Thus � �r−1 j=1(xj − tn) � ˜g is a desired polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case r = a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 1, z(w) for w ∈ Sa+1 n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) z(w) = \uf8eb \uf8ed a � j=1 (tw(j) − tn) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sa+1 n , h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Note that Γ(Sa+1 n , h) is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Two vertices v, w ∈ Sa+1 n lie in the same connected component if and only if {v(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , v(a)} = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)} ⊂ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For K := {k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , ka} ⊂ [n − 1], we consider the element ρK defined by ρK = a � j=1 ya,kj, where ya,k(w) = � tk − tw(a+1) (k ∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) 0 (k /∈ {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, since w(a + 1) = n for w ∈ Sa+1 n , we have ρK(w) = ��a j=1(tw(j) − tn) (K = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}) 0 (K ̸= {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(a)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence ρK coincides with the product in the big parentheses of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) on the connected component (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) {w ∈ Sa+1 n | w([a]) = K} REGULAR SEMISIMPLE HESSENBERG VARIETIES 15 and vanishes on the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since n /∈ K and w(j) = n for w ∈ Sj n, ρK also vanishes on �a j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' On the other hand, the element g in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='3) restricted to the connected component (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4) is obtained as the restriction of a polynomial ˜gK in H2(Γ(Sn, h)) similarly to Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, we obtain a desired polynomial f as � K⊂[n−1], |K|=a ρK˜gK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case a + 2 ≤ r ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) z(w) = (tw(r−1) − tw(r))g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 2, Γ(Sr n, h) is not connected and two vertices v, w ∈ Sr n lie in the same connected component if and only if {v(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , v(r − 1)} = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)} ⊂ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' For A ⊂ [n − 1] with |A| = r − 1, we have τA(w) = � tw(r−1) − tw(r) (A = {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)}) 0 (A ̸= {w(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(r − 1)}) by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence, τA coincides with the factor of the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='5) on the connected component {w ∈ Sr n | w([r − 1]) = A}, and vanishes on the other connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Since n /∈ A and w(j) = n for w ∈ Sj n, τA also vanishes on �r−1 j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, similarly to Case 2, we obtain a desired polynomial f as � A⊂[n−1], |A|=r−1 τA˜gA, where ˜gA is a polynomial in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' The case b + 1 ≤ r ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' In this case, z(w) for w ∈ Sr n decomposes as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) z(w) = \uf8eb \uf8ed r−1 � j=b (tn − tw(j)) \uf8f6 \uf8f8 g(w), g ∈ H∗(Γ(Sr n, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Similarly to Case 1, X(hr) is connected and g is the restriction of a polynomial ˜g in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We consider the element y∗ b+1,n ∈ H∗(Γ(Sn, h)) in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2, which is defined as y∗ b+1,n(w) = � tn − tw(b) (n ∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (n /∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Then \uf8eb \uf8edy∗ b+1,n r−1 � j=b+1 (tn − xj) \uf8f6 \uf8f8 (w) = ��r−1 j=b(tn − tw(j)) (n ∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}) 0 (n /∈ {w(b + 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' , w(n)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Hence y∗ b+1,n �r−1 j=b+1(tn − xj) coincides with the product in the big parentheses of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='6) on Sr n, and vanishes on �r−1 j=1 Sj n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Therefore, � y∗ b+1,n �r−1 j=b+1(tn − xj) � ˜g is a desired polynomial f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Finally we give a proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='4 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='2 that when h is of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), the elements in {xi, ya,k, τA, ti | i, k ∈ [n], A ⊂ [n], a + 1 ≤ |A| < b} span H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Through the isomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1), one can find generators of H2(Γ0(Sr n, h)) which corre- spond to the generators of H2(Γ0(Sn−1, hr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' They are given as restrictions of xi for i ∈ [n], i ̸= r, ti for i ∈ [n], and the following elements in H2(Γ(Sn, h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When 1 ≤ r ≤ a, ya,k for k ∈ [n − 1], τA⊔{n} for A ⊂ [n − 1], a ≤ |A| < b − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' MASUDA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' SATO Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When r = a + 1, for a connected component Γ0(Sa+1 n , h) which contains σ ∈ Sa+1 n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' τB⊔σ([a+1]) for B ⊂ σ([n]\\[a + 1]), 1 ≤ |B| < b − (a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When a + 1 < r ≤ b, for a connected component Γ0(Sr n, h) which contains σ ∈ Sr n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ya,k for k ∈ [n − 1], τA for A ⊂ σ([r − 1]), a + 1 ≤ |A| < r − 1, τB⊔σ([r]) for B ⊂ σ([n] \\ [r]), 1 ≤ |B| < b − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Case 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' When b < r ≤ n, ya,k for k ∈ [n − 1], τA for A ⊂ [n − 1], a + 1 ≤ |A| < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' ✷ Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' We thank Yunhyung Cho for his help on moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Masuda was supported in part by JSPS Grant- in-Aid for Scientific Research 22K03292 and a HSE University Basic Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' This work was partly supported by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Abe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Harada, T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=', n), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content='1 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' 27–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE2T4oBgHgl3EQfPwZL/content/2301.03762v1.pdf'} +page_content=' Abe, T.' metadata={'source': 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