The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
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DOI：
10.1093/imrn/rnx275
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发表时间：
2015-12
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影响因子：
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通讯作者：
Hiraku Abe;M. Harada;T. Horiguchi;M. Masuda
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文献类型：
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作者：
Hiraku Abe;M. Harada;T. Horiguchi;M. Masuda
设$n$是一个固定的正整数，$h: \{1,2,\ldots,n\} \right row \{1,2,\ldots,n\}$是一个海森伯格函数。本文的主要结果有两个方面。首先，我们给出了一种系统的方法，以一种简单的方式依赖于Hessenberg函数$h$，用对应的正则幂零Hessenberg变量$Hess(\mathsf{N},h)$的系数$\mathbb{Q}$的上同环$h ^\ast(Hess(\mathsf{N},h) $的生成器和关系来显式表示。我们的结果概括了在特殊情况下的已知结果，如彼得森变化，也允许我们回答Mbirika和Tymoczko提出的一个问题。此外，我们的生成器列表实际上形成了一个正则序列，允许我们在参数中使用交换代数中的技术。我们的第二个主要结果给出了正则幂零Hessenberg变量的上同环$H^*(Hess(\mathsf{S}， H))$与正则半单Hessenberg变量的上同环$S_n$-不变子$H^*(Hess(\mathsf{S}， H))$ {S_n}$之间的同构性（关于$S_n$-作用于$H^*(Hess(\mathsf{S}， H))$）。我们的第二个主要结果表明$\ mathm {dim}_{\mathbb{Q}} H^k(Hess(\mathsf{N}， H)) = \ mathm {dim}_{\mathbb{Q}} H^k(Hess(\mathsf{S}， H))^{S_n}$对于所有$k$，从而部分证明了组合学中的Shareshian-Wachs猜想，该猜想又与著名的Stanley-Stembridge猜想有关。最近，Brosnan和Chow给出了Shareshian-Wachs猜想的一个证明，但在我们的特殊情况下，我们的方法通过更初等的考虑得到了一个更强的结果（即环的同构）。本文提供了我们之前在研究公告中记录的结果的详细证明。
Let $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$ a Hessenberg function. The main results of this paper are twofold. First, we give a systematic method, depending in a simple manner on the Hessenberg function $h$, for producing an explicit presentation by generators and relations of the cohomology ring $H^\ast(Hess(\mathsf{N},h))$ with $\mathbb{Q}$ coefficients of the corresponding regular nilpotent Hessenberg variety $Hess(\mathsf{N},h)$. Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring $H^*(Hess(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $S_n$-invariant subring $H^*(Hess(\mathsf{S},h))^{S_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $S_n$-action on $H^*(Hess(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that $\mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{N},h)) = \mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{S},h))^{S_n}$ for all $k$ and hence partially proves the Shareshian-Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley-Stembridge conjecture. A proof of the full Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.