The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A

摘要

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 aretwofold. First, we give a systematic method, depending in a simple manner onthe Hessenberg function $h$, for producing an explicit presentation bygenerators and relations of the cohomology ring $H^\ast(Hess(\mathsf{N},h))$with $\mathbb{Q}$ coefficients of the corresponding regular nilpotentHessenberg variety $Hess(\mathsf{N},h)$. Our result generalizes known resultsin special cases such as the Peterson variety and also allows us to answer aquestion posed by Mbirika and Tymoczko. Moreover, our list of generators infact forms a regular sequence, allowing us to use techniques from commutativealgebra in our arguments. Our second main result gives an isomorphism betweenthe cohomology ring $H^*(Hess(\mathsf{N},h))$ of the regular nilpotentHessenberg variety and the $S_n$-invariant subring$H^*(Hess(\mathsf{S},h))^{S_n}$ of the cohomology ring of the regularsemisimple Hessenberg variety (with respect to the $S_n$-action on$H^*(Hess(\mathsf{S},h))$ defined by Tymoczko). Our second main result impliesthat $\mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{N},h)) =\mathrm{dim}_{\mathbb{Q}} H^k(Hess(\mathsf{S},h))^{S_n}$ for all $k$ and hencepartially proves the Shareshian-Wachs conjecture in combinatorics, which is inturn related to the well-known Stanley-Stembridge conjecture. A proof of thefull Shareshian-Wachs conjecture was recently given by Brosnan and Chow, but inour special case, our methods yield a stronger result (i.e. an isomorphism ofrings) by more elementary considerations. This paper provides detailed proofsof results we recorded previously in a research announcement.