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

Abstract

Let $n$ be a fixed positive integer and $h:\{1,2,\dots ,n\}\to \{1,2,\dots ,n\}$ 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 $H^{\ast }(Hess(\mathsf{N},h))$ of the regular nilpotent Hessenberg variety and the $S_n$-invariant subring $H^{\ast }(Hess(\mathsf{S},h){)}^{S_n}$ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the $S_n$-action on $H^{\ast }(Hess(\mathsf{S},h))$ defined by Tymoczko). Our second main result implies that ${\dim }_{\mathbb{Q}}{H}^k(Hess(\mathsf{N},h))={\dim }_{\mathbb{Q}}{H}^k(Hess(\mathsf{S},h){)}^{S_n}$ 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.

42 pages. Major expositional revision. We streamlined and tightened exposition to reduce from previous 52 pages to 42 pages

References in corpus (3)

• A Schubert calculus recurrence from the noncomplex W-action on G/B
• Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
• The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement

Cited by in corpus (5)

• A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra
• Hessenberg varieties and hyperplane arrangements
• The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\dots ,n)$
• Torus actions, localization and induced representations on cohomology
• Weyl group symmetries of the toric variety associated with Weyl chambers