[Paper Review] The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A

Hiraku Abe, Megumi Harada+2 more|Osaka City University (Osaka City University)|Dec 30, 2015

Advanced Combinatorial Mathematics35 references12 citations

TL;DR

This paper provides a complete, explicit presentation of the rational cohomology ring of regular nilpotent Hessenberg varieties in type A using generators and relations that depend simply on the Hessenberg function h. It proves an isomorphism between the cohomology ring of the regular nilpotent Hessenberg variety and the S_n-invariant subring of the cohomology ring of the corresponding regular semisimple Hessenberg variety, thereby establishing a stronger result than the recent proof of the Shareshian-Wachs conjecture in this special case.

ABSTRACT

Let $n$ be a fixed positive integer and $h: \{1,2,\ldots,n\} ightarrow \{1,2,\ldots,n\}$ 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.

Motivation & Objective

• To provide a systematic, explicit presentation of the rational cohomology ring of regular nilpotent Hessenberg varieties in Lie type A using generators and relations.
• To generalize known results for special cases such as the Peterson variety and resolve a question posed by Mbirika and Tymoczko.
• To establish a ring isomorphism between the cohomology of regular nilpotent Hessenberg varieties and the S_n-invariant subring of the cohomology of regular semisimple Hessenberg varieties.
• To contribute to the combinatorial Stanley-Stembridge conjecture by partially proving the Shareshian-Wachs conjecture via an explicit, elementary method.

Proposed method

• The authors construct an explicit set of generators for the cohomology ring H*(Hess(N,h)) with Q coefficients, where the generators depend directly and simply on the Hessenberg function h.
• They prove that the set of generators forms a regular sequence, enabling the use of commutative algebra techniques in the cohomological analysis.
• A recursive definition of certain polynomials f_{i,j}(w) is used to relate the generators to elementary symmetric polynomials and the variables u_j.
• The proof of the main isomorphism relies on a detailed inductive argument on pairs (i,j) using properties of elementary symmetric polynomials and the recursive structure of the f_{i,j}(w) functions.
• The authors use Tymoczko’s S_n-action on the cohomology of regular semisimple Hessenberg varieties to define the invariant subring and establish the isomorphism.
• The key technical step involves proving a generating function identity (equation A.3) that relates the f_{i,j}(w) to sums over elementary symmetric polynomials and the b_{k,j} generators.

Experimental results

Research questions

• RQ1 Can a uniform, explicit presentation of the cohomology ring of regular nilpotent Hessenberg varieties in type A be given using generators and relations that depend only on the Hessenberg function h?
• RQ2 Does the cohomology ring of a regular nilpotent Hessenberg variety admit a ring isomorphism to the S_n-invariant subring of the cohomology ring of the corresponding regular semisimple Hessenberg variety?
• RQ3 How does the structure of the cohomology ring relate to the Shareshian-Wachs conjecture on chromatic quasisymmetric functions and S_n-representations?
• RQ4 Can the known result that the Betti numbers of the regular nilpotent and semisimple Hessenberg varieties are equal be strengthened to a ring isomorphism in this setting?
• RQ5 Is there a systematic method to compute the cohomology ring of regular nilpotent Hessenberg varieties that generalizes the Peterson variety case?

Key findings

• The cohomology ring H*(Hess(N,h)) admits an explicit presentation by generators and relations that depend simply on the Hessenberg function h.
• The set of generators forms a regular sequence, enabling the application of commutative algebra tools such as the Koszul complex and regular sequence techniques.
• There is a ring isomorphism between H*(Hess(N,h)) and the S_n-invariant subring H*(Hess(S,h))^{S_n}, which implies equality of Betti numbers in all degrees.
• The isomorphism provides a stronger result than the recent proof of the Shareshian-Wachs conjecture, as it establishes a ring isomorphism rather than just equality of Betti numbers.
• The result partially proves the Shareshian-Wachs conjecture and provides a more elementary, geometric proof in the special case of regular nilpotent Hessenberg varieties.
• The paper resolves a question posed by Mbirika and Tymoczko regarding the structure of the cohomology ring of regular nilpotent Hessenberg varieties.