[Paper Review] The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\ldots,n)$

Hiraku Abe, Tatsuya Horiguchi+1 more|arXiv (Cornell University)|Apr 4, 2017

Advanced Combinatorial Mathematics26 references12 citations

TL;DR

This paper provides an explicit presentation of the cohomology rings of regular semisimple Hessenberg varieties in type $A_{n-1}$ with Hessenberg function $h = (h(1), n, \ldots, n)$, using generators and relations. The main result gives a ring structure compatible with the $σ_n$-representation on cohomology and naturally specializes to Borel's presentation of the flag variety cohomology, while also yielding an explicit presentation of the $σ_n$-invariant subring.

ABSTRACT

We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $\mathfrak{S}_n$-invariant subring of the cohomology ring.

Motivation & Objective

• To determine the cohomology ring structure of regular semisimple Hessenberg varieties in type $A_{n-1}$ for Hessenberg functions of the form $h = (h(1), n, \ldots, n)$.
• To provide an explicit presentation of the cohomology ring in terms of generators and relations, extending Borel's presentation of the flag variety.
• To establish compatibility of the ring structure with the $σ_n$-representation on cohomology constructed by Tymoczko.
• To derive an explicit presentation of the $σ_n$-invariant subring of the cohomology ring for such Hessenberg varieties.

Proposed method

• Utilize the GKM presentation of equivariant cohomology to analyze the ordinary cohomology ring of the Hessenberg variety.
• Construct a ring homomorphism from a polynomial ring $π[X_1, \ldots, X_n]$ to the cohomology ring $H^*(X(h))$, mapping $X_k$ to the class $\check{x}_k$.
• Identify the kernel of this homomorphism using relations derived from the GKM relations and the structure of the Hessenberg variety.
• Prove that the induced map is an isomorphism by comparing Hilbert series and using torsion-freeness of the cohomology ring.
• Leverage the $σ_n$-action on cohomology to describe the invariant subring, using symmetric polynomials and the relation $\check{x}_1 \prod_{\ell=2}^{h(1)} (\check{x}_1 - \check{x}_\ell) = 0$.
• Compare the resulting presentation with known results on the cohomology of regular nilpotent Hessenberg varieties in $ϵ$-coefficients, showing compatibility with existing ideals.

Experimental results

Research questions

• RQ1 What is the explicit presentation of the cohomology ring of a regular semisimple Hessenberg variety with Hessenberg function $h = (h(1), n, \ldots, n)$ in type $A_{n-1}$?
• RQ2 How does the cohomology ring structure relate to the $σ_n$-representation on cohomology, and is it compatible with Tymoczko's construction?
• RQ3 Can the $σ_n$-invariant subring of the cohomology ring be explicitly described in terms of symmetric polynomials and additional relations?
• RQ4 Does the ring presentation naturally specialize to Borel's presentation of the flag variety cohomology?
• RQ5 Is the presentation of the $σ_n$-invariant subring compatible with the known presentation of the cohomology ring of regular nilpotent Hessenberg varieties?

Key findings

• The cohomology ring $H^*(X(h))$ for $h = (h(1), n, \ldots, n)$ is isomorphic to $\mathbb{Z}[X_1, \ldots, X_n] / (e_i(X_1, \ldots, X_n), X_1 \prod_{\ell=2}^{h(1)} (X_1 - X_\ell) \mid 1 \leq i \leq n)$, where $e_i$ is the $i$-th elementary symmetric polynomial.
• The ring presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety when $h(1) = n$, recovering the standard presentation via elementary symmetric polynomials.
• The $σ_n$-invariant subring $H^*(X(h))^{σ_n}$ is isomorphic to $\mathbb{Z}[X_1, \ldots, X_n] / (e_i(X_1, \ldots, X_n), X_1 \prod_{\ell=2}^{h(1)} (X_1 - X_\ell) \mid 1 \leq i \leq n)$, confirming compatibility with the $σ_n$-action.
• The Hilbert series of $H^*(X(h))^{σ_n}$ is $\frac{1 - q^{h(1)}}{1 - q} \prod_{j=1}^{n-1} \frac{1 - q^j}{1 - q}$, matching the expected Poincaré series of the invariant subring.
• The presentation of the $σ_n$-invariant subring is compatible with the known presentation of the cohomology ring of regular nilpotent Hessenberg varieties in $\mathbb{Q}$-coefficients, as both ideals are equal in $\mathbb{Q}[X_1, \ldots, X_n]$.
• The map from $\mathbb{Z}[X_1, \ldots, X_n]$ to $H^*(X(h))^{σ_n}$ is an isomorphism, proven via Hilbert series comparison and torsion-freeness of the cohomology ring.