The second cohomology of regular semisimple Hessenberg varieties from GKM theory

TL;DR

This paper explicitly describes the second cohomology of regular semisimple Hessenberg varieties using GKM theory, revealing its structure as a symmetric group module and providing a new formula for its isomorphism class.

Contribution

It offers an explicit GKM-theoretic presentation of the second cohomology and a novel formula for its symmetric group module structure, extending understanding of Hessenberg varieties.

Findings

Explicit generators and relations for second cohomology

A new formula for the symmetric group module structure

Discussion of potential higher degree generalizations

Abstract

We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak{S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak{S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho-Hong-Lee but they are equivalent. We also discuss its higher degree generalization.