Bases of the equivariant cohomologies of regular semisimple Hessenberg varieties

TL;DR

This paper constructs explicit bases for the cohomology of regular semisimple Hessenberg varieties using Bialynicki-Birula decomposition, enabling detailed analysis of symmetric group actions and resolving longstanding conjectures.

Contribution

It provides a combinatorial description of basis classes, computes symmetric group actions, and explicitly constructs permutation modules for permutohedral varieties, solving open problems.

Findings

Explicit bases for Hessenberg cohomology constructed

Symmetric group actions on bases explicitly computed

Permutation module decompositions explicitly realized

Abstract

We consider bases for the cohomology space of regular semisimple Hessenberg varieties, consisting of the classes that naturally arise from the Bialynicki-Birula decomposition of the Hessenberg varieties. We give an explicit combinatorial description of the support of each class, which enables us to compute the symmetric group actions on the classes in our bases. We then successfully apply the results to the permutohedral varieties to explicitly write down each class and to construct permutation submodules that constitute summands of a decomposition of cohomology space of each degree. This resolves the problem posed by Stembridge on the geometric construction of permutation module decomposition and also the conjecture posed by Chow on the construction of bases for the equivariant cohomology spaces of permutohedral varieties.