The Betti numbers of regular Hessenberg varieties are palindromic

TL;DR

This paper proves that the Betti numbers of regular Hessenberg varieties are palindromic for all reductive algebraic groups, extending previous results known for $GL_n(C)$ and confirming a key geometric property.

Contribution

The authors extend the palindromic Betti number property from $GL_n(C)$ to all reductive algebraic groups, broadening the understanding of Hessenberg varieties.

Findings

Betti numbers of regular Hessenberg varieties are palindromic for all reductive groups

Extension of Brosnan and Chow's result beyond $GL_n(C)$

Supports conjectures about symmetry in Hessenberg variety cohomology

Abstract

Recently Brosnan and Chow have proven a conjecture of Shareshian and Wachs describing a representation of the symmetric group on the cohomology of regular semisimple Hessenberg varieties for $GL_n(\mathbb{C})$. A key component of their argument is that the Betti numbers of regular Hessenberg varieties for $GL_n(\mathbb{C})$ are palindromic. In this paper, we extend this result to all reductive algebraic groups, proving that the Betti numbers of regular Hessenberg varieties are palindromic.