The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture

TL;DR

This paper introduces abelian Hessenberg varieties and provides an inductive formula for their cohomology representations, demonstrating that a graded version of the Stanley-Stembridge conjecture holds in this abelian case.

Contribution

It defines abelian Hessenberg varieties and derives an inductive formula for their cohomology, extending previous results and confirming a graded Stanley-Stembridge conjecture case.

Findings

An inductive formula for the $S_n$-representation on cohomology.

Verification of a graded Stanley-Stembridge conjecture in the abelian case.

New formulas for Poincaré polynomials of abelian Hessenberg varieties.

Abstract

We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the $S_n$-representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko. Our result implies that a graded version of the Stanley-Stembridge conjecture holds in the abelian case, and generalizes results obtained by Shareshian-Wachs and Teff. Our proof uses previous work of Stanley, Gasharov, Shareshian-Wachs, and Brosnan-Chow, as well as results of the second author on the geometry and combinatorics of Hessenberg varieties. As part of our arguments, we obtain inductive formulas for the Poincar\'e polynomials of regular abelian Hessenberg varieties.