Divided difference operators for Hessenberg representations

TL;DR

This paper uses divided difference operators to decompose the Tymoczko dot action on the equivariant cohomology of Hessenberg varieties, linking it to chromatic quasisymmetric functions and categorifying modular relations.

Contribution

It introduces a novel decomposition method for the Tymoczko dot action using divided difference operators, connecting geometric representation theory with combinatorial symmetric functions.

Findings

Decomposition of the Tymoczko dot action into sub-representations.

Categorification of the modular relation between chromatic quasisymmetric functions.

New insights into the structure of Hessenberg varieties' equivariant cohomology.

Abstract

The equivariant cohomology ring of a regular semisimple Hessenberg variety in type A is a free module over the equivariant cohomology ring of a point. When equipped with Tymoczko's dot action, it becomes a twisted representation of the symmetric group, and the character of this representation is given by the chromatic quasisymmetric function of an indifference graph. In this note, we use divided difference operators to decompose this representation as a direct sum of sub-representations in a way that categorifies the modular relation between chromatic quasisymmetric functions.