Chromatic quasisymmetric functions and Hessenberg varieties

TL;DR

This paper explores deep connections between enumerative combinatorics, symmetric functions, and algebraic geometry, focusing on q-analogs of Eulerian polynomials, refined chromatic symmetric functions, and Hessenberg varieties.

Contribution

It introduces new links between combinatorial q-analogs, refined symmetric functions, and geometric representations of Hessenberg varieties.

Findings

A new q-analog of generalized Eulerian polynomials.

Refinement of Stanley's chromatic symmetric functions.

Insights into Tymoczko's representation on Hessenberg variety cohomology.

Abstract

We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko's representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.