A unipotent realization of the chromatic quasisymmetric function

TL;DR

This paper connects combinatorial symmetric functions, specifically chromatic quasisymmetric functions and LLT polynomials, to the representation theory of finite general linear groups, providing new algebraic and geometric insights.

Contribution

It introduces a unipotent realization of these symmetric functions via complex characters of $ ext{GL}_n( ext{F}_q)$, linking combinatorics, algebra, and geometry.

Findings

Characters are elementary and induced from unipotent upper triangular groups

Provides a Hopf algebraic approach to induction

Connects characters to Hessenberg varieties and reinterprets existing theorems

Abstract

This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant $\mathrm{GL}_{n}(\mathbb{F}_{q})$ characters and Hessenberg varieties and a re-interpretation of known theorems and conjectures about the relevant symmetric functions in terms of $\mathrm{GL}_{n}(\mathbb{F}_{q})$.