A unipotent realization of the chromatic quasisymmetric function

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 ${\mathrm{GL}}_n({\mathbb{F}}_q)$.

References in corpus (9)

• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• Hopf monoids from class functions on unitriangular matrices
• LLT polynomials, chromatic quasisymmetric functions and graphs with cycles
• A Supercharacter Analogue for Normality
• A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions
• Chromatic symmetric functions of Dyck paths and q-rook theory
• The combinatorics of ${\mathrm{GL}}_n$ generalized Gelfand--Graev characters
• Hessenberg varieties associated to ad-nilpotent ideals