A unipotent realization of the chromatic quasisymmetric function

Gagnon, Lucas

doi:10.2140/ant.2024.18.1737

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A unipotent realization of the chromatic quasisymmetric function

Lucas Gagnon

Vol. 18 (2024), No. 9, 1737–1766

DOI: 10.2140/ant.2024.18.1737

Abstract

We realize two families of combinatorial symmetric functions via the complex character theory of the finite general linear group ${GL}_{n} (𝔽_{q})$ : chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated ${GL}_{n} (𝔽_{q})$ characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups ${UT}_{n} (𝔽_{q})$ . 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 ${GL}_{n} (𝔽_{q})$ characters and Hessenberg varieties and a reinterpretation of known theorems and conjectures about the relevant symmetric functions in terms of ${GL}_{n} (𝔽_{q})$ .

Keywords

chromatic quasisymmetric function, LLT polynomial, unipotent group, combinatorial Hopf algebra, supercharacter

Mathematical Subject Classification

Primary: 05E05, 05E10, 16T30

Secondary: 05E14, 20C33

Milestones

Received: 21 February 2023

Revised: 26 July 2023

Accepted: 18 September 2023

Published: 19 September 2024

Authors

Lucas Gagnon
Department of Mathematics University of Colorado Boulder Boulder, CO United States

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Vol. 18, No. 9, 2024