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A unipotent realization of the chromatic quasisymmetric function

Lucas Gagnon

Vol. 18 (2024), No. 9, 1737–1766
Abstract

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

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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