English

A unipotent realization of the chromatic quasisymmetric function

Combinatorics 2024-09-25 v3 Representation Theory

Abstract

This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group GLn(Fq)\mathrm{GL}_{n}(\mathbb{F}_{q}): chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated GLn(Fq)\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 UTn(Fq)\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 GLn(Fq)\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 GLn(Fq)\mathrm{GL}_{n}(\mathbb{F}_{q}).

Keywords

Cite

@article{arxiv.2211.06981,
  title  = {A unipotent realization of the chromatic quasisymmetric function},
  author = {Lucas Gagnon},
  journal= {arXiv preprint arXiv:2211.06981},
  year   = {2024}
}