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 $\mathrm{GL}{n}(\mathbb{F}{q})$.