A unipotent realization of the chromatic quasisymmetric function

Abstract

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