A $\mathrm{GL}({\mathbb{F}}_q)$-compatible Hopf algebra of unitriangular class functions

Abstract

This paper constructs a novel Hopf algebra $\mathsf{cf}({\mathrm{UT}}_{\bullet })$ on the class functions of the unipotent upper triangular groups ${\mathrm{UT}}_n({\mathbb{F}}_q)$ over a finite field. This construction is representation theoretic in nature and uses the machinery of Hopf monoids in the category of vector species. In contrast with a similar known construction, this Hopf algebra has the property that induction to the finite general linear group induces a homomorphism to Zelevinsky's Hopf algebra of ${\mathrm{GL}}_n({\mathbb{F}}_q)$ class functions. Furthermore, $\mathsf{cf}({\mathrm{UT}}_{\bullet })$ contains a Hopf subalgebra which is isomorphic to a known combiantorial Hopf algebra, previously used to prove a conjecture about chromatic quasisymmetric functions. Some additional Hopf algebraic properties are also established.