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

Abstract

This paper constructs a novel Hopf algebra cfpUT ‚ q on the class functions of the unipotent upper triangular groups UT n pF q 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 GL n pF q q class functions. Furthermore, cfpUT ‚ q 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.