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

TL;DR

This paper introduces a new Hopf algebra structure on class functions of unipotent upper triangular groups over finite fields, linking it to known algebraic frameworks and revealing new algebraic properties.

Contribution

It constructs a novel Hopf algebra on class functions of unipotent groups, connecting induction to Zelevinsky's algebra and embedding a known combinatorial Hopf algebra.

Findings

Induction induces a homomorphism to Zelevinsky's Hopf algebra.

Contains a Hopf subalgebra isomorphic to a known combinatorial Hopf algebra.

Establishes additional algebraic properties of the constructed Hopf algebra.

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.