The ring of stable characters over $\text{GL}_\bullet(q)$

TL;DR

This paper investigates the algebra of stable class functions on the family of general linear groups over a fixed finite field, revealing four natural bases and clarifying their properties, including a proven conjecture.

Contribution

It introduces and compares four natural bases for the algebra of stable class functions on $ ext{GL}_ullet(q)$, connecting representation theory, combinatorics, and previous conjectures.

Findings

Four different linear bases for the algebra of stable class functions.

Identification of stable irreducible characters with simple $ ext{VI}$-modules.

Proof of a conjecture from prior work.

Abstract

For a fixed prime power $q$, let $\text{GL}_\bullet(q)$ denote the family of groups $\text{GL}_N(q)$ for $N \in \mathbb{Z}_{\geq 0}$. In this paper we study the $\mathbb{C}$-algebra of "stable" class functions of $\text{GL}_\bullet(q)$, and show it admits four different linear bases, each arising naturally in different settings. One such basis is that of stable irreducible characters, namely, the class functions spanned by the characters corresponding to finitely generated simple $\mathrm{VI}$-modules in the sense of [arXiv:1408.3694,arXiv:1602.00654]. A second one comes from characters of parabolic representations. The final two, one originally defined in [arXiv:1803.04155] and the other in [arXiv:2110.11099], are more combinatorial in nature. As corollaries, we clarify many properties of these four bases and prove a conjecture from [arXiv:2106.11587].