Stability of the centers of group algebras of $GL_n(q)$

TL;DR

This paper investigates the stability properties of the centers of group algebras of $GL_n(q)$, revealing a universal stable center with consistent structure constants across all dimensions.

Contribution

It introduces a stable center for the group algebra of $GL_n(q)$ with structure constants independent of $n$, extending known stability results to this setting.

Findings

Structure constants of the associated graded algebras are independent of n.

A universal stable center with positive integer structure constants is established.

Several structure constants of the stable center are explicitly computed.

Abstract

The center $\mathscr{Z}_n(q)$ of the integral group algebra of the general linear group $GL_n(q)$ over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras $\mathscr{G}_n(q)$ are independent of $n$, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras $\mathscr{G}_n(q)$ for all $n$. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author.