Stability of the centers of group algebras of general affine groups $GA_n(q)$

TL;DR

This paper investigates the structure and stability of the centers of group algebras of general affine groups over finite fields, establishing a universal stable algebra with consistent structure constants across different dimensions.

Contribution

It introduces a new notion of matrix type in $GA_n(q)$, describes conjugacy class representatives, and proves the stability of the algebraic structure constants in the centers of their group algebras.

Findings

The center $ ext{Z}[ ext{GA}_n(q)]$ is a filtered algebra with stable structure constants.

The associated graded algebra's structure constants are independent of $n$.

A universal stable center algebra $ ext{G}(q)$ with positive integer structure constants is established.

Abstract

The general affine group $GA_n(q)$ consisting of invertible affine transformations of an affine space of codimension one in the vector space $\mathbb{F}_q^n$ over a finite field $\mathbb{F}_q$, can be viewed as a subgroup of the general linear group $GL_{n}(q)$ over $\mathbb{F}_q$. In the article, we introduce the notion of the type of each matrix in $GA_n(q)$ and give an explicit representative for each conjugacy class. Then the center $\mathscr{A}_n(q)$ of the integral group algebra $\mathbb{Z}[GA_n(q)]$ is proved to be a filtered algebra via the length function defined via the reflections lying in $GA_n(q)$. We show in the associated graded algebras $\mathscr{G}_n(q)$ the structure constants with respect to the basis consisting of the conjugacy class sums are independent of $n$. The structure constants in $\mathscr{G}_n(q)$ is further shown to contain the structure constants in the graded algebras introduced by the first author and Wang for $GL_n(q)$ as special cases. The stability leads to a universal stable center $\mathscr{G}(q)$ with positive integer structure constants only depending on $q$ which governs the algebras $\mathscr{G}_n(q)$ for all $n$.