Stability of the centers of group algebras of general affine groups $G{A}_n(q)$

Abstract

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

32 pages