Wedderburn decomposition of the rational group algebras of $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$

TL;DR

This paper provides explicit formulas for the Wedderburn decomposition of rational group algebras of SL_2(q) and PSL_2(q), including counts of simple modules of each dimension.

Contribution

It derives combinatorial formulas for the Wedderburn decomposition of these group algebras, depending only on q, and classifies simple modules by dimension.

Findings

Explicit formulas for Wedderburn decomposition depending only on q.

Number of non-isomorphic simple modules of each dimension determined.

Formulas applicable to both SL_2(q) and PSL_2(q).

Abstract

In this article, we derive explicit combinatorial formulas, depending only on $q$, for the Wedderburn decomposition of the rational group algebras of the finite linear groups $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$. Furthermore, we also determine the number of pairwise non-isomorphic simple $\mathbb Q G$-modules of each possible dimension for $G$ being either $\operatorname{SL}_2(q)$ or $\operatorname{PSL}_2(q)$.