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

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 ${\mathrm{SL}}_2(q)$ and ${\mathrm{PSL}}_2(q)$. Furthermore, we also determine the number of pairwise non-isomorphic simple $\mathbb{Q}G$-modules of each possible dimension for $G$ being either ${\mathrm{SL}}_2(q)$ or ${\mathrm{PSL}}_2(q)$.