On rational representations and rational group algebra of ${\mathrm{GL}}_2(q)$

In this article, we study rational representations of $G={\mathrm{GL}}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character $$\mathrm{\Omega }(\chi )=m_{\mathbb{Q}}(\chi )\sum _{\sigma \in \mathrm{Gal}(\mathbb{Q}(\chi )\mathrm{/}\mathbb{Q})}{\chi }^{\sigma },$$ for some irreducible complex character $\chi$ of $G$, where $m_{\mathbb{Q}}(\chi )$ denotes the Schur index of $\chi$ over $\mathbb{Q}$, with the converse also holding. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of distinct degrees. Furthermore, we present a method to construct an irreducible rational matrix representation $\rho$ of $G$ affording the character $\mathrm{\Omega }(\chi )$, where $\chi$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results from the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$.