Wedderburn decomposition of special linear groups

Question

$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field with char $0$ or $k = \mathbb{Q}$. From Artin Wedderburn's theorem, if $k$ is algebraically closed, then for a finite group $G$ such that $\operatorname{char}(k)$ does not divide $\card G$, $$kG \cong \prod_{i=1}^m M_{n_i}(k). $$

Do we have any information on $n_i$s (like bounds or even explicit characterization in terms of n and p) when the group is $\SL(n, F_p)$? Indeed, I know some immediate consequences of the Wedderburn theorem, such as $m$ is equal to a number of conjugacy classes of $G$, and $\card G = \sum_i n_i^2$. But, I want to know if there is something specific to $\SL$ groups in the literature. I also want to know if there is some literature on decompositions of $\mathbb{Q}\SL(n,\mathbb{F_p})$.

Answer 1

$\DeclareMathOperator\M{M}\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\End{End}$As has been mentioned in the comments, the question for algebraically closed fields of characteristic $0$ is equivalent to the question on the degrees of the irreducible characters.

The question for $k=\mathbb{Q}$ is answered by, in addition, knowing the fields of character values and the Schur indices of the irreducible characters. Specifically, if $\chi$ is an irreducible character, $m$ is its degree (so that the corresponding Artin–Wedderburn component of $\mathbb{C}[G]$ is $\M_m(\mathbb{C})$), $d$ is the degree of the field $\mathbb{Q}(\chi)$ of character values of $\chi$, and $s$ is the Schur index of $\chi$, then the character $$ s\cdot\!\!\sum_{\sigma\in \Gal(\mathbb{Q(\chi)}/\mathbb{Q})}{}^\sigma\chi $$ is the character of a simple $\mathbb{Q}[G]$-module, $M$, say, and the corresponding Artin–Wedderburn component of $\mathbb{Q}[G]$ is $\M_{m'}(D)$, where $m'=m/(ds)$, and $D\cong\End M$.

For $G={\rm SL}_n(\mathbb{F}_q)$, all of the necessary information is contained in A. Turull, The Schur Indices of the Irreducible Characters of the Special Linear Groups, J. Algebra 235.