# Artin-Wedderburn decomposition of $\mathbb{F}_2[S_5]/J$

If $p$ is a prime that divides the order of a finite group $G$ and $k$ is the field with $p$ elements then we can form the group algebra $kG$ and quotient out by the Jacobson radical $J$ to obtain a semisimple artinian algebra $kG/J$ which is therefore a direct sum of matrix rings of the form $M_{n_i}(D_i)$ where $D_i$ is a finite extension of $k$. I'm interested in calculating the numbers $n_i$ and $d_i = \dim_k(D_i)$ for particular cases.

The problem I'm trying to solve now is to compute those numbers in the case of $k=\mathbb{F}_2$ and $G=S_5$. Trying to do as in Artin-Wedderburn decomposition of a particular group ring (where the case $\mathbb{F}_5 S_3$ is considered) does not help very much because $|S_5|=120$ is quite big. Also, I thought those numbers would appear in the Atlas page of $S_5$ (representations in characteristic 2) http://brauer.maths.qmul.ac.uk/Atlas/v3/alt/A5/ but I'm quite sure they don't, or at least not in the way I would expect. Do you have any hint?

To summarize, my question is: let $J$ be the Jacobson radical of $\mathbb{F}_2 S_5$, how do you compute the Artin-Wedderburn decomposition of the artinian semisimple algebra $\mathbb{F}_2 S_5/J$? That is, how do you compute the numbers that I called $n_i$ and $d_i$ above?

Thank you!

First of all, the symmetric group is split over every field, or in other words, the division algebras that appear are just $\mathbf{F}_2$ and $d_i=1$ for all $i$. Consequently, the $n_i$ are the dimensions of the irreducible representations of $\mathbf{F}_2[ S_5]$.
The irreducible representations of $\mathbf{F}_p [S_n]$ are indexed by the $p$-restricted partitions of $n$; these are the integer partitions for which successive parts differ by at most $p-1$. For $p=2$ and $n=5$ these are $$(1,1,1,1,1), (2,1,1,1), (2,2,1).$$ Each of these is a quotient of a Specht module (obtained by modular reduction of a lattice in $\mathbf{Q}$-irreducible) whose dimensions can be obtained from the hook-length formula and are $1, 4$, and $5$.
Now we use the combinatorial classification of blocks: two irreducibles are in the same block if and only if the corresponding partitions have the same $p$-core (obtained for $p=2$ by removing as many dominoes as possible). In our case that means the first two Specht modules remain irreducible, while the third irreducible must be of dimension $4$. The modular irreducibles are then of dimensions $1$, $4$, and $4$, which implies that the quotient by the radical is of dimension $33$.