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?

Any good reference about this would be of great help.

Thank you!


It's very helpful here to use the existing theory of modular representations of symmetric groups. This example is a nice first application that would be hard by totally naive methods but is fairly straightforward given the (by now) standard theory.

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 $33$.

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    $\begingroup$ Thank you :) may I ask you, if you had to recommend a reference would you recommend Gordon James, Adalbert Kerber, The Representation Theory of the Symmetric Group? Or another reference? $\endgroup$ – Martino Apr 2 '17 at 20:03
  • $\begingroup$ @Martino Sure, that's a classic (and very good) reference for this. $\endgroup$ – Stephen Apr 4 '17 at 12:24

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