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Quite simply:

What is known about the indecomposable modules of the symmetric group $S_n$ in positive characteristic?

Since the answer is likely to be "very little", perhaps I should rather ask

What is known about the indecomposable modules of $S_4$ in characteristic $2$ ?

References appreciated.

Thanks!

Pierre

Pierre
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    This is not exactly what you asked for, but maybe you will the reference useful. Thrall and Nesbitt have a series of papers about modular representation theory of the symmetric group. The reference for the first paper of the series is: Thrall, R. M.; Nesbitt, C. J. On the modular representations of the symmetric group. Ann. of Math. (2) 43, (1942). 656–670, MR0007418 (4,134a). – Leandro Vendramin Feb 24 '12 at 14:24
  • @Pierre: I don't know how hard your questions are, but in general the answer probably just got harder with the withdrawal today of a joint preprint on arXiv by Bowman-De Visscher-Orellana on decomposition of tensor products of Specht modules. One or two of the authors have been involved earlier in parallel questions about tensor products of Weyl modules in small cases, which certainly illustrate how difficult the problem is there. I wouldn't expect miracles in the search for indecomposable modules. – Jim Humphreys Nov 15 '12 at 15:02
  • Here are some tools to compute bases and dimension: https://mathoverflow.net/questions/458191/what-is-dim-d-lambda-for-the-symmetric-group – Jackson Walters Nov 15 '23 at 19:55

1 Answers1

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Well you certainly asked the right question. If the Sylow subgroup is cyclic then there is a nice way to write down all the (finitely many) indecomposable modules, see for example the last chapter of Alperin's "Local Representation Theory" or 6.5 in Benson's "Representations and Cohomology." If $p$ is odd and one is not in the cyclic defect case then the representation type is wild, and classifying the indecomposable modules is hopeless in some precise sense that it is at least as difficult as for a polynomial algebra on two noncommuting variables.

In between these two is something called "tame" representation type, which happens for symmetric groups only for $p=2$ and $n=4, 5$, where there is a dihedral Sylow. (Or also for larger $n$ but but blocks of weight $2$). However by work of Jost it is known that all these blocks are Morita equivalent to either the principal block of $S_4$ or $S_5$. The main reference for this situation is Erdmann's LNM 1428 although there is a lot of work since. See also for example the paper "Representation type of Hecke algebras of type A" by Erdmann-Nakano that gives the quiver and relations for the principal blocks of $S_4$ and $S_5$ in characteristic two.

As an aside, even in the wild type case I find it an interesting question to ask if one can classify indecomposable, self-dual modules that have Specht filtrations.

daveh
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  • Thanks for this answer! I've looked at the LNM by Erdmann, it certainly is full of information. I've noticed that Benson's book on representations and cohomology contains a description of the indecomposable modules for the dihedral groups, which I realize thanks to you is crucial to the general question.

    Yet I haven't found the same thing for $S_4$. I wonder how much harder it is.

     – Pierre Feb 24 '12 at 20:59