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Hey there,

several weeks ago, there was a discussion on the Brauer hom (see Is the Brauer correspondence injective ? ). I like to investigate this hom when being applied to simple modules: Let $k$ be an alg. closed field of positive characteristic and let $G$ be a finite group. Assume that $P$ is a $p$-subgroup of $G$. What can be said about the image of the Brauer hom on a simple $kG$-module $S$ with vertex $P$, i.e. what can be said about $Br_P(S):=\frac{S^P}{\sum_{Q<P}tr^P_Q(S^Q)}$, where $S^Q$ denotes the $Q$-invariants of $S$ and $tr^P_Q:S^Q\rightarrow S^P, x\mapsto \sum_{g\in P/Q}gx$ denotes the trace map. Does the fact that $S$ is simple add any properties to the image as for instance indecomposable modules with trivial source do?

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I think this question has been considered by P.J. Webb in connection with Mackey functors. –  Geoff Robinson Aug 30 '12 at 4:53
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up vote 6 down vote accepted

I can point to two places where Brauer quotients of simple modules have been considered. The earliest appears in section 15 of the paper P.J. Webb and J. Thévenaz, The structure of Mackey functors, Trans. Amer. Math. Soc. 347 (1995), 1865-1961, especially Example 15.8 and results 15.4 - 15.7 which lead up to it. Those results are stated for a module V which need not be simple, but in particular they apply when V is, in fact, simple. You probably need to have studied Mackey functors quite a bit to make sense of section 15, unfortunately.

The later place where these things were considered is much more approachable, and occurs in the work of Luis Valero-Elizondo, and also work he did with Radha Kessar. His work is described in his thesis (http://www.math.umn.edu/~webb/PhDStudents/ValeroThesis.pdf) and in subsequent papers, of which the following is very direct: L. Valero-Elizondo, A computer approach to Alperin's conjecture for the symmetric groups, J. Symbolic Comput. 11 (1997), 1-7. For some reason this paper does not seem to appear on MathSciNet. You should also look at his papers in J. Algebra 236 (2001), no. 2, 796–818, Bol. Soc. Mat. Mexicana (3) 8 (2002), no. 2, and with Kessar in Bol. Soc. Mat. Mexicana (3) 10 (2004), no. 1, 53–62.

What Valero-Elizondo does is to compute the Brauer quotients of simple modules for symmetric groups with respect to the p-subgroups which arise as the subgroup part of weight, in the sense of Alperin. He finds in many cases that there is a unique 'weight subgroup' for which the Brauer quotient is a simple projective module, and that this gives an explicit correspondence between simple modules and weights in the cases he is able to consider. This approach is tantalizing and has not, to my knowledge, been taken further.

To come back to the Mackey functors which Thévenaz and I considered, we were interested in obtaining the composition factors of the functor obtained from a simple module by taking fixed points. We had shown that this functor has a simple socle and were interested in looking at its semisimple quotient, motivated by Alperin's weight conjecture. The question which Valero-Elizondo worked on was closely related to this, and can be viewed as a version of the same thing without so much machinery. As described in our paper, it turns out that finding the semisimple quotient is a question about the Brauer quotient plus some further information.

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Peter, welcome to MathOverflow! –  Patricia Hersh Sep 3 '12 at 17:03
    
Bienvenido a MO, Peter! –  Hailong Dao Sep 3 '12 at 17:32
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