General Background

Take $G$ to be a finite group, and say $V$ is s $G$-representation over $\mathbb{Q}_p$. By picking a $G$-invariant lattice $L\subset V$ we can get an $\mathbb{F}_p$ representation by looking at $L/pL$, which we call a reduction of $V$.

It is not hard to show that if we semisimplify the reduction, we get an answer which depends only on $V$, and not on the choice of $L$. However, I am interested in how decomposable we can make the reduction. That is, I want to get as large a subrepresentation of $L/pL$ as possible which is semisimple.

Specific Question

Say $V$ is absolutely irreducible, and its reduction for a $G$-invariant lattice $L\subset V$ is a non-trivial extension of 2 simple modules $A,B$, so that we have $$0\rightarrow A\rightarrow L/pL\rightarrow B\rightarrow 0.$$

I know that it is not always possible to pick a different $G$-invariant $L_0$ such that $L_0/pL_0=A\oplus B$.

My question is this: Can we do better by taking powers of $V$? That is, does there exist an integer $n$ and a $G$-invariant lattice $L_1\subset V^n$ such that $L_1/pL_1$ has a semisimple subrepresentation which is a summand of at least $n+1$ simple objects?

I would also appreciate any reference for studying which reductions one can get from a $\mathbb{Q}_p$ representation.

Thanks for your help!

  • $\begingroup$ My impression is that almost anything can happen in such a reduction process, though perhaps I'm too pessimistic. In any case, it's hard to point to encouraging rresults in the literature of modular representation theory, including Jon Alperin's work. $\endgroup$ – Jim Humphreys Feb 6 '14 at 19:12
  • $\begingroup$ By working over a splitting field with enough ramification of p, you can make the reduction (mod p) completely reducible. This is treated in Feit's book "Representations of Finite Groups" $\endgroup$ – Geoff Robinson Feb 7 '14 at 3:39
  • $\begingroup$ @GeoffRobinson: Cool! I did not know that. Just to be clear, you mean base change $V$ to some (wildly ramified) p-adic field K, and then theres some lattice $L$ in $V$ whose reduction mod $p$ is completely reducible? As an $\mathbb{F}_p$ representation or a $O_K/p$ representation? $\endgroup$ – jacob Feb 7 '14 at 6:09
  • $\begingroup$ Well certainly competely reducible over the large splitting field of characteristic p. I would have to think about the case of the prime field, but probably OK $\endgroup$ – Geoff Robinson Feb 7 '14 at 6:43

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