Modular reductions of simple characters

Question

Given a (splitting) $p$-modular system $(K, \mathcal{O}, k)$ for a finite group $G$, any given simple character $\chi$ is afforded by some $KG$ module $V_\chi$, and there in general many non-isomorphic $\mathcal{O}G$ modules $B$ (free as an $\mathcal{O}$ module) such that $B\otimes_{\mathcal{O}} K\cong V_\chi$ (call $B$ an $\mathcal{O}$-form of $\chi$). Further, the decomposition matrix tells us the composition factors of $B\otimes_\mathcal{O} k$. My question is, what further restrictions are known on the isomorphism class of $B\otimes k$, given $\chi$?

In particular, it is not hard to see that an $\mathcal{O}$-form $B$ can be chosen such that $B\otimes k$ is indecomposable. Is the reduction of every $\mathcal{O}$-form of $\chi$ indecomposable?

Any advice would be greatly appreciated, everything I've read to date seems to skirt this question.

Answer 1

The answer to your basic question is certainly no, though it would take me some time to provide convincing examples. Decades ago I raised a similar question with experts like Walter Feit and Jon Alperin, who assured me that almost anything is possible when choosing the form $B$: the modular reduction might be indecomposable or completely reducible or whatever. (But lifting idempotents attached to indecomposable projectives from characteristic $p$ to characteristic $0$ does lift projective covers effectively to indecomposable modules over the ring $\mathcal{O}$. That's a different matter.)

When Brauer emphasized only the preservation of composition factors under this process of reduction mod $p$ (what we would now describe in terms of Brauer characters), he must have had insight into this question. But as you point out, it's not discussed in the literature such as the Curtis-Reiner books. Though it seemed to me to be a real question at the time, I guess people familiar with the theory had already moved on beyond it.

I will check some of the old correspondence I kept from that period, but it was before the age of email and wasn't all preserved.

ADDED: I don't find a paper trail at this point, perhaps because most of this occurred in conversations. During the 1980s both Jantzen and I were inspired by the success of the Deligne-Lusztig work on irreducible characters of finite groups of Lie type. This led to multiplicity results on reduction mod $p$ in the defining characteristic, but module structure remains tricky. Since the characters tend to fall into families (such as "principal series" and "discrete series"), the concrete work by Curtis and Carter-Lusztig on principal series modules in characteristic $p$ raised questions about the possible existence of intrinsic discrete series modules, etc. Jantzen found reasonable filtrations in some indecomposable projective modules, for instance, suggesting such structure.

Anyway, the specific examples given in the few books which cover ordinary and modular representations via reduction mod $p$ (Curtis-Reiner, Serre especially) are small and don't quite illustrate all possibilities. But they do indicate caution about expecting too much control over the reduction process.

Answer 2

I am working from memory, but I believe that there is a proof in Feit's Book "Representations of finite groups" that if the prime $p$ is sufficiently ramified in the dvr then it can even be the case that the reduction (mod $p$) of (a module affording) $\chi$ can be completely reducible. In any case, that is a true statement.

Answer 3

Here is a concrete example that the reduction of such a form can be decomposable. Let $$G = \langle t, s \mid t^4 =s^2 = 1, t^s=t^{-1} \rangle$$ be the dihedral group of order $8$ and $\chi$ the irreducible character of degree $2$. Let $p=2$. Consider first the natural representation affording $\chi$, namely $$ t \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad s \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. $$ The reduction of $\mathcal{O}^2=M$ with respect to this representation is indecomposable, as is easily seen. Now assume that $1-i$ (where $i=\sqrt{-1}$) is a prime element of $\mathcal{O}$. Let $B\subseteq \mathcal{O}^2$ be the $\mathcal{O}$-submodule $$ B= \langle \begin{pmatrix} 1\\ i \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \end{pmatrix} \rangle, $$ which is the unique maximal $\mathcal{O}G$-submodule of $M$, by the way. The action of $t$ and $s$ on $B$ with respect to the new basis is given by the matrices $$ t\mapsto \begin{pmatrix} -i & -1-i \\ 0 & i \end{pmatrix},\quad s \mapsto \begin{pmatrix} i & 1+i \\ 1-i & -i \end{pmatrix}. $$ Thus $t$ and $s$ act as identity on the reduction mod $2$, and thus $B\otimes_{\mathcal{O}}k \cong k^2$ is completely reducible.

It follows that $B$ has $|k|+1$ maximal $\mathcal{O}G$-submodules. Each of these happens to have indecomposable reduction mod $2$ and a unique maximal submodule, namely $(1-i)B$. When $\mathcal{O}$ is not ramified, however, then there are only two non-isomorphic forms and both have indecomposable reduction mod $2$.