TITLE: Modular reductions of simple characters
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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 $2$.