# Representations over $\mathbb{Z}_p$

Hi, I would like to work with indecomposable representations over a commutative ring and start with $R=\mathbb{Z}_p$

Notations: $G$ a finite group, $S$ a subgroup, $R=\mathbb{Z}_p$ the p-adic ring, and $W$ an indecomposable $R[S]$ module

1. The Mackey criterion tells if an induced representation is simple or not. How can one decide if it is indecomposable? I know just theorems which tell " If ... then $\operatorname{Ind}_S^G(W)$ is indecomposable". Is there something for the other direction to have "$\operatorname{Ind}(W)$ is indecomposable if and only if" ?

2. Let $V$ be some finite generate $\mathbb{Z}_p[G]$ module. Is the decomposition of $V$ into indecomposables unique? I know that usually it would be necessary that the ring is artinian.

Thanks

edit: Is there maybe a theorem, which tells when an indecomposable representation from a normal subgroup extends to the whole group when working over $R$?

• Q1: If the theorems you know give you conditions for the indecomposability of an induced representation, then didn't they solve your problem ? – Ralph May 2 '13 at 14:14
• Thank you for your answer. My Problem in Q1 is that I know when it is indecomposable, but not when it decomposes. The reason is that I do not know a theorem with an "if and only if" part. So if the assumptions of the theorem (e.j. from Ward, 1968) are not fulfiled, I do not get a result. – user33618 May 2 '13 at 14:37

## 1 Answer

Concerning 2: Yes, the decomposition is unique, since the Krull-Schmidt theorem applies to finitely generated modules over $\mathbb{Z}_pG$. This follows by a theorem of Swan (Induced Representations and Projectives. Ann. of Math. 71(1960), 552-578. Remark after Prop. 6.1):

Let $R$ be a commutative complete local ring and let $A$ be an $R$-algebra that is finitely generated as $R$-module. Then the Krull-Schmidt theorem holds for finitely generated $A$-modules.

An alternative reference is Curtis, Reiner: Representation Theory of Finite Groups and Associative Algebras. Theorem 76.26.