1
$\begingroup$

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 $R$?

$\endgroup$
2
  • $\begingroup$ Q1: If the theorems you know give you conditions for the indecomposability of an induced representation, then didn't they solve your problem ? $\endgroup$ – Ralph May 2 '13 at 14:14
  • $\begingroup$ 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. $\endgroup$ – user33618 May 2 '13 at 14:37
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.