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I am trying to understand the analogies between linear representations and permutation representations:

A representation $\rho:G \to GL(V)$ is irreducible if it has no invariant subspace apart from the trivial ones. It is indecomposable if we cannot write the representation as a direct sum of invariant subspaces.

I was reading the article "Representations of Finite Groups as Permutation Groups" by Aschbacher. There he claims that the indecomposable representations - i.e. now in the context of permutation groups - are exactly the transitive groups. I think I understand this since we can decompose an intransitive representation into its transitive constituents. However, he goes on writing that the indecomposable irreducible ones are the transitive primitive groups.

I want to understand how this reflects the properties of irreducible linear representations. I would already call a transitive representation irreducible since there cannot be any subsets that are fixed because of transitivity. Of course primitivity means that no blocks are fixed. But why is this the right analogue for irreducibility?

In fact I might ask a more general question, which might also answer the previous one. On wikipedia it is mentioned that permutation representations and linear representation can be regarded as a special case of representations into a fixed category. How would one define decomposability and irreducibility in this context.

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up vote 4 down vote accepted

Irreducibility means that no nontrivial subrepresentations exist. It also means that no nontrivial quotient representations exist.

Transitivity means that no nontrivial subactions exist. But primitive transitivity means that no nontrivial quotient actions exist, and unlike the case above this is not equivalent to transitivity.

In general, there are two ways to define a simple object, one using subobjects and one using quotient objects. These two definitions agree in an abelian category, which includes the case of representations but not the case of group actions, and disagree in general.

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that's awesome! – wood Sep 9 '11 at 10:21

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