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 http://en.wikipedia.org/wiki/Group_representation#Representations_in_other_categories
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.
Best Answer
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.