Irreducible iff simple


Let’s try to be explicit about this, since I feel like I may keep beating around the bush. The reason that we can say things about reducibility of a representation of a group by looking at the simplicity of the modules over the group ring is that they are really the same thing. By this I mean that a k-representation is irreducible (completely reducible) if and only if the corresponding kG-module is simple (semisimple).

Proof: Let \sigma:G\to GL(V) be an irreducible k-representation. Suppose that V^\sigma is not simple. Then there is a proper non-trivial submodule W\subset V^\sigma. By virtue of being a submodule, W is stable under the action of \sigma. i.e. as a vector subspace it is \sigma-invariant, and hence the representation was reducible, a contradiction. Thus V^\sigma was simple. The reverse implication works precisely the same way.

Corollary 1: Maschke’s Theorem tells us that if char(k) does not divide |G|, and if V is a vector space over k, then any representation \sigma : G\to GL(V) is completely reducible.

I know this was a sort of silly post, but I had lots of different things floating around in different worlds, and needed to really clarify that I could not only switch between them, but I could do it in a nice way.

Now I’ve set up the motivation I wanted for Artin-Wedderburn, since it will classify how semisimple rings decompose, which in turn will help us look at how to decompose our representations.

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