# 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.