Maschke's averaging lemma

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This fact is related to: linear representation theory
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Statement

Let G be a finite group and k a field whose characteristic does not divide the order of G. Let \rho: G \to GL(V) be a representation of G over k. If W is an invariant subspace for \rho, then there exists an invariant subspace W' of \rho, that is complementary to W. In other words, any subrepresentation is a direct summand of the whole representation.

Generalizations

Proof

The idea is to take any complementary subspace to W in V, and consider the induced projection from V to W. Call this projection p.

Now, from the representations of G on V and W, we also get a representation of G on Hom(V,W). Call this representation \alpha. Then consider the sum:

\frac{1}{|G|} \sum_{g \in G} \alpha(g).p.

Note that the expression is well-defined iff the order of G is not a multiple of the characteristic of k.

It turns out that \alpha(g).p is identity restricted to W for every g, hence the average is also identity when restricted to W. Further, the image of V under this map is entirely in W. Hence the map is a projection from V to W.

The kernel of that projection is W', and that kernel is easily seen to be invariant.