Maschke's averaging lemma

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 $G$ (this includes characteristic zero fields). 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

• Maschke's averaging lemma for abelian groups

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.

Applications, corollaries and results

• The result Representation of finite group over field of characteristic zero decomposes as direct sum of irreducible subrepresentations is essentially an immediate corollary, only requiring an induction argument.