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I was wondering if anyone here could pitch a plausible geometric interpretation of Maschke's Theorem for $FG$-modules (or at least for a particular instance of its conclusions.) It seems reasonable that there should be some sort of picture associated with averaging a projection over a finite group, but the few examples that have occurred to me seem uninteresting; when is the direct sum decomposition in question sensible but not obvious?

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Isn't Maschke's theorem in a way simply about orthogonal projections (in the cases, where $F$ allows that to be geometrically meaningful)? First we replace the inner product with another one that is also $G$-invariant. Then if $V$ is a $G$-invariant subspace, $V^\perp$ is another? – Jyrki Lahtonen Oct 17 '11 at 17:42
That does seem to be a very natural way of thinking about it. The proof I first came across avoids any mention of the standard inner product, but I guess the process of passing from an ordinary complementary subspace (such as $V^{\perp}$) to the kernel of an $FG$-homomorphism is basically the same. Thanks! – user17794 Oct 17 '11 at 23:16

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