Maschke’s theorem is a fundamental result that will make our project of understanding the representation theory of finite groups — and of symmetric groups in particular — far simpler. It tells us that every representation of a finite group is completely reducible.
We saw last time that in the presence of an invariant form, any reducible representation is decomposable, and so any representation with an invariant form is completely reducible. Maschke’s theorem works by showing that there is always an invariant form!
Let’s start by picking any form whatsoever. We know that we can do this by picking a basis of and declaring it to be orthonormal. We don’t anything fancy like Gram-Schmidt, which is used to find orthonormal bases for a given inner product. No, we just define our inner product by saying that — the Kronecker delta, with value when its indices are the same and otherwise — and extend the only way we can. If we have and then we find
so this does uniquely define an inner product. But there’s no reason at all to believe it’s -invariant.
We will use this arbitrary form to build an invariant form by a process of averaging. For any vectors and , define
Showing that this satisfies the definition of an inner product is a straightforward exercise. As for invariance, we want to show that for any we have . Indeed:
where the essential second equality follows because as ranges over , the product ranges over as well, just in a different order.
And so we conclude that if is a representation of then we can take any inner product whatsoever on and “average” it to obtain an invariant form. Then with this invariant form in hand, we know that is completely reducible.
Why doesn’t this work for our counterexample representation of ? Because the group is infinite, and so the averaging process breaks down. This approach only works for finite groups, where the average over all only involves a finite sum.