Projecting Onto Invariants
Given a -module , we can find the -submodule of -invariant vectors. It’s not just a submodule, but it’s a direct summand. Thus not only does it come with an inclusion mapping , but there must be a projection . That is, there’s a linear map that takes a vector and returns a -invariant vector, and further if the vector is already -invariant it is left alone.
Well, we know that it exists, but it turns out that we can describe it rather explicitly. The projection from vectors to -invariant vectors is exactly the “averaging” procedure we ran into (with a slight variation) when proving Maschke’s theorem. We’ll describe it in general, and then come back to see how it applies in that case.
Given a vector , we define
This is clearly a linear operation. I say that is invariant under the action of . Indeed, given we calculate
since as ranges over , so does , albeit in a different order. Further, if is already -invariant, then we find
so this is indeed the projection we’re looking for.
Now, how does this apply to Maschke’s theorem? Well, given a -module , the collection of sesquilinear forms on the underlying space forms a vector space itself. Indeed, such forms correspond to correspond to Hermitian matrices, which form a vector space. Anyway, rather than write the usual angle-brackets, we will write one of these forms as a bilinear function .
Now I say that the space of forms carries an action from the right by . Indeed, we can define
It’s straightforward to verify that this is a right action by . So, how do we “average” the form to get a -invariant form? We define
which — other than the factor of — is exactly how we came up with a -invariant form in the proof of Maschke’s theorem!