Dimensions of Hom Spaces
Now that we know that hom spaces are additive, we’re all set to make a high-level approach to generalizing last week’s efforts. We’re not just going to deal with endomorphism algebras, but with all the -spaces.
Given -modules and , Maschke’s theorem tells us that we can decompose our representations as
where the are pairwise-inequivalent irreducible -modules with degrees . I’m including all the irreps that show up in either decomposition, so some of the coefficients or may well be zero. This is not a problem, since it just means direct-summing on a trivial module.
So let’s use additivity! We find
Now to calculate these summands, we can pick a basis for and and use the same sorts of methods we did to calculate commutant algebras. We find that if — — then there are no -morphisms at all, even if we include multiplicities. On the other hand, if we find that an intertwinor between and has the form , where is an complex matrix. That is, as a vector space it’s isomorphic to the space of matrices.
and its dimension is
Notice that any for which or doesn’t count for anything.
As a special case, we consider the endomorphism algebra . This time we assume that none of the are zero. We find:
Just like before, we can calculate the center, which goes summand-by-summand. Each summand is (isomorphic to) a complete matrix algebra, so we know that its center is isomorphic to . Thus we find that the center of is the direct sum of copies of , and so has dimension .
As one last corollary, let be irreducible and let be any representation. Then we calculate the dimension of the -space:
That is, the dimension of the space of intertwinors is exactly the multiplicity of in the representation .