First off, we want to calculate the character of . If — as a left -module — has character and has character , then we know that the inner tensor product has character
Next, we recall that the submodule of invariants can be written as
Now, we know that , and thus the dimension of our space of invariants is the dimension of the space. We’ve seen that this is the multiplicity of the trivial representation in , which we’ve also seen is the inner product . We calculate:
This may not be as straghtforward and generic a result as the last one, but it’s at least easily calculated for any given pair of modules and .