The Dimension of the Space of Tensors Over the Group Algebra
Now we can return to the space of tensor products over the group algebra and take a more solid pass at calculating its dimension. Key to this approach will be the isomorphism .
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
.
No comments yet.
Leave a Reply