(Fake) Frobenius Reciprocity
Now, naïvely we might hope that induction and restriction would be inverse processes. But this is clearly impossible, since if we start with a -module with dimension , it restricts to an -module which also has dimension . Then we can induce it to a -module with dimension . This can’t be the original representation unless , which is a pretty trivial example indeed.
So, instead we have the following “reciprocity” relation. If is a character of the group and is a character of the subgroup , we find that
Where the left inner product is that of class functions on , while the right is that of class functions on . We calculate the inner products using our formula
where we have also used the fact that is a class function on , and that is defined to be zero away from .
As a special case, let and be irreducible characters of and respectively, so the inner products are multiplicities. For example,
is the multiplicity of in the representation obtained by inducing to a representation of . On the other hand,
is the multiplicity of in the representation obtained by restricting down to . The Frobenius reciprocity theorem asserts that these multiplicities are identical.
Now, why did I call this post “fake” Frobenius reciprocity? Well, this formula gets a lot of press. But really it’s a pale shadow of the real Frobenius reciprocity theorem. This one is a simple equation that holds at the level of characters, while the real one is a natural isomorphism that holds at the level of representations themselves.