Different Formula For the Character of an Induced Representation (Pt. II)
Point of Post: This post is a continuation of this one.
Now, note that if then and can never happen. Indeed, if then which is impossible since . Thus, if then by definition either or for every and thus by definition for every either or and so evidently for every . Moreover, along the same lines of reasoning we can consider the sum only over since the term for associated to a is zero since . Thus by we know that
from where the fact that is an orthonormal basis follows. Note though that evidently commutes with . Indeed, since is a -representation and one has that
But, since is orthonormal it’s a common fact the matrix entry on the diagonal associated to is but we may rewrite this as
But, note that if one has that it is impossible to simultaneously have and and so or is for every and so evidently
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.