Different Formula For the Character of an Induced Representation (Pt. I)
Point of post: In this post we derive a formula for the induced character of a representation different than the one derived in the previous post.
Motivation In our last post we derived a formula for the induced character of a representation. The interesting thing is that we derived this formula using the ‘type’ of induced representation we derived second. A natural question is that if we can use our first equivalent form of induced representations to derive another formula for induced characters. In this post we do precisely that.
Let be a finite group, , and a representation. Consider then the first form of the induced representation: . If is the character for we define, analogously to the previous post, the induced character to be the character of . We now compute in terms of . But first we make an observation:
Lemma: Let be a group and a conjugacy class in . Then, for any there exists distinct conjugacy classes in such that .
Proof: Indeed we let be the elements of where is the conjugacy class in of . Clearly then for each we have that (since ) and so . Conversely, if then and so .
Moreover, it’s clear that this decomposition is unique. Thus, we may state unambiguously:
Theorem: Let be a finite group, , and a representation of with character . Consider then the first construction of the induced representation . Then, if is any conjugacy class and has the unique decomposition (as described in the lemma) as where is a conjugacy class in then
(where we recall that since a character is a class function by and we mean the unique value each of those characters takes on the conjugacy class).
Proof: We begin by considering the usual extension of to given by
and recall that is in fact a homomorphism of -algebras. Then we see that
Let then be a basis for and a transversal for . Define then for by
what we claim then is that if for and then is an orthonormal basis for . Since which we know is equal to it clearly suffices to prove orthonormality. To do this we note that
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.