Induced Representation is ‘Inductive’
Point of Post: In this post we use the Frobenius Reciprocity theorem to show that the construction of Induced representation is inductive, in the sense that for any .
An obvious question is the following: given groups there are two ways we can create a representation on from a representation on . Namely, we take a representation of of we can then obviously consider the induced representation , but we could also consider creating a -representation by considering and then successively considering the -representation . The obvious question though is that if we do this, do we get two different representation. The answer turns out to be no, in other words it’s true that given any -representation the representations and are the same representation. In general this may be a very messy task, but thanks to the Frobenius Reciprocity theorem this becomes almost obvious in the sense that since is the adjoint of it suffices to show that the map is the adjoint of and we will be done by the uniqueness of the adjoint. The fact that this is true is often stated that inducing representations is inductive.
Induced Representation is Inductive
We begin by noting the following fact. Namely, let be finite and . Let then , , and be the restriction maps from to , to , and to respectively. We note then the obvious fact that . The reason this is ‘obvious’ is that the map is just restriction. From this we get that the analogous result for the maps , and . Namely:
Theorem: Let be a finite group and , then .
Proof: We note that for any and we have by (double) application of the Frobenius Reciprocity theorem
Thus, is adjoint to . But, by Frobenius Reciprocity we know that is an adjoint to and since the adjoint is unique we may conclude that .
A corollary of this which is mostly of novel interest is that:
Theorem: Let then
where (resp. ) are obviously the notation to indicate the the function is itself when in (resp. ) and when not in (resp. ).
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.