## Composition of the Restriction Map and the Induction Map

**Point of Post: **In this post we discuss what happens when we compose the maps and .

*Motivation*

So in our recent discussion of the induction map and a very obvious question is…what happens if we compose the two maps? Namely, we have that and so what does the map look like? This is a question which was first solved by Mackey. In this post we prove a more general result, which in particular finds where .

*What is ?*

We begin by define a way to correspond class functions on subgroups of and class functions on their conjugates. Namely, if and we can define a map given by , in other words . This is evidently well defined map, in the sense that since given we have that

and thus by a previous theorem. With this in mind we can state the following theorem:

**Theorem: ***Let and let be a transveral of the set of double cosets then for any one has that*

**Proof: **We use the orbit-stabilizer theorem we write as

But, by a previous theorem we may rewrite this as

but it’s a common fact that that where is a transversal for which by a previous observation is equal to . Thus, if we let be the union over of then by we have that

and thus forms a transversal for . Thus, by definition for any and we have that

where has its usual meaning. But, since is a transversal fro the above may be rewritten

and since was arbitrary the conclusion follows.

From this we get the obvious corollary:

**Theorem: ***Let be a finite group, and a representation. Then, if is the usual induced representation then for any *

*where is given by .*

**Proof: **This is obvious since the above theorem proves that these two representations of admit the same character.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Math. Soc., 1996. Print

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