## Induced Representations

**Point of Post: **In this post we introduce the notion of induced representations and prove some of the basic properties about them (the fact that they are representations, etc.). We also define some preliminary notions

*Motivation*

In the past we’ve encountered the situation where we had a representation and we restricted to so that where , this is usually denoted by . In this post we pursue an essentially dual concept where given a representation we define a representation on –in essence the representation on the subgroup ‘induces’ a representation on the full group. This shall prove to be an indispensable tool in all that comes. The most apparent reason is that it is often much easier to produce representations on smaller groups and we can use this to create representations on the larger groups.

*Induced Representations*

Let be a finite group and . Let then be a representation of . Consider then the set with the usual inner product structure with normalization constant . Define then to be the set . We claim then that . Indeed, let and then for any and one has that

from where the conclusion follows. We next note that for any and any the function given by is an element of . Indeed, for any and one has that

Thus, the map defined by is well-defined. What we next claim is that . Indeed, we note that for any and any one has

The last thing we claim is that in fact the mapping is a group morphism. Indeed, for any and any one has that . But, since was arbitrary it follows that . But, since was arbitrary it follows that , and finally since were arbitrary the conclusion follows. Summing this all up:

**Theorem: ***Let be a finite group and and a representation. Then,the set given by*

* *

*is a subspace of . Moreover, the map given by the formula is a representation of .*

The representation described in this theorem as is called the *representation induced on by *and is generally denoted .

As a final simple exercise to end this post we find a connection between the degree of and the degree of . Indeed:

**Theorem: ***Let be a finite group and . Then, if is a representation of and is the corresponding induced representation on then .*

**Proof: **Let denote the representation space of as described above. Let be a transversal of the set of cosets . Define then the map by , we claim that is an isomorphism. Indeed, we note that for any and one has that

To see that is injective note that if then for . But, for any we know that for some and some and so . Thus, since was arbitrary it follows that and so is trivial. Thus, is injective. To see that is surjective we let be arbitrary. We define then by where is the unique representation of as the product of some element of and an element of . Evidently and since we see that and thus . But, since was arbitrary it follows that . Therefore, is an isomorphism as claimed and so we may conclude that

**References:**

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

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