## Another Way of Looking at Induced Representations (Pt. I)

**Point of Post: **In this post we discuss another way to view induced representations and and show that it is actually equivalent to our previous definition since the two different constructed representations of the ambient group will be equivalent.

*Motivation*

Of course, as is true for most things in mathematics, the notion of induced representation has several equivalent formulations. In this post we describe another one which may seem, at first, to be a simpler construction than our previous one. In fact, both are equally useful and come up in different aspects of the coming theory. Thus, it is wise to have both of these (among many more) equivalent formulations sitting in our back pocket.

*Induced Representations Again*

Let be a finite group and . Assume that there is a representation . We begin our construction by fixing a transversal for the set of cosets . We then consider then the direct sum of copies of indexed by the elements of where each copy is distinguished. Said more concretely for each we define to be equal to and consider the space given the usual inner product structure on direct sum spaces with a normalization factor of –we will think of the elements of this space as where and is a formal symbol ‘tagging’ . We then note that since the set forms a partition of for each and there exists unique and such that , we denote this by and this as . We then define, for each the map

We claim that in fact is in . Indeed, to see that is a linear transformation we note that for any and

To see that it’s unitary we merely note that

and thus is in as claimed. Our last claim is that the map is a homomorphism. To do this we first claim that for any and any one has that and . To see this we merely note that if then . Then, for any and any one has that

and since was arbitrary we may conclude that and thus is a homomorphism as claimed. Summing this all up we have that:

**Theorem: ***Let be a finite group, , a transversal of , and a representation of . Then, if the space (as described above) is given the usual inner product structure on direct sums with a normalization factor then the map *

*(where are the unique elements of respectively such that ) is a representation.*

We shall call this representation .

**References:**

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

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