## *-representations of the Group Algebra

**Point of post: **This is a continuation of this post

We now show that not only is every mapping from of that form a -representation but that every -representation is of this form for some representation . More formally:

**Theorem: ***Let be a finite group, the group algebra on , and a pre-Hilbert space. Then, if is a -representation then there exists some representation such that for each one has*

*.*

**Proof: **Define by . We first claim that in fact . Indeed, we recall that for finite-dimensional spaces unitarity of is equivalent to . Note though that for each one has

but a quick computation shows that:

so that and so

and similarly from where and thus unitarity of follows. Thus, to prove that is a representation it suffices to show that it’s a homomorphism, but this follows immediately from the fact that for any

To finish the argument it suffices to note that for any we have that

as desired.

*Remark: *The representation is called the *representation induced by .*

This shows that essentially there is a one-to-one correspondence between -representations of on and representations of on .

**References:**

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

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Print

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