## *-representations of the Group Algebra

**Point of post: **In this post we discuss the notion of -representations on the group algebra of a finite group .

*Motivation*

In our last few posts we’ve been discussing the notion of the group algebra over a finite group and parsing some of it’s complicated definitions to get some relatively routine results. In this post we’ll show (at least start to) why the group algebra in all its at-first-glance complexity is worth considering in our study of group representations.

*-representations*

Let be a finite group and the group algebra on . If is a pre-Hilbert space then a mapping is called a *-representation *of if the following conditions hold for every

Notice that conditions and say that is an associative unital algebra embedding into .

Our first theorem gives us the first clue how group representations of on and -representations of on are intimately linked. In particular:

**Theorem: ***Let be a finite group, a pre-Hilbert space and * *a representation of on . Then, the mapping*

*is a -representation of on . *

**Proof: **Assume first that is given by

for some representation . We claim that is a -representation of on . Indeed, to see that satisfies we must merely note that for any we have that

To see that satisfies we merely note that for any we have that

to see that satisfies note that

Lastly to prove we merely note that

The conclusion follows.

**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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