## Left Regular Representation

**Point of post: **In this post we shall discuss the notion of the left and right regular representations on the group algebra

*Motivation*

We saw in our last post that -representations of the group algebra on a pre-Hilbert space are, in a sense, one-to-one. In this post we give a specific example of a -representation of on itself which will then induce a representation of on . This representation will have a canonical form once we parse it out. This representation will be called (and one very similar to it) will be called the *left regular representation *of on .

*Canonical Inner Product on *

As it stands we have yet to put an inner product on the group algebra of some finite group . We begin by defining the *canonical inner product on . *It’s defined by

The first order of business is to verify that this is, in fact, an inner product on .

**Theorem: ***Let be a finite group and the group algebra over . Then,*

*is an inner product on . *

**Proof: **To see that is linear in the first entry let and . Then,

and so it’s linear in the first entry. It’s conjugate symmetric since

and lastly it’s positive-definite since

and evidently from this it’s clear that if and only if . The conclusion follows.

One last thing to notice about this inner product is that it makes an orthonormal basis for . Indeed, we already know it’s a basis and if then

and if then

* *

**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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