## Consequence of the Decomposition of the Group Algebra Into Matrix Algebras

**Point of post: **In this post we shall use our previous results concerning the decomposition of the group algebra to prove that if one writes an element of in a particular way then that form is conducive to computing polynomials.

*Motivation*

We saw in our last post that the group algebra is isomorphic to a direct sum of matrix algebras. We shall use this fact to derive an interesting fact about the group algebra. Namely, we know that for every choice of matrix entry functions one has that the group algebra is a direct sum of the subalgebras of the form where . Thus, every element has a decomposition of the form where . We shall show that with this decomposition it is much simpler to calculate for some polynomial . Namely, we’ll show the awesome result that is actually equal to . In fact, this isn’t surprising as we shall see that each shall act analogously to sitting inside .

*Ramifications of the Group Algebra Decomposition*

We recall that the map

where is a unitary algebra isomorphism where we’ve fixed some set of representative matrix entry functions. We will use this to derive a very, very useful result concerning the deomposition of the group algebra. Namely, if we let

(which we know are subalgebras) then we have that . Thus, for every we may write uniquely as where . But, since is an isomorphism of algebras we know that this means that

Note though that (where the notation is defined as before for general algebras). Indeed, for each coordinate we have that

where the last equality is clear since for some suitable constants and using theorthogonality relation between the matrix entry functions. It clearly follows then that for any polynomial with no constant term that

and since is an algebra homomorphism this can be rewritten

Putting all this together we get the following interesting theorem:

**Theorem:** *Let and the unique representation of in terms of . Then, for any with no constant term one has*

* *

* *

**Proof: **From the above and the fact that is clearly an algebra homomorphism we have that

The conclusion follows.

**References:**

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

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