Bases for group algebras

The matrices of the irreducible representations provide one with a special set of group algebra elements. We define 

Equation (5.19) gives the algebra basis as a sum over the group elements. Using Eq. (5.18) we may also write the group elements as a sum over the algebra basis. 

In the theory of operators over vector spaces Eq. (5.22) is said to give the resolution of the identity, since by Eq. (5.20) each (e ) = e , and is idempotent. 

We note another important property of these bases. Irreducible representation matrices may be obtained from the by using the relation 

NB In the inverse we have interchanged the index labels of the irreducihle representation matrix. 

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