Orthogonality is the most fundamental theme in representation theory, as in
Fourier analysis. We will show how to construct an orthonormal basis of
functions on the finite group $G$ out of the "matrix coefficients'' of
irreducible representations. For many purposes, one may work with a smaller
set of computable functions, the *characters* of the group, which give
an orthonormal basis of the space of *class functions*, that is,
functions that are constant on conjugacy classes.

In some books, such as Lang's *Algebra*, character theory is developed
from the theory of semisimple rings (Wedderburn theory). We will follow a
different method, in which we directly prove the orthogonality relations for
matrix coefficients. We prefer this method because of our desire to emphasize
convolution, and because the approach that we use here is also valid for
compact topological groups, such as compact Lie groups.

The proofs are long so we've included a summary of the results of at the end
of Section 2.4. A look at these should show that although
the *proofs* are *long*, and do involve important concepts, such
as Schur's Lemma, the *results* are *simple*. And in
Section 2.7 we bring the subject back to earth by computing
character tables for particular groups.