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.