## Class Functions

Our first observation about characters takes our work from last time and spins it in a new direction.

Let’s say and are conjugate elements of the group . That is, there is some so that . I say that for any -module with character , the character takes the same value on both and . Indeed, we find that

We see that is not so much a function on the group as it is a function on the set of conjugacy classes , since it takes the same value for any two elements in the same conjugacy class. We call such a complex-valued function on a group a “class function”. Clearly they form a vector space, and this vector space comes with a very nice basis: given a conjugacy class we define to be the function that takes the value for every element of and the value otherwise. Any class function is a linear combination of these , and so we conclude that the dimension of the space of class functions in is equal to the number of conjugacy classes in .

The space of class functions also has a nice inner product. Of course, we could just declare the basis to be orthonormal, but that’s not quite what we’re going to do. Instead, we’ll define

The basis isn’t orthonormal, but it *is* orthogonal. However, we can compute:

Incidentally, this is the reciprocal of the size of the centralizer of any . Thus if we pick a in each we can write down the orthonormal basis .