# Class function (algebra)

In mathematics, especially in the fields of group theory and representation theory of groups, a **class function** is a function on a group *G* that is constant on the conjugacy classes of *G*. In other words, it is invariant under the conjugation map on *G*. Such functions play a basic role in representation theory.

## Characters

The character of a linear representation of *G* over a field *K* is always a class function with values in *K*. The class functions form the center of the group ring *K*[*G*]. Here a class function *f* is identified with the element
.

## Inner products

The set of class functions of a group *G* with values in a field *K* form a *K*-vector space. If *G* is finite and the characteristic of the field does not divide the order of *G*, then there is an inner product defined on this space defined by
where |*G*| denotes the order of *G*. The set of irreducible characters of *G* forms an orthogonal basis, and if *K* is a splitting field for *G*, for instance if *K* is algebraically closed, then the irreducible characters form an orthonormal basis.

In the case of a compact group and *K* = **C** the field of complex numbers, the notion of Haar measure allows one to replace the finite sum above with an integral:

When *K* is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.

## References

- Jean-Pierre Serre,
*Linear representations of finite groups*, Graduate Texts in Mathematics**42**, Springer-Verlag, Berlin, 1977.