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.
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
The set of class functions of a group G with values in a field K form a K-vector space.
where |G| denotes the order of G and bar is conjugation in the field K. 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: