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
\sumgf(g)g
The set of class functions of a group with values in a field form a -vector space. If is finite and the characteristic of the field does not divide the order of, then there is an inner product defined on this space defined by
\langle\phi,\psi\rangle=
| 1 | |
| |G| |
\sumg\phi(g)\overline{\psi(g)},
When is a compact group and is the field of complex numbers, the Haar measure can be applied to replace the finite sum above with an integral:
\langle\phi,\psi\rangle=\intG\phi(t)\overline{\psi(t)}dt.
When is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.
