Class function (algebra)

Not to be confused with a class function in set theory.

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.

See also

References

This article is issued from Wikipedia - version of the 8/4/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.