Class function
title: "Class function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["group-theory"] topic_path: "general/group-theory" source: "https://en.wikipedia.org/wiki/Class_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 \sum_{g \in G} f(g) g.
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 \langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \overline{\psi(g)}, where G denotes the order of G and the overbar denotes conjugation in the field K. The set of irreducible characters of G forms an orthogonal basis. Further, if K is a splitting field for Gfor instance, if K is algebraically closed, then the irreducible characters form an orthonormal basis.
When G 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 = \int_G \phi(t) \overline{\psi(t)}, dt.
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.
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