Class function

In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f 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 $\sum_{g \in G} f(g) g$.

Inner products

The set of class functions of a finite group G with values in a field K form a K-vector space. If the characteristic of the field does not divide the order of G, then there is an inner product defined on this space defined by $\langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \psi(g^{-1})$ 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: $\langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt.$

When restricted to real linear combinations of characters, 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.