Class function

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.

## Contents

* 1 Characters
* 2 Inner products
* 3 See also
* 4 References

## 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 ∑g∈Gf(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 ⟨ϕ,ψ⟩=1|G|∑g∈Gϕ(g)ψ(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 G—for instance, if K is algebraically closed, then the irreducible characters form an orthonormal basis.

When G is a compact group and K = C is the field of complex numbers, the Haar measure can be applied to replace the finite sum above with an integral: ⟨ϕ,ψ⟩=∫Gϕ(t)ψ(t)‾dt. In this setting, the irreducible characters form a Hilbert basis of the Hilbert space of square-integrable class functions, by the Peter–Weyl theorem.

When K is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.

## See also

* Brauer's theorem on induced characters

## References

* Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, Berlin, 1977.

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