Class function

In mathematics, a class function in group theory 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 we identify a class function f 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. 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. In the case that K has characteristic 0, the set of irreducible characters of G forms an orthogonal basis. If K is algebraically closed, we can say further that they form an orthonormal basis.

In the case of a compact group, the notion of Haar measure allows one to replace the finite sum above with an integral: $\langle \phi ,\psi \rangle =\int _{G}\phi (t)\psi (t^{-1})\,dt$.

References

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