Class function

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 ${\displaystyle \sum _{g\in G}f(g)g}$.

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 ${\displaystyle \langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{g\in G}\phi (g)\psi (g^{-1})}$ 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: ${\displaystyle \langle \phi ,\psi \rangle =\int _{G}\phi (t)\psi (t^{-1})\,dt.}$

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

• Brauer's theorem on induced characters

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