Class function

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.