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.
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 <math> \sum_{g \in G} f(g) g</math>.
Inner products
The set of class functions of a group with values in a field form a -vector space. If is finite and the characteristic of the field does not divide the order of , then there is an inner product defined on this space defined by <math>\langle \phi , \psi \rangle = \frac{1}{|G|} \sum_{g \in G} \phi(g) \overline{\psi(g)},</math> where denotes the order of and the overbar denotes conjugation in the field . The set of irreducible characters of forms an orthogonal basis. Further, if is a splitting field for for instance, if is algebraically closed, then the irreducible characters form an orthonormal basis.
When 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: <math>\langle \phi, \psi \rangle = \int_G \phi(t) \overline{\psi(t)}\, dt.</math> 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 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.