class function

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 $\sum_ f(g) g$.

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 $\langle \phi , \psi \rangle = \frac \sum_ \phi(g) \psi(g^)$ 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: $\langle \phi, \psi \rangle = \int_G \phi(t) \psi(t^{-1})\, dt.$

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

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.

Group theory