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

Inner products

The set of class functions of a group Template:Mvar with values in a field Template:Mvar form a Template:Mvar-vector space. If Template:Mvar is finite and the characteristic of the field does not divide the order of Template:Mvar, then there is an inner product defined on this space defined by $⟨\varphi ,\psi ⟩=\frac{1}{|G|}\sum _{g\in G}\varphi (g)\overline{\psi (g)},$ where Template:Math denotes the order of Template:Mvar and the overbar denotes conjugation in the field Template:Mvar. The set of irreducible characters of Template:Mvar forms an orthogonal basis. Further, if Template:Mvar is a splitting field for Template:MvarTemplate:--for instance, if Template:Mvar is algebraically closed, then the irreducible characters form an orthonormal basis.

When Template:Mvar is a compact group and Template:Math is the field of complex numbers, the Haar measure can be applied to replace the finite sum above with an integral: $⟨\varphi ,\psi ⟩=\underset{G}{\int }\varphi (t)\overline{\psi (t)}dt.$

When Template:Mvar 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.