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class function

{{distinguish|text=a class function in set theory}}

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 {{mvar|G}} with values in a field {{mvar|K}} form a {{mvar|K}}-vector space. If {{mvar|G}} is finite and the characteristic of the field does not divide the order of {{mvar|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 {{math|{{!}}G{{!}}}} denotes the order of {{mvar|G}} and the overbar denotes conjugation in the field {{mvar|K}}. The set of irreducible characters of {{mvar|G}} forms an orthogonal basis. Further, if {{mvar|K}} is a splitting field for {{mvar|G}}{{--}}for instance, if {{mvar|K}} is algebraically closed, then the irreducible characters form an orthonormal basis.

When {{mvar|G}} is a compact group and {{math|K {{=}} C}} 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 {{mvar|K}} is the real numbers or the complex numbers, the inner product is a non-degenerate Hermitian bilinear form.

See also

References

Category:Group theory