Class function

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In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function f on a group G, such that f is constant on the conjugacy classes of G. In other words, f is invariant under the conjugation map on G. Such functions play a basic role in representation theory.

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 $\langle \phi ,\psi \rangle ={\frac {1}{|G|}}\sum _{{g\in G}}\phi (g)\psi (g^{{-1}})$ 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.

References

Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, Berlin, 1977.

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