class function

A class function is a function on a group that cannot tell apart conjugate elements — it gives the same value to g and to hgh^{-1}. Such functions are exactly the ones that 'respect symmetry' in the deepest sense, and characters are the prototypical examples. The space of class functions is the natural arena in which all of character theory takes place.

Formally, a class function on G with values in a field k (usually C) is a function f: G -> k that is constant on conjugacy classes: f(hgh^{-1}) = f(g) for all g, h in G. Equivalently, f is a function on the set of conjugacy classes of G. For a finite group the class functions over C form a vector space whose dimension equals the number of conjugacy classes, with a natural basis given by the indicator functions of the classes.

The central theorem is that the irreducible characters form an orthonormal basis of this space of class functions (over C, for finite G). Hence every class function is a unique linear combination of irreducible characters; those combinations with nonnegative-integer coefficients are exactly the characters of genuine representations (called virtual characters when the coefficients are allowed to be negative integers). This is why character theory reduces representation problems to finite-dimensional linear algebra.