Class function

Let $G$ be a group and let $A$ be a set. A class function on $G$ with values in $A$ is a function $f\colon G\to A$ such that for all $x,g\in G$, $f(xgx^{-1})=f(g).$ Equivalently, $f$ is constant on each conjugacy class of $G$.

Class functions are precisely the functions invariant under the conjugation action of $G$ on itself. In representation theory, the character of a finite-dimensional complex representation is a fundamental example of a class function.

Examples:

• Any constant function $f(g)=a$ is a class function.
• Fix a conjugacy class $C\subseteq G$. The indicator function $1_C\colon G\to\{0,1\}$ defined by $1_C(g)=1$ if $g\in C$ and $0$ otherwise is a class function.
• In $S_n$, the sign map $\mathrm{sgn}\colon S_n\to\{\pm1\}$ is a class function (it depends only on the cycle type, hence is constant on conjugacy classes).