Class function

Definition

Symbol-free definition

A class function on a group is defined as a function (to any set) that takes the same value on any two conjugate elements. Equivalently, it is a function on the group that is constant on conjugacy classes, and hence descends to a function from the set of conjugacy classes.

Definition with symbols

A class function on a group $G$ is a function $f$ from $G$ to some set $X$ such that $f(g)=f(hgh^{-1})$ for any $g,h\in G$.

Particular cases

Conjugacy classes of images are class functions

Let $\rho :G\to H$ be a homomorphism. Then the function that sends each $g$ to the conjugacy class of $\rho (g)$ is a class function. This follows from the fact that if two elements in $G$ are conjugate, their images in $H$ are also conjugate.

Characters of linear representations are class functions

Further information: Character

For any linear representation, the character of that linear representation, viz the map that sends each group element to the trace of the corresponding linear operator, is a class function. This follows from the fact that the character depends only on the conjugacy class of the linear operator corresponding to the group element.

The vector space of class functions

Let ${\mathcal {C}}_{G}$ be the set of class functions of $G$ onto a field $K$. Then ${\mathcal {C}}_{G}$ is a $K$-vector space. Suppose $G$ has conjugacy classes ${\mathcal {O}}_{1},\dots ,{\mathcal {O}}_{r}$. One such basis of ${\mathcal {C}}_{G}$ is the basis of the indicator functions of each of the conjugacy classes $\{\mathbf {1} _{{\mathcal {O}}_{i}}:1\leq i\leq r\}$.

If $K=\mathbb {C}$ we can define an inner product by $\langle f_{1},f_{2}\rangle _{G}={\frac {1}{|G|}}\sum _{g\in G}{\overline {f_{1}(g)}}f_{2}(g)$, where ${\bar {z}}$ denotes the complex conjugate of $z$.