Induced class function

Definition

Suppose $G$ is a group and $H$ is a subgroup of finite index in $G$. Suppose $f$ is a class function on $H$ (i.e., a function on $H$ that is constant on each conjugacy class of $H$). Then, the induced class function on $G$, denoted $\operatorname {Ind} _{H}^{G}(f)$ is defined by the following summation over a left transversal $S$ of $H$ in $G$:

$\!\operatorname {Ind} _{H}^{G}(f)=x\mapsto \sum _{s\in S}f_{0}(s^{-1}xs)$

where $f_{0}(g)$ is $f(g)$ if $g\in H$ and $0$ otherwise.

Note that this is well-defined (independent of $S$_ precisely because $f$ is a class function on $H$, so replacing $s$ by $sh,h\in H$, gives an element conjugate via $H$ and hence the same value of $f_{0}$.

Facts

• Character of induced representation equals induced class function from character: If $f$ is the character of a linear representation, then $\operatorname {Ind} _{H}^{G}(f)$ is the character of the induced representation from $H$ to $G$. it is often called the induced character from $f$.
• Similarly, if $f$ is a virtual character, the induced class function is also a virtual character.