The induced $R$-linear $G$-module ${\mathrm{Ind}}_H^{G}W$ as $H$-covariant functions on $G$

Let $R$ be a commutative ring, let $G$ be a group, let $H\le G$ be a subgroup (Subgroup), and let $W$ be an $R$-linear $H$-module (An $R$-linear action of $G$ on a left $R$-module, and a $G$-module over $R$).

The induced $R$-linear $G$-module is the set

${\mathrm{Ind}}_H^{G}W:=\{f\in W^G:f(gh)=h^{-1}\cdot f(g)\text{ for all }g\in G,h\in H\},$

where $W^G$ is the set of all functions $G\to W$ (The set $B^A$ of all functions $A\to B$).

Pointwise addition and scalar multiplication make ${\mathrm{Ind}}_H^{G}W$ an $R$-module:

$(f+f^{\mathrm{\prime }})(g):=f(g)+f^{\mathrm{\prime }}(g),(rf)(g):=rf(g)\mathrm{.}$

The left action of $G$ on this module is

$(x\cdot f)(g):=f(x^{-1}g)(x,g\in G)\mathrm{.}$

This action is well defined on the displayed subset because

$(x\cdot f)(gh)=f(x^{-1}gh)=h^{-1}\cdot f(x^{-1}g)=h^{-1}\cdot (x\cdot f)(g),$

and each operator $f\mapsto x\cdot f$ is $R$-linear by the pointwise definitions. Thus ${\mathrm{Ind}}_H^{G}W$ is an $R$-linear $G$-module.

• The covariance condition is written on the right, so the values of an induced function are determined by one value on each left coset $gH$.

• When $R=k$ is a field and $W$ is finite-dimensional, this construction is the induced representation of $G$ from the representation of $H$ on $W$.