Definition:Induced Representation

Introduction
Let $(G,\cdot)$ be a group.

Let $(K,+,\cdot)$ be a field.

Let $H$ be a subgroup of $G$.

Let $(W,\pi)$ be a representation of $H$ over $K$.

(Group Actions)
Let $X=\left\lbrace x_1,x_2,\ldots,x_n \right\rbrace$ be a representative set of $G/H$ (i.e. $X$ contains exactly one element from each element of $G/H$).

Define the vector space:


 * $V = \displaystyle \operatorname{\bigoplus}\limits_{\mathop x \in X} (x\otimes W)$

Notice that $X$ can be realized as a basis for $K[G/H]$.

Fix $g \in G$.

Since the $x_i$ are distinct, $\exists j \in S_n$ such that


 * $\forall x_i \in X \exists! h_i \in H: g x_i = x_{j(i)}h_i$

Notice that the $j$ is unique since $g x_i$ can only belong to one Coset this formula works for all $i$.

Thus the $h_i$ is unique since


 * $h_i = x_{j(i)} ^{-1} g x_i$

For all $\vec{w}_i \in W$ define


 * $\displaystyle \rho_{g}(\sum\limits_{\mathop i=1}^n x_i\vec{w}_i):= \sum\limits_{\mathop i=1}^n x_{j(i)} \pi_{h_i} (\vec{w}_i)$

Then equip $V$ with


 * $\rho:G\to \operatorname{Aut}(V),g\mapsto \rho_{g}$

Definition ($K[G]$-module)
Consider $W$ as a $K[H]$-module.

Then define


 * $\displaystyle \operatorname{Ind}_{H}^{G} W := K[G] \otimes_{K[H]} W$

Define the action by $K[G]$ as the following:

Suppose $\vec{u},\vec{v} \in K[G]$, $\vec{w} \in W$.

Then $\vec{u} (\vec{v} \otimes \vec{w}) = \vec{u}\vec{v} \otimes \vec{w}$