\begin{definition}[Definition:Induced Representation]
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:
:$V = K[G/H] \bigotimes_{H} W \cong \displaystyle \operatorname{\bigoplus}\limits_{\mathop x \in X} (x\otimes W)$
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
:$\displaystyle \rho_{g}(\sum\limits_{\mathop i=1}^n x_i \otimes\vec{w}_i):= \sum\limits_{\mathop i=1}^n x_{j(i)} \otimes \pi_{h_i} (\vec{w}_i)$
Then equip $V$ with 
:$\rho:G\to \operatorname{Aut}(V),g\mapsto \rho_{g}$
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\end{definition}