Definition:Induced Representation

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Introduction

Let $\left({G, \circ}\right)$ be a group.

Let $\left({K, +, \times}\right)$ be a field.

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

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


(Group Actions)

Let $X = \left\{ {x_1, x_2, \ldots, x_n}\right\}$ be a representative set of $G /H$.

That is, such that $X$ contains exactly one element from each element of $G / H$.

Let the vector space $V$ be defined as:

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

Notice that $X$ can be realized as a basis for $K \left[{G / H}\right]$.

Fix $g \in G$.

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

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

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:

$h_i = x_{j \left({i}\right)}^{-1} g x_i$

For all $\vec w_i \in W$ define:

$\displaystyle \rho_g \left({\sum \limits_{i \mathop = 1}^n x_i\vec{w}_i}\right) := \sum\limits_{i \mathop = 1}^n x_{j \left({i}\right)} \pi_{h_i} \left({\vec w_i}\right)$

Then equip $V$ with:

$\rho: G \to \operatorname {Aut} \left({V}\right), g \mapsto \rho_g$


Definition ($K \left[{G}\right]$-module)

Consider $W$ as a $K \left[{H}\right]$-module.

Then define:

$\displaystyle \operatorname{Ind}_H^G W := K \left[{G}\right] \otimes_{K \left[{H}\right]} W$

Define the action by $K \left[{G}\right]$ as the following:

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

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