Definition:Induced Representation

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Introduction

Let $\struct {G, \circ}$ be a group.

Let $\struct {K, +, \times}$ be a field.

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

Let $\struct {W, \pi}$ be a representation of $H$ over $K$.

(Group Actions)

Let $X = \set {x_1, x_2, \ldots, x_n}$ 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:

$\ds V = \operatorname{\bigoplus}_{x \mathop \in X} \paren {x \otimes W}$

Notice that $X$ can be realized as a basis for $K \sqbrk {G / H}$.

Fix $g \in G$.

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

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

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:

$h_i = x_{\map j i}^{-1} g x_i$

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

$\ds \map {\rho_g} {\sum_{i \mathop = 1}^n x_i \vec w_i} := \sum_{i \mathop = 1}^n x_{\map j i} \map {\pi_{h_i} } {\vec w_i}$

Then equip $V$ with:

$\rho: G \to \Aut V, g \mapsto \rho_g$

Definition ($K \sqbrk G$-module)

Consider $W$ as a $K \sqbrk H$-module.

Then define:

$\ds \operatorname{Ind}_H^G W := K \sqbrk G \otimes_{K \sqbrk H} W$

Define the action by $K \sqbrk G$ as the following:

Suppose $\vec u,\vec v \in K \sqbrk G$, $\vec w \in W$.

Then $\vec u \paren {\vec v \otimes \vec w} = \vec u \vec v \otimes \vec w$

Sources

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