Induction from Subgroups

Induced representations provide a systematic method to construct representations of a group from representations of its subgroups, forming a fundamental tool in representation theory.

DefinitionInduced Representation

Let $H$ be a subgroup of $G$ and $(\rho, W)$ a representation of $H$ over a field $\mathbb{F}$ . The induced representation $\text{Ind}_H^G(W)$ is the space of functions $f: G \to W$ satisfying: $f(hg) = \rho(h)(f(g)) \text{ for all } h \in H, g \in G$

The group $G$ acts by right translation: $(\pi(g)f)(x) = f(xg) \text{ for } g, x \in G$

This is a representation of $G$ on the $\mathbb{F}$ -vector space $\text{Ind}_H^G(W)$ .

Equivalently, if we choose a set of left coset representatives $\{g_1, \ldots, g_n\}$ for $H$ in $G$ (so $G = \bigsqcup_{i=1}^n g_i H$ ), then: $\text{Ind}_H^G(W) \cong \bigoplus_{i=1}^n W_i$ where each $W_i \cong W$ as vector spaces, and the action is determined by how $G$ permutes cosets and how $H$ acts on each copy of $W$ .

ExampleBasic Examples

Trivial subgroup: If $H = \{e\}$ and $W = \mathbb{F}$ (trivial representation of $H$ ), then $\text{Ind}_{\{e\}}^G(\mathbb{F}) \cong \mathbb{F}[G]$ is the regular representation.

Permutation representation: If $H \leq G$ and $W$ is the trivial representation of $H$ , then $\text{Ind}_H^G(\mathbb{F})$ is the permutation representation on the coset space $G/H$ .

$S_3$ from $S_2$ : Let $H = S_2 \subset S_3$ (the subgroup fixing element 3). Inducing the trivial representation of $H$ gives a 3-dimensional representation of $S_3$ (the permutation representation on $\{1,2,3\}$ ).

TheoremDimension Formula

If $G$ is finite and $[G:H] = n$ , then: $\dim \text{Ind}_H^G(W) = [G:H] \cdot \dim W = n \cdot \dim W$

Proof: Choose $n$ coset representatives $g_1, \ldots, g_n$ . Each function $f$ is determined by its values $f(g_i) \in W$ , giving an isomorphism $\text{Ind}_H^G(W) \cong W^n$ as vector spaces.

DefinitionRestriction

The dual operation to induction is restriction. For a representation $V$ of $G$ , the restriction $\text{Res}_H^G(V)$ is simply $V$ viewed as an $H$ -representation via the inclusion $H \hookrightarrow G$ .

This forms an adjoint pair with induction (Frobenius reciprocity).

Remark

Induction is a fundamental construction that:

Builds large representations from small ones
Provides a systematic method to construct all irreducibles (in many cases)
Satisfies remarkable reciprocity properties (Frobenius reciprocity)
Generalizes to compact groups via integration over coset spaces
Connects to important concepts in number theory (automorphic induction) and physics (gauge theory)

The philosophy: to understand representations of $G$ , study representations of subgroups $H$ and use induction to "lift" them to $G$ .