Induced representation

A construction Ind_H^G that extends a representation of a subgroup H to a representation of the whole group G.

Induced representation

Definition (finite groups)

Let $G$ be a finite group, $H\le G$ a subgroup, and $\sigma:H\to \mathrm{GL}(W)$ a finite-dimensional complex representation of $H$ .

The induced representation $\mathrm{Ind}_H^G \sigma$ is the representation of $G$ on the vector space

\mathrm{Ind}_H^G W =\Bigl\{\, f:G\to W \ \Bigm|\ f(hg)=\sigma(h)\,f(g)\ \text{for all }h\in H,\ g\in G \Bigr\},

with $G$ -action given by right translation:

(g_0\cdot f)(g)=f(gg_0)\qquad (g_0,g\in G).

This space is finite-dimensional, and one has the dimension formula

\dim(\mathrm{Ind}_H^G W)=[G:H]\cdot \dim(W),

where $[G:H]$ is the index of $H$ in $G$ .

Equivalent module-theoretic description

Using the group algebra , induction can be realized as

\mathrm{Ind}_H^G W \;\cong\; \mathbb{C}[G]\otimes_{\mathbb{C}[H]} W,

where $\mathbb{C}[G]$ is viewed as a $(\mathbb{C}[G],\mathbb{C}[H])$ -bimodule and $\otimes$ is the tensor product over $\mathbb{C}[H]$ .

Relationship with restriction (Frobenius reciprocity)

Let $\mathrm{Res}_H^G V$ denote the restricted representation of a $G$ -representation $V$ to $H$ . Then induction is left adjoint to restriction: there is a natural vector space isomorphism

\mathrm{Hom}_G(\mathrm{Ind}_H^G W,\; V)\;\cong\;\mathrm{Hom}_H(W,\; \mathrm{Res}_H^G V),

where $\mathrm{Hom}$ denotes module homomorphisms in the appropriate categories.

Examples

Example 1: Inducing the trivial representation gives a permutation representation

Let $\sigma$ be the trivial $1$ -dimensional representation of $H$ . Then $\mathrm{Ind}_H^G \sigma$ is naturally isomorphic to the permutation representation of $G$ on the set of left cosets $G/H$ .

Special case: if $H=\{e\}$ , then $G/H\cong G$ and $\mathrm{Ind}_{\{e\}}^G \mathbf{1}$ is the regular representation .

Example 2: $S_3$ induced from an $S_2$ subgroup

Let $G=S_3$ and $H\cong S_2$ be the stabilizer of $3$ (so $[G:H]=3$ ). Induce the trivial representation of $H$ :

\mathrm{Ind}_{S_2}^{S_3} \mathbf{1}.

This is the $3$ -dimensional permutation representation of $S_3$ on $\{1,2,3\}$ . It decomposes as

\mathrm{Ind}_{S_2}^{S_3}\mathbf{1}\ \cong\ \mathbf{1}\ \oplus\ \rho_{\mathrm{std}},

i.e. a $1$ -dimensional invariant subspace (spanned by $(1,1,1)$ ) plus the $2$ -dimensional standard representation (compare complete reducibility ).

Example 3: Cyclic example $C_4$ induced from $C_2$

Let $G=C_4=\langle t\rangle$ and $H=\langle t^2\rangle\cong C_2$ . Induce the trivial character of $H$ . The induced representation has dimension $[C_4:C_2]\cdot 1 = 2$ .

Over $\mathbb{C}$ , $C_4$ has four $1$ -dimensional characters, and precisely two of them restrict trivially to $H$ : the trivial character and the character sending $t\mapsto -1$ . Accordingly,

\mathrm{Ind}_{C_2}^{C_4}\mathbf{1}\ \cong\ \mathbf{1}\ \oplus\ \chi,

where $\chi(t)=-1$ . This illustrates how induction in abelian groups often decomposes as a direct sum of characters extending the given subgroup character.