Induced representation

Equivalently, $\operatorname{Ind}_H^G W$ can be realized on

with action

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

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

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 $H$:

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

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,

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