Restricted representation

Let $G$ be a group, $H\le G$ a subgroup, and let $(V,\rho)$ be a (finite-dimensional) representation of $G$ over a field $k$, i.e. a homomorphism

Definition (restriction)

The restricted representation of $(V,\rho)$ from $G$ to $H$ is the representation

where $\rho|_H: H\to \mathrm{GL}(V)$ is the composite $H \hookrightarrow G \xrightarrow{\rho} \mathrm{GL}(V)$.

Equivalently, if $V$ is a $kG$-module (via the group algebra $k[G]$), then $\mathrm{Res}^G_H V$ is the same vector space $V$ regarded as a $kH$-module by restricting scalars along the inclusion $k[H]\hookrightarrow k[G]$.

• Any $G$-subrepresentation is in particular an $H$-subrepresentation after restriction.
• If $k=\mathbb C$ and $\chi_V$ is the character of $V$, then the character of $\mathrm{Res}^G_H V$ is simply the pointwise restriction: $\chi_{\mathrm{Res}^G_H V}(h)=\chi_V(h)\quad (h\in H).$

Restriction is a functor $\mathrm{Res}^G_H:\mathrm{Rep}_k(G)\to \mathrm{Rep}_k(H)$, and it pairs naturally with induction (see Frobenius reciprocity).

Examples

1. Restricting the sign representation $S_3\to A_3$.
   Let $G=S_3$, $H=A_3\cong C_3$. The 1-dimensional sign representation $\mathrm{sgn}:S_3\to\{\pm 1\}\subset \mathbb C^\times$ becomes trivial on $A_3$ (every element of $A_3$ is even). Hence

   $$\mathrm{Res}^{S_3}_{A_3}(\mathrm{sgn}) \cong \mathbf{1}_{A_3}.$$

2. Restricting the standard 2D representation $S_3\to A_3$.
   Let $V$ be the 2-dimensional irreducible (standard) representation of $S_3$ over $\mathbb C$. Restricting to $A_3=\langle (123)\rangle$, the element $(123)$ acts as a rotation of order 3 on $V$. Over $\mathbb C$, this restriction splits as a direct sum of the two nontrivial 1-dimensional characters of $C_3$:

   $$\mathrm{Res}^{S_3}_{A_3}(V)\;\cong\;\chi_\omega \oplus \chi_{\omega^2},$$

   where $\omega=e^{2\pi i/3}$.

3. Restricting the regular representation to a subgroup.
   Let $G=S_3$ and let $H=\langle (12)\rangle\cong C_2$. Consider the left regular representation $k[G]$ with basis $\{e_g:g\in G\}$ and action $h\cdot e_g=e_{hg}$. Under restriction to $H$, the set $G$ decomposes into 3 left cosets of $H$, each of size 2. Each coset is an $H$-orbit isomorphic (as an $H$-set) to $H$ acting on itself by left translation, so as $kH$-modules:

   $$\mathrm{Res}^{S_3}_{H}\big(k[G]\big)\;\cong\;k[H]\oplus k[H]\oplus k[H].$$