Subrepresentation

Definition

Let $(V,\rho)$ be a group representation of a group $G$ over a field $k$. A subrepresentation of $V$ is a $k$-subspace $W\subseteq V$ such that

Equivalently, $W$ is a $G$-invariant subspace of $V$. In that case, restricting $\rho$ gives a representation $(W,\rho|_W)$.

In the group algebra viewpoint, subrepresentations are exactly $k[G]$-submodules.

Basic properties

• The inclusion $i:W\hookrightarrow V$ is a homomorphism of representations (a $G$-equivariant linear map ).
• If $W$ is a subrepresentation, one can form the quotient vector space $V/W$, which carries an induced $G$-action by $\overline{v}\mapsto \overline{\rho(g)v}$ (well-defined precisely because $W$ is $G$-stable).

Subrepresentations are the objects whose absence (except $0$ and $V$) defines irreducibility .

Examples

1. Permutation representation of $S_3$ on $k^3$.
   Let $V=k^3$ with $S_3$ acting by permuting coordinates:

   $$\rho(\sigma)(e_i)=e_{\sigma(i)}.$$

   Then the line

   $$W_{\mathrm{triv}}=\mathrm{span}\{(1,1,1)\}$$

   is $S_3$-invariant, hence a subrepresentation (it is isomorphic to the trivial representation). Also the subspace

   $$W_{\mathrm{std}}=\{(x_1,x_2,x_3)\in k^3:\ x_1+x_2+x_3=0\}$$

   is invariant. When $\mathrm{char}(k)\nmid 3$, one has a direct sum decomposition

   $$k^3 = W_{\mathrm{triv}} \oplus W_{\mathrm{std}},$$

   exhibiting complete reducibility in this case (see completely reducible representation ).

2. A canonical 1-dimensional subrepresentation inside the regular representation.
   In the regular representation of $G$ on $k[G]$, the vector

   $$\Omega=\sum_{g\in G} g \in k[G]$$

   spans a $G$-stable line: for any $h\in G$,

   $$h\cdot \Omega = \sum_{g\in G} hg = \sum_{g\in G} g = \Omega.$$

   Thus $k\Omega$ is a subrepresentation isomorphic to the trivial representation.

3. An invariant line without an invariant complement (modular phenomenon).
   Let $G=C_p=\langle g\rangle$ and $k=\mathbb{F}_p$. Define a 2-dimensional representation on $V=k^2$ by

   $$\rho(g)=\begin{pmatrix}1&1\\0&1\end{pmatrix}.$$

   Then $W=\mathrm{span}\{e_1\}$ is $G$-stable since $\rho(g)e_1=e_1$. This is a subrepresentation, but (as discussed in complete reducibility ) it has no $G$-stable complement in $V$.

See also: irreducible representation , restriction to a subgroup (different notion).