Definition:Reducible Linear Representation

Definition
Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation.

$\rho$ is reducible iff there exists a non-trivial proper vector subspace $W$ of $V$ such that:
 * $\forall g \in G: \rho \left({g}\right) \left({W}\right) \subseteq W$

That is, such that $W$ is invariant for every linear operator in the set $\left\{{\rho \left({g}\right): g \in G}\right\}$.

Also see

 * Irreducible Linear Representation: a linear representation which is not reducible.