# Definition:Reducible Linear Representation

## Definition

Let $\rho: G \to \GL V$ be a linear representation.

$\rho$ is reducible if and only if there exists a non-trivial proper vector subspace $W$ of $V$ such that:

$\forall g \in G: \map {\map \rho g} W \subseteq W$

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