Definition:Irreducible (Representation Theory)/Linear Representation

Definition

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

Then $\rho$ is irreducible if and only if it is not reducible.

That is, if and only if there exists no non-trivial proper vector subspace $W$ of $V$ such that:

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