Definition:Reducible Linear Representation

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

$\rho$ is reducible iff there exists $W$ a proper subspace 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 $\left\{{\rho \left({g}\right): g \in G}\right\}$.

Irreducible Linear Representation
$\rho$ is called irreducible iff it is not reducible.

$G$-modules
In Equivalence of Representation Definitions, it is shown that representations and $G$-modules are bijective. Then a $G$-modules is said irreducible iff the correspondent representation is irreducible.