## Irreducible representations

Let $\rho:G\to \text{GL}(V)$ be a linear representation of $G$. We say that it is irreducible or simple if $V$ is not 0 and if no vector subspace of $V$ is stable under $G$, except of course 0 and $V$. This is equivalent to saying $V$ is not the direct sum of two representations, except for the trivial decomposition $V=0\oplus V$.