Subrepresentations, direct sums of representations, and irreducibility

Let $\rho :G\to \mathrm{GL}(V)$ be a finite-dimensional representation of $G$ over a field $k$ (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree).

A subrepresentation is a linear subspace $W\le V$ (Linear subspace of a vector space) such that $\rho (g)(W)\subseteq W$ for every $g\in G$. Equivalently, $g\cdot w\in W$ for all $g\in G$ and $w\in W$.

If $\sigma :G\to \mathrm{GL}(W)$ is another representation, the direct sum representation $\rho \oplus \sigma$ on $V\oplus W$ is defined by $g\cdot (v,w):=(\rho (g)v,\sigma (g)w)\mathrm{.}$

The representation $\rho$ is irreducible when $V\mathrm{\ne }0$ and its only subrepresentations are $0$ and $V$ itself.

• The zero representation on the zero vector space is not irreducible, by the explicit nonzero requirement.

• A decomposition $V=U\oplus W$ into nonzero subrepresentations is exactly a nontrivial decomposition of the representation as a direct sum.