Subrepresentation

In representation theory, a subrepresentation of a representation ${\displaystyle (\pi ,V)}$ of a group G is a representation ${\displaystyle (\pi |_{W},W)}$ such that W is a vector subspace of V and ${\displaystyle \pi |_{W}(g)=\pi (g)|_{W}}$.

A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations.

If ${\displaystyle (\pi ,V)}$ is a representation of G, then there is the trivial subrepresentation:

${\displaystyle V^{G}=\{v\in V\mid \pi (g)v=v,\,g\in G\}.}$