Subrepresentation

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

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

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

If $f:V\to W$ is an equivariant map between two representations, then its kernel is a subrepresentation of $V$ and its image is a subrepresentation of $W$.