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 $$W$$.