Subrepresentation Explained

(\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