In representation theory, a subrepresentation of a representation
of a group G is a representation
such that W is a vector subspace of V and
.
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
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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f399be6349f94a1d1ae3bd831106bd1bf6de2471)
If
is an equivariant map between two representations, then its kernel is a subrepresentation of
and its image is a subrepresentation of
.
References