In representation theory, a subrepresentation of a representation ( π , V ) {\displaystyle (\pi ,V)} of a group G is a representation ( π | W , W ) {\displaystyle (\pi |_{W},W)} such that W is a vector subspace of V and π | W ( g ) = π ( g ) | W {\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 ( π , V ) {\displaystyle (\pi ,V)} is a representation of G, then there is the trivial subrepresentation:
V G = { v ∈ V ∣ π ( g ) v = v , g ∈ G } . {\displaystyle V^{G}=\{v\in V\mid \pi (g)v=v,\,g\in G\}.}If f : V → W {\displaystyle f:V\to W} is an equivariant map between two representations, then its kernel is a subrepresentation of V {\displaystyle V} and its image is a subrepresentation of W {\displaystyle W} .
- Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.