Subrepresentation

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

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

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

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

References

• 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.