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 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:
\( {\displaystyle V^{G}=\{v\in V|\pi (g)v=v,\,g\in G\}.} \)
References
Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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