Subrepresentation
In representation theory, a subrepresentation of a representation [math]\displaystyle{ (\pi, V) }[/math] of a group G is a representation [math]\displaystyle{ (\pi|_W, W) }[/math] such that W is a vector subspace of V and [math]\displaystyle{ \pi|_W(g) = \pi(g)|_W }[/math]. 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 [math]\displaystyle{ (\pi, V) }[/math] is a representation of G, then there is the trivial subrepresentation:
- [math]\displaystyle{ V^G = \{ v \in V \mid \pi(g)v = v, \, g \in G \}. }[/math]
If [math]\displaystyle{ f: V \to W }[/math] is an equivariant map between two representations, then its kernel is a subrepresentation of [math]\displaystyle{ V }[/math] and its image is a subrepresentation of [math]\displaystyle{ W }[/math].
References
- Fulton, William; Harris, Joe (1991) (in en-gb). 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. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9.
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