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Subrepresentation

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Source updated December 24, 2023

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

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

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

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

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This article is adapted from the Wikipedia article “Subrepresentation”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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