Subrepresentation

This article gives a basic definition in the following area: linear representation theory
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This article describes a notion of representation, or a group action on a certain kind of object.
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Let $V$ be a vector space, $G$ be a group.

A subrepresentation of a representation $(\rho ,V)$ of a group $G$ is a representation $(\rho |_{W},W)$ of $G$ where $W\leq V$ is a vector subspace, and for all $g\in G,w\in W,\rho |_{W}(g)(w)\in W$.

A subrepresentation $W$ of a representation $V$ is called proper if $W\neq 0$ and $W\neq V$.

Related concepts

We say a representation is irreducible if it has no proper subrepresentations.

Results

Maschke's theorem shows that a finite-dimensional representation can be expressed as a direct sum of irreducible subrepresentations under a certain condition, namely that the field $k$ that the representation is over must have characteristic not dividing the order of $|G|$ (including characteristic zero).