Invariant subspace for a linear representation

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This article gives a basic definition in the following area: linear representation theory
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Definition

Definition with symbols

Let G be a group and \rho: G \to GL(V) be a linear representation of G. An invariant subspace for \rho is a linear subspace W of V such that for any x \in W, we have \rho(x) \in W.

Given an invariant subspace of V for \rho, we can define a subrepresentation of \rho on this invariant subspace (in fact, subrepresentations correspond to invariant subspaces. In other words, we can define an action of G on W be restricting the action on V.