Invariant subspace for a linear representation

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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 $V$.