# Definition:Irreducible (Representation Theory)

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## Definition

### Linear Representation

Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation.

Then $\rho$ is **irreducible** if and only if it is not reducible.

That is, if and only if there exists **no** non-trivial proper vector subspace $W$ of $V$ such that:

- $\forall g \in G: \rho \left({g}\right) \left({W}\right) \subseteq W$

### G-Module

A $G$-module is **irreducible** if and only if the corresponding linear representation is irreducible.

## Also see

In Correspondence between Linear Group Actions and Linear Representations, it is shown that linear representations and $G$-modules are bijective.