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.