Definition:Irreducible (Representation Theory)
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.
Write the above in more rigorous language: either "they are equivalent" (in the sense that there exists a bijection between them) or "there exists a bijection between them".
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