# Definition:Irreducible (Representation Theory)

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## Definition

### Linear Representation

Let $\rho: G \to \GL V$ 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:

- $\forall g \in G: \map {\map \rho g} W \subseteq W$

### 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.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: 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".You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |