[[Group representation]]
# Reducibility of representations

A ([[Unitary representation|unitary]]) representation $\Gamma : G \to \mathrm{GL}(V)$ is

- **irreducible** iff $V$ has no non-trivial $\Gamma$-[[Invariant subspace]]. See [[Irrep]]
- **reducible** iff $V$ has at least one non-trivial $\Gamma$-[[Invariant subspace]], thus carrying a **subrepresentation** #m/def/rep
- **completely reducible** iff $\Gamma$ can be written as the [[Direct sum of representations|direct sum]] of irreps. #m/def/rep


## Properties

- A unitary representation is an irrep iff it is not the direct sum of nontrivial representations.
- [[Character irreducibility criterion]]

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