Group representation

Reducibility of representations

A (unitary) representation $Γ : 𝐺 \to G L (𝑉)$ is

irreducible iff $𝑉$ has no non-trivial $Γ$ -Invariant subspace. See Irrep
reducible iff $𝑉$ has at least one non-trivial $Γ$ -Invariant subspace, thus carrying a subrepresentation #m/def/rep
completely reducible iff $Γ$ can be written as the 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

#state/tidy | #lang/en | #SemBr