Group representation

Reducibility of representations

A (unitary) representation Γ :𝐺 →GL(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 rep
• completely reducible iff Γ can be written as the direct sum of irreps. 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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