Irreducible operators applied to an irreducible orthonormal basis

Let $𝑈 : 𝐺 \to G L (𝑉)$ be a unitary representation with its decomposition into irreps. Let ${O 𝜇 𝑖} 𝑑 𝜇 𝑖 = 1$ be irreducible operators transforming in $Γ 𝜇$ and ${| 𝑒 𝜈 𝛼 𝑗 ⟩} 𝑑 𝜈 𝑗 = 1$ be an irreducible orthonormal basis transforming in $Γ 𝜈$ . Then $O 𝜇 𝑖 | 𝑒 𝜈 𝛼 𝑗 ⟩$ transform under $𝐺$ in the product representation $Γ 𝜇 \otimes 𝜈$ (within the same carrier space)

𝑈 (𝑔) O 𝜇 𝑖 | 𝑒 𝜈 𝛼 𝑗 ⟩ = 𝑈 (𝑔) O 𝜇 𝑖 𝑈 (𝑔) † 𝑈 (𝑔) | 𝑒 𝜈 𝛼 𝑗 ⟩ = \sum 𝑘 O 𝑘 Γ 𝜇 𝑘 𝑖 (𝑔) \sum ℓ | 𝑒 𝜈 𝛼 ℓ ⟩ Γ 𝜈 ℓ 𝑗 = \sum 𝑘, ℓ O 𝑘 | 𝑒 𝜈 𝛼 ℓ ⟩ Γ 𝜇 𝑘 𝑖 (𝑔) Γ 𝜈 ℓ 𝑗

which may then be expanded in a decomposed reälization using the Clebsch-Gordan coëfficients.

O 𝜇 𝑖 | 𝑒 𝜈 𝛼 𝑗 ⟩ = | 𝑖, 𝑗 ⟩ = \sum 𝜆; 𝛽, ℓ | 𝛼, 𝜆, ℓ ⟩ ⟨ 𝛼, 𝜆, ℓ | 𝑖, 𝑗 ⟩ 𝜇 \otimes 𝜈