Some tips on the decomposition of tensor product representations of compact connected lie groups

Let ?₁, ?₂, ?₃, ?₄ be irreducible representations of a compact connected semisimple Lie group G with highest weights Λ₁, Λ₂, Λ₃, Λ₄, respectively. Let ?₁?₂ (resp. ?₃?₄) be irreducible representations of G with highest weights Λ₁·Λ₂ (resp. Λ₃·Λ₄). It is assumed that one knows the Clebsch-Gordan series of ?₁??₃ and ?₂??₄ (resp. of S²?₁, S²?₂, A²?₁ and A²?₂). Then we formulate a result (Theorem 2) which gives information on the decomposition of ?₁?₂??₃?₄ (resp. of S²(?₁?₂) and A²(?₁?₂)). Though this result is not complete, it is useful because it delivers the information very quickly and in a very simple manner.