Components of V(ρ)⊗ V(ρ)

Original · EN

Kostant asked the following question: Let g be a simple Lie algebra over the complex numbers. Let λ be a dominant integral weight. Then, V(λ) is a component of V(ρ)⊗ V(ρ) if and only if λ≤ 2ρ under the usual Bruhat-Chevalley order on the set of weights. We give an affirmative answer to this question up to a saturation factor. In particular, the question is answered affirmatively for the special linear Lie algebras due to the Saturation Theorem of Knutson-Tao.