Components of V(ρ)\otimes V(ρ)

Abstract

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $\hat{I}»$ be a dominant integral weight. Then, $V(\hat{I}»)$ is a component of $V(\ddot{I})\otimes V(\ddot{I})$ if and only if $\hat{I}»\le 2\ddot{I}$ 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.

7 pages