Components of V(\rho)\otimes V(\rho)

TL;DR

This paper investigates the components of tensor products of representations of simple Lie algebras, providing an affirmative answer to Kostant's question up to a saturation factor, especially for special linear Lie algebras.

Contribution

It offers a partial resolution to Kostant's question on tensor product components, extending the result with a saturation factor and confirming it for special linear Lie algebras.

Findings

Confirmed Kostant's question up to a saturation factor.

Established the result for special linear Lie algebras using the Saturation Theorem.

Extended the understanding of tensor product components in representation theory.

Abstract

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