On a Convexity Property of Tensor Products of Irreducible, Rational Representations of SL(n)

The aim of this note is to point out a convexity property with respect to the root lattice for the support of the highest weights that occur in a tensor product of irreducible rational representations of SL(n) (or more generally any Lie group over which the saturation property holds) over the complex numbers. The observation is a consequence of the convexity properties of the saturation cone and the validity of the saturation conjecture for SL(n).