On a convexity property of tensor products of irreducible, rational representations of $SL(n)$

TL;DR

This paper reveals a convexity property related to the support of highest weights in tensor products of irreducible rational representations of SL(n), based on the convexity of the saturation cone and the saturation conjecture.

Contribution

It identifies a new convexity property of the support of highest weights in tensor products of irreducible representations of SL(n).

Findings

Support of highest weights forms a convex set.

Convexity follows from saturation cone properties.

Saturation conjecture validity underpins the result.

Abstract

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 $SL(n)$ 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 $SL(n)$.