Root polytope and partitions

Abstract

Given a crystallographic reduced root system and an element (Formula presented.) of the lattice generated by the roots, we study the minimum number (Formula presented.), called the length of (Formula presented.), of roots needed to express (Formula presented.) as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots. The map (Formula presented.) turns out to be the upper integral part of a piecewise-linear function with linearity domains the cones over the facets of this convex hull. In order to show this relation, we investigate the integral closure of the monoid generated by the roots in a facet. We study also the positive length, i.e., the minimum number of positive roots needed to write an element, and we prove that the two notions of length coincide only for the types (Formula presented.) and (Formula presented.).