Given a crystallographic reduced root system and an element γ of the lattice generated by the roots, we study the minimum number |γ|, called the length of γ , of roots needed to express γ as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots. The map γ → |γ| 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 A and C.
Root polytope and partitions
CHIRIVI', Rocco
2014-01-01
Abstract
Given a crystallographic reduced root system and an element γ of the lattice generated by the roots, we study the minimum number |γ|, called the length of γ , of roots needed to express γ as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots. The map γ → |γ| 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 A and C.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
