Root polytope and partitions (Q2256054)

Consider the lattice \(L\) generated by the roots of a crystallographic reduced root system. Viewed as a monoid, \(L\) comes with a natural word-length function, namely the minimum number of roots needed to write \(x \in L\) as a sum of roots. The authors study this word-length function by way of the \textit{root polytope} \(P\) associated to \(L\), the convex hull of the roots. The main result expresses the word-length function in terms of the facets (i.e., codimension-1 faces) of \(P\).