Triangulations of root polytopes

Abstract

Let Φ be an irreducible crystallographic root system and P its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of P. We also prove that, on each orbit of facets under the action of the Weyl gruop, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.