Triangulations of root polytopes

TL;DR

This paper presents a uniform method to triangulate the facets of root polytopes associated with irreducible crystallographic root systems, ensuring unimodularity within each Weyl group orbit.

Contribution

It introduces a universal construction for facet triangulations of root polytopes applicable to all root types, with unimodularity properties under Weyl group actions.

Findings

Triangulations are uniform across all root types.

Triangulations are unimodular within each Weyl orbit.

Construction applies to all irreducible crystallographic root systems.

Abstract

Let $\Phi$ be an irreducible crystallographic root system and $\mathcal P$ its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of $\mathcal P$. 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.