Triangulations of root polytopes

Abstract

Let Φ be an irreducible crystallographic root system and P its root polytope, i.e., the convex hull of Φ. 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 group, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.Definition 5.1. Let J ⊆ I. We say that J is bipartite if it has an initial section J i , and a final section J f such thatIf the above conditions hold, we say that (3) is called a separating hyperplane, for the bipartitionNote that, by definition, if J has a proper bipartition, then it has at least two elements. If J is also saturated, then it contains two crossing pairs and these provide at least three elements in J. The definition also implies that, if {J i , J f } is a bipartition of J, then J i J f and J f J i are an initial and a final section of J, respectively, since the complement of an initial section is a final section, and vice-versa, and we havewhere by we denote disjoint union. Finally, we note that if J is saturated, also all the subsetsDefinition 5.2. For each subset S of Φ + , we define the restricted relations S and ∼ S on S as follows. For all β 1 , β 2 ∈ S we set: (1) β 1 S β 2 if there exists a middle pair {γ 1 , γ 2 } between β 1 and β 2 contained in S;Obviously, for any S ⊆ Φ + and β 1 , β 2 ∈ S, the relation β 1 S β 2 implies β 1 β 2 . Hence, if S is ∼closed, then, for all β 1 , β 2 ∈ S, we have β 1 ∼ β 2 if and only if β 1 ∼ S β 2 .The first of following lemmas is clear, hence we omit the proof.Lemma 5.3. Let S ⊆ Φ + . If S is saturated, then S is ∼closed.Lemma 5.4. Let I be an abelian ideal in Φ, Ψ a root subsystem of Φ, and Ψ 1 , . . . , Ψ k be the irreducible components of Ψ. Moreover, let J be a ∼closed subset of Φ such that J ⊆ I ∩ Ψ, and let R ⊆ J. Then, R is reduced in Φ if and only if R ∩ Ψ i is reduced in Ψ i for all i ∈ {i, . . . , k}.