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.
Highlights
Let Φ be an irreducible crystallographic root system in a Euclidean space E, Φ+ a positive system of Φ, and W the Weyl group of Φ
The construction is formally equal for both root types, though the proofs are distinct and based on the special combinatorics of these two root systems and their maximal abelian ideals
The first main result in this paper is that the maximal reduced subsets in a facet ideal provide a triangulation of the corresponding standard parabolic facet
Summary
Let Φ be an irreducible crystallographic root system in a Euclidean space E, Φ+ a positive system of Φ, and W the Weyl group of Φ. The first main result in this paper is that the maximal reduced subsets in a facet ideal provide a triangulation of the corresponding standard parabolic facet. For each standard parabolic facet F of P, let IF be the corresponding facet ideal: IF = F ∩ Φ, and TF = {Conv(R) | R ⊆ IF , R maximal reduced }, where Conv(R) is the convex hull of R. The standard parabolic facets of P naturally correspond to the irreducible standard equal rank root subsystems of Φ [6]. The results of [6] imply that the facet ideal Iα (α ∈ Π, Φα irreducible), is an abelian nilradical (see Subsection 2.4) in the root subsystem Φ+α. This proof provides an algorithm for the explicit computation of the triangulations for each root type, which will be done in a future paper
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
