Root polytopes and Abelian ideals

Abstract

We study the root polytope P Φ of a finite irreducible crystallographic root system Φ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system Φ. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of P Φ and analyze its relation with the facets of P Φ . For Φ of type A n or C n , we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of P Φ . We show that this triangulation restricts to a triangulation of the positive root polytope P + Φ .

Simion’s Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

“…In recent paper, Cellini and Marietti [6] used abelian ideals to produce a triangulation for various root polytopes. In the case of type A, their construction yields once again a lexicographic triangulation of each face.…”

Fundamental polytopes of metric trees via parallel connections of matroids

“…Then, by definition, the fundamental polytope of this space is the convex hull of the vectors e i,j = 1 i − 1 j , where i = j ∈ [n]. This is also called the root polytope of type A n−1 , and its face numbers have been computed via algebraic-combinatorial considerations by Cellini and Marietti [4,Proposition 4.3]. Of course, one could compute these numbers by computing the Möbius function of the corresponding matroid, i.e., the uniform matroid U n−1 n .…”

Root Polytopes and Borel Subalgebras

“…We call this the positive root polytope and, if confusion may arise, we call P Φ the complete root polytope. In this paper we only consider the complete root polytope; our results have direct applications to the study of the positive root polytopes of types A n and C n (see [4]). In [1], some properties of the complete root polytopes are provided for the classical types through a case by case analysis, using the usual coordinate descriptions of the root systems.…”