Root polytopes and Abelian ideals

Abstract

We study the root polytope script P signΦ 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 script P signΦ and analyze its relation with the facets of script P signΦ. 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 script P signΦ. We show that this triangulation restricts to a triangulation of the positive root polytope script P signΦ+. © 2013 Springer Science+Business Media New York.