Root polytopes and abelian ideals

Abstract

We study the root polytope ${\mathcal{P}}_{\Phi }$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of ${\mathcal{P}}_{\Phi }$ and analyze its relation with the facets of ${\mathcal{P}}_{\Phi }$. For $\Phi$ 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 ${\mathcal{P}}_{\Phi }$. We show that this triangulation restricts to a triangulation of the positive root polytope ${\mathcal{P}}_{\Phi }^{+}$.

41 pages, revised version, accepted for publication in Journal of Algebraic Combinatorics

References in corpus (6)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Root polytopes and Borel subalgebras
• Abelian ideals of a Borel subalgebra and long positive roots
• On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces
• Root polytope and partitions
• A curious identity and the volume of the root spherical simplex

Cited by in corpus (6)

• Root polytopes and Borel subalgebras
• Fundamental polytopes of metric trees via parallel connections of matroids
• Polar Root Polytopes that are Zonotopes
• Cross-section Lattices of $\mathcal{J}$-irreducible Monoids and Orbit Structures of Weight Polytopes
• Root polytope and partitions
• Root systems, affine subspaces, and projections