Root polytopes and Borel subalgebras

Abstract

Let $\Phi$ be a finite crystallographic irreducible root system and ${\mathcal{P}}_{\Phi }$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope ${\mathcal{P}}_{\Phi }$, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.

revised version, accepted for publication in IMRN

References in corpus (4)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Root polytopes and abelian ideals
• Polar Root Polytopes that are Zonotopes
• Root polytope and partitions

Cited by in corpus (10)

• Root polytopes and abelian ideals
• Faces and maximizer subsets of highest weight modules
• Standard parabolic subsets of highest weight modules
• Faces of highest weight modules and the universal Weyl polyhedron
• Polar Root Polytopes that are Zonotopes
• Cross-section Lattices of $\mathcal{J}$-irreducible Monoids and Orbit Structures of Weight Polytopes
• Root polytope and partitions
• Moving between weights of weight modules
• The Degree of a Tropical Root Surface of Type $A$
• Root systems, affine subspaces, and projections