Polar Root Polytopes that are Zonotopes

Abstract

Let ${\mathcal{P}}_{\Phi }$ be the root polytope of a finite irreducible crystallographic root system $\Phi$, i.e., the convex hull of all roots in $\Phi$. The polar of ${\mathcal{P}}_{\Phi }$, denoted ${\mathcal{P}}_{\Phi }^{\ast }$, coincides with the union of the orbit of the fundamental alcove under the action of the Weyl group. In this paper, we establishes which polytopes ${\mathcal{P}}_{\Phi }^{\ast }$ are zonotopes and which are not. The proof is constructive.

12 pages

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• Root systems, affine subspaces, and projections