Polar Root Polytopes that are Zonotopes

Let $\mathcal P_Φ$ be the root polytope of a finite irreducible crystallographic root system $Φ$, i.e., the convex hull of all roots in $Φ$. The polar of $\mathcal P_Φ$, denoted $\mathcal P_Φ^*$, 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 $\mathcal P_Φ^*$ are zonotopes and which are not. The proof is constructive.

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