Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the singularities of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions e_i − e_j.

Finally we interpret Mizera’s formula for the biadjoint scalar amplitude m(𝕀_n, 𝕀_n), restricted to a certain dimension n − 2 subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots e₁ − e₂, … , e_n−2 − e_n−1.