Honeycomb Tessellations and Graded Permutohedral Blades

TL;DR

This work develops a comprehensive theory of permutohedral blades, unifying matroid subdivisions, tropical geometry, and scattering amplitudes. It introduces a canonical, graded framework for graduated blades and establishes a flag factorization of blade functions into tripods, yielding a Minkowski-sum decomposition and a triangulation-independent structure. The paper also connects blades to tropical hypersurfaces, configuration-space cohomology, and circle-configuration moduli, and provides detailed enumerative results for the graded blade basis, including symmetry results later proved by Donghyun Kim. These results reveal rich combinatorial and geometric structure with potential implications for nonplanar amplitude moves and combinatorial scattering equations. The work thus opens multiple avenues for future exploration in geometry, topology, and mathematical physics, including deeper connections to Parke–Taylor factors and Δ-algebras.

Abstract

This paper investigates enumerative aspects of permutohedral blades, which provide a generalization of the notion of the tropical hyperplane arrangement. Blade provide the combinatorial underpinning of generalized biadjoint scalar scattering amplitudes in work of Cachazo, Early, Guevara and Mizera (CEGM). We construct a graded basis for a vector space of indicator functions of blades, with the grading determined by the dimension of the support of the function. We prove a Minkowski sum decomposition law into lines and tripods and we explore connections to the cohomology ring of certain moduli spaces and a non-planar analog of the square move for plabic graphs. We give a closed formula for the graded dimension of the basis. It is shown in an Appendix by Donghyun Kim that the coefficients appearing in the numerator of the generating function for the graded dimension are symmetric, and that they sum to $\frac{(2j)!}{j!}$. Unimodality is still open.

Figures (15)

• Figure 1: Left: the tripod $((1,2,3))$. Right: the blade $((1,2,3,4))$ as a Minkowski sum of the two tripods $((1,2,3))$ and $((1,3,4))$.
• Figure 2: One choice of a (hexagonal) period; opposite edges of the red hexagon are identified. Red edges are segments of affine hyperplanes, placed at $x_1-x_2 \in 3+6\mathbb{Z},\ x_2-x_3 \in 1+6\mathbb{Z},\ x_3-x_1\in 2+6\mathbb{Z}$. Black edges in the tiling are segments parallel to the three root directions $e_i-e_j$. Near the vertices of the (three) weight permutohedra (with black edges of lengths respectively 1,2,3) the black shell coincides with a blade. The blade $((1,2,3))$ is translated to for instance the point $(3,1,2)$, while the blade $((1,3,2))$ is translated to for instance $(0,1,5)$; compare to Figures \ref{['fig:weightpermtessellation']} and \ref{['fig:weightpermtessellation2']}.
• Figure 3: Part of the blade $((1,2,3,4))$. It is the 2-skeleton of a complete fan, here viewed inside a tetrahedral frame. Note that here the edges in the 1-skeleton of the blade are not perpendicular to the facets of the tetrahedron, but are rather parallel to its edges.
• Figure 4: Tessellating with three generalized (weight) permutohedra. Here rather than taking rational-valued coordinates for the point $y$, we have dilated the Weyl alcoves by a factor 6; then $y=(3,1,2)$ is reflected across the (red) affine reflection hyperplanes placed at $x_1-x_2 \in 3+6\mathbb{Z},\ x_2-x_3 \in 1+6\mathbb{Z},\ x_3-x_1\in 2+6\mathbb{Z}$.
• Figure 6: The blade relation $\gamma_{1,2,3} + \gamma_{1,3,2} = 1_{12}+1_{23}+1_{31} -1$.

Theorems & Definitions (21)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof