Honeycomb Tessellations and Graded Permutohedral Blades

TL;DR

This paper explores the combinatorial and algebraic properties of permutohedral blades, generalizing tropical hyperplane arrangements, and connects these structures to moduli space cohomology and scattering amplitudes.

Contribution

It introduces a graded basis for indicator functions of blades, proves a Minkowski sum decomposition, and links these to cohomology and non-planar square moves.

Findings

Constructed a graded basis for blade indicator functions.

Proved a Minkowski sum decomposition law.

Derived a closed formula for the graded dimension.

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.