Honeycomb tessellations and canonical bases for permutohedral blades

Abstract

This paper studies two families of piecewise constant functions which are determined by the $(n-2)$-skeleta of collections of honeycomb tessellations of $\mathbb{R}^{n-1}$ with standard permutohedra. The union of the codimension $1$ cones obtained by extending the facets which are incident to a vertex of such a tessellation is called a blade. We prove ring-theoretically that such a honeycomb, with 1-skeleton built from a cyclic sequence of segments in the root directions $e_i-e_{i+1}$, decomposes locally as a Minkowski sum of isometrically embedded components of hexagonal honeycombs: tripods and one-dimensional subspaces. For each triangulation of a cyclically oriented polygon there exists such a factorization. This consequently gives resolution to an issue proposed and developed by A. Ocneanu, to find a structure theory for an object he discovered during his investigations into higher Lie theories: permutohedral blades. We introduce a certain canonical basis for a vector space spanned by piecewise constant functions of blades which is compatible with various quotient spaces appearing in algebra, topology and scattering amplitudes. Various connections to scattering amplitudes are discussed, giving new geometric interpretations for certain combinatorial identities for one-loop Parke-Taylor factors. We give a closed formula for the graded dimension of the canonical blade basis. We conjecture that the coefficients of the generating function numerators for the diagonals are symmetric and unimodal.Comment: Added references; new section on configuration space