Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the poles 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 $m(\mathbb{I}_n,\mathbb{I}_n)$, 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 $e_1-e_2,\ldots, e_{n-2}-e_{n-1}$.Comment: Section 5 added, with proof for Theorem 2

Honeycomb tessellations and canonical bases for permutohedral blades

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

Striking NYNEX

[Excerpt] The four-month strike by 60,000 telephone workers at NYNEX in 1989 was one of the largest and most significant anti-concession struggles of the decade. In an era when many unions have lost highly publicized contract fights and been forced to make give-backs, the NYNEX strikers successfully resisted management demands that they pay hundreds and eventually thousands of dollars a year for their medical coverage. They also defeated the company\u27s drive for new forms of flexible compensation designed to replace base wage increases and COLAs with lump-sum payments and profit-sharing. Successful union resistance to these concessions would not have been possible without an unprecedented pre-strike program of membership education and internal organizing. The contract campaign conducted by the 30 NYNEX local unions within the Communications Workers of America (CWA) and their allies in NYNEX units represented by the International Brotherhood of Electrical Workers (IBEW) reflects CWA\u27s nationwide commitment to rankand- file mobilization through the one-on-one approach

High Tech Professionals Are Hard to Organize Too

[Excerpt] It is unlikely that any technical and professional employees will be organized in non-union high tech firms until more blue-collar production workers become union members. There are, however, some high technology companies which already have heavily unionized blue-collar workforces. Two industrial unions have recently tried to recruit new members among the engineering and computer personnel at such firms. The experiences of the Communications Workers of America (CWA) at AT&T Technologies and the International Union of Electronic, Electrical, Technical, Salaried, & Machine Workers (IUE) at Raytheon indicate that the obstacles facing unions in this type of high tech organizing are formidable