Weighted blade arrangements and the positive tropical Grassmannian

Abstract

In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on ${\Delta }_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type $S{L}_n$ that enjoys a cyclic symmetry. When placed at the center of a simplex, a blade induces a decomposition into $n$ maximal cells which are known as Pitman-Stanley polytopes. We introduce a complex $(B_{k,n},\partial )$ of weighted blade arrangements and we prove that the positive tropical Grassmannian surjects onto the top component of the complex, such that the induced weights on blades in the faces ${\Delta }_{2,n-(k-2)}$ of ${\Delta }_{k,n}$ are (1) nonnegative and (2) their support is weakly separated. We finally introduce a hierarchy of elementary weighted blade arrangements for all hypersimplices which is minimally closed under the boundary maps $\partial$, and apply our result to classify up to isomorphism type all rays of the positive tropical Grassmannian ${\text{Trop}}_{+}G(3,n)$ for $n\le 9$.

30 pages, 1 figure

References in corpus (5)

• Tropical fans, scattering equations and amplitudes
• Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams
• Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra
• Minimal Kinematics: An All $k$ and $n$ Peek into ${\mathrm{Trop}}^{+}\mathrm{G}(k,n)$
• On the inner structure of a permutation: Bicolored Partitions and Eulerians, Trees and Primitives

Cited by in corpus (2)

• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra