[Paper Review] Weighted blade arrangements and the positive tropical Grassmannian

Nick Early|arXiv (Cornell University)|May 25, 2020

Polynomial and algebraic computation28 references4 citations

TL;DR

This paper introduces a weighted blade arrangement complex $(\mathfrak{B}_{k,n}, \partial)$ that provides a combinatorial framework for studying the positive tropical Grassmannian $\text{Trop}^+G(k,n)$. It proves that the positive tropical Grassmannian surjects onto the top homology component of this complex, with blades in faces $\Delta_{2,n-(k-2)}$ having nonnegative weights and weakly separated supports. The key contribution is a conjecture that minimal weighted blade arrangements generate rays of $\text{Trop}^+G(k,n)$, enabling classification of rays for $\text{Trop}_+G(3,n)$ up to $n \leq 9$.

ABSTRACT

In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on $Δ_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type $SL_n$ 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 $Δ_{2,n-(k-2)}$ of $Δ_{k,n}$ 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 $ ext{Trop}_+ G(3,n)$ for $n\le 9$.

Motivation & Objective

• Develop a combinatorial framework using weighted blade arrangements to study the positive tropical Grassmannian $\text{Trop}^+G(k,n)$.
• Characterize the image of the positive tropical Grassmannian within the space of weighted blade arrangements via positivity and orthogonality constraints.
• Conjecture that minimal weighted blade arrangements induce coarsest positroidal subdivisions and generate rays of $\text{Trop}^+G(k,n)$.
• Classify all rays of $\text{Trop}_+G(3,n)$ up to isomorphism for $n \leq 9$ using the proposed framework.
• Establish linear independence for a set of planar kinematic invariants arising from the generalized biadjoint scalar formalism.

Proposed method

• Construct a chain complex $(\mathfrak{B}_{k,n}, \partial)$ of weighted blade arrangements on the vertices of the hypersimplex $\Delta_{k,n}$, where blades are tropical hypersurfaces from affine roots of type $SL_n$.
• Define a basis for the space of height functions over $\Delta_{k,n}$, which induces multi-split subdivisions via lower envelopes.
• Prove that the positive tropical Grassmannian surjects onto the top homology component of the complex, with weights on blades in $\Delta_{2,n-(k-2)}$ being nonnegative and supported on weakly separated collections.
• Introduce a hierarchy of elementary weighted blade arrangements closed under boundary maps $\partial$, conjecturing they generate rays of $\text{Trop}^+G(k,n)$.
• Apply the framework to classify rays of $\text{Trop}_+G(3,n)$ for $n \leq 9$, using known computations from [7, 11, 21] and translating them into blade configurations.
• Utilize the generalized biadjoint scalar formalism and planar basis to relate blade arrangements to scattering amplitudes and kinematic invariants.

Experimental results

Research questions

• RQ1 How can weighted blade arrangements be used to characterize the positive tropical Grassmannian $\text{Trop}^+G(k,n)$?
• RQ2 What conditions ensure that a weighted blade arrangement induces a coarsest positroidal subdivision of $\Delta_{k,n}$?
• RQ3 How do nonnegative weights and weak separation of supports constrain the image of $\text{Trop}^+G(k,n)$ in the blade complex?
• RQ4 What is the complete classification of rays of $\text{Trop}_+G(3,n)$ for $n \leq 9$ using this framework?
• RQ5 How do the kinematic invariants derived from the generalized biadjoint scalar formalism relate to blade arrangements and their linear independence?

Key findings

• The positive tropical Grassmannian $\text{Trop}^+G(k,n)$ surjects onto the top homology component of the weighted blade complex $(\mathfrak{B}_{k,n}, \partial)$, with induced weights on blades in $\Delta_{2,n-(k-2)}$ being nonnegative.
• Blades in the faces $\Delta_{2,n-(k-2)}$ of $\Delta_{k,n}$ have supports that are weakly separated, satisfying a key combinatorial constraint.
• A hierarchy of elementary weighted blade arrangements is minimally closed under the boundary map $\partial$, suggesting a foundational role in constructing the 1-skeleton of $\text{Trop}^+G(k,n)$.
• The conjecture is that any such minimal element induces a coarsest positroidal subdivision and thus generates a ray of $\text{Trop}^+G(k,n)$.
• Using this framework, all rays of $\text{Trop}_+G(3,n)$ are classified up to isomorphism for $n \leq 9$, with the first appearance of two-tripod and three-tripod configurations at $n=8$ and $n=9$, respectively.
• Linear independence is established for a set of planar kinematic invariants derived from the generalized biadjoint scalar amplitude, confirming their role in the blade-based formalism.