Combinatorics of $m=1$ Grasstopes

TL;DR

This paper investigates the combinatorial structure of m=1 Grasstopes, a generalization of the amplituhedron, revealing their cell decomposition via hyperplane arrangements and proposing a new matroid-based perspective.

Contribution

It characterizes m=1 Grasstopes as unions of hyperplane arrangement cells with sign variation conditions and introduces a matroid-based framework for Grasstopes.

Findings

Characterization of m=1 Grasstopes as unions of hyperplane arrangement cells

Extension of sign variation conditions to Grasstopes

Proposal of a matroid-based notion of Grasstopes

Abstract

A Grasstope is the image of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a linear map $\text{Gr}(k,n)\dashrightarrow \text{Gr}(k,k+m)$. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. We study Grasstopes in the $m=1$ case and show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition, extending work of Karp and Williams. Inspired by this characterization, we also suggest a notion of a Grasstope arising from an arbitrary oriented matroid.