Combinatorics of $m=1$ Grasstopes

TL;DR

This work extends the study of Grasstopes, the images of the totally nonnegative Grassmannian under linear maps, to the case $m=1$ by establishing a sign-variation criterion that characterizes tame Grasstopes as unions of cells in an affine hyperplane arrangement. It proves that, for maps well-defined on $ ext{Gr}_{ geq 0}(k,n)$, the Grasstope consists of points in $ extbf{P}^k$ with $ ext{var}$-type constraints on twistor coordinates, and it analyzes open versus closed rational Grasstopes when base loci are present. The authors classify Grasstopes into tame, wild, and rational types, provide explicit examples illustrating each case (including a Möbius-strip boundary in the rational case), and generalize the framework to oriented matroids, defining Grasstopes for non-realizable configurations. They also investigate extremal counts of Grasstope regions, deriving bounds and using computational data to highlight when region counts are arrangement-independent, while noting non-realizable examples where upper bounds fail to be attained. Overall, the paper broadens the combinatorial and topological understanding of Grasstopes beyond the tame, amplituhedron-like setting, with implications for both geometry and matroid theory.

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.

Figures (5)

• Figure 1: Affine chart in which the tame Grasstope is bounded. The six lines corresponding to the rows of $Z$ are colored red, orange, yellow, green, and blue, in order, with orientations given by arrows. The shaded portion of the figure is the Grasstope, which consists exactly of the regions with at least two sign changes.
• Figure 2: The six lines corresponding to the rows of $Z$ are pictured, with orientations given by the arrows. The regions can then be labelled by sign patterns. The shaded portion of the figure is the Grasstope, and it consits exactly of those regions with at least two sign changes.
• Figure 3: The six lines corresponding to the rows of $Z$ are pictured as described, with orientations given by the arrows. The regions can then be labelled by sign patterns. The shaded portion of the figure is the Grasstope, and it consits exactly of those regions with at least two sign changes. In this case, the shaded region is a Möbius strip.
• Figure 4: An oriented hyperplane arrangement and its sign labels.
• Figure 5: The Grasstope of a totally positive matrix with two rows negated. The six lines are cyclically ordered with orientations indicated by arrows. Every region has at least two sign changes, so the Grasstope is all of $\mathbb P^2$.

Theorems & Definitions (40)

• Definition 2.1: Grasstopes
• Definition 2.2: Tame Grasstope
• Proposition 2.3
• proof
• Corollary 2.4
• Definition 2.5: Wild Grasstope
• Definition 2.6: Rational Grasstope
• Remark 2.7
• Definition 2.8: Sign variation
• Proposition 2.9: karp, Lemma 4.1