# Signaling oscillations: molecular mechanisms and functional roles

Pablo Casani-Galdon<EMAIL_ADDRESS>Department of Medicine and Life
Sciences, Universitat Pompeu Fabra, Barcelona Biomedical Research Park, Dr.
Aiguader 88, 08003 Barcelona, Spain  Jordi Garcia-Ojalvo
<EMAIL_ADDRESS>Department of Medicine and Life Sciences, Universitat
Pompeu Fabra, Barcelona Biomedical Research Park, Dr. Aiguader 88, 08003
Barcelona, Spain

###### Abstract

Mounting evidence shows that oscillatory activity is widespread in cell
signaling. Here we review some of this recent evidence, focusing on both the
molecular mechanisms that potentially underlie such dynamical behavior, and
the potential advantages that signaling oscillations might have in cell
function. The biological processes considered include tissue maintenance,
embryonic development and wound healing. With the aid of mathematical
modeling, we show that a common principle, namely delayed negative feedback,
underpins this wide variety of phenomena.

##  I Introduction

Signaling allows cells to sense, integrate and respond to cues from their
environment, and most importantly, to coordinate their behavior with
neighboring cells, thereby enabling cellular populations to self-organize in
space and time. The textbook view of cellular signaling usually considers,
either explicitly or implicitly, that the signals received by cells are
constant in time. However, the cellular environment is commonly dynamical _in
vivo_ , and such time-dependent character is bound to have influenced the
evolution of signaling circuits.

One way in which signaling circuits can cope with the dynamical, and usually
poorly predictable, external signals that cells receive, is by being dynamical
themselves [1, 2]. In agreement with this expectation, examples of oscillatory
and pulsatile signaling have begun to be uncovered in recent years [3, 4].
Early instances of oscillatory signaling were reported in the last decades of
the past century, in processes including glycolysis in muscle and yeast cells,
and cAMP signaling in Dyctiostelium [5, 6, 7].

Two questions arise in this context. First, what are the molecular mechanisms
underlying such self-sustained dynamical behavior? Second, what are the roles
of signaling oscillations in cells? Regarding the latter question, the
function of dynamical signaling is still unclear in many cases. In some
situations, oscillations could arise as a byproduct of adaptation, with no
relevant biological function [3]. In other cases, however, oscillations have
been found to be crucial for the proper operation of cells, such as in the
response of p53 to DNA damage [8], the maintenance of the neural progenitor
state [9], the segmentation of vertebrates [10, 11], and the growth of
bacterial biofilms [12].

Beyond their functional role, the existence of oscillations in certain
signaling systems may help us to identify and dissect the molecular circuits
underlying these cellular processes: while a biochemical system can reach a
steady state in many different ways, the number of potential circuits and
parameter values that can produce oscillations with the right properties
(period, amplitude, and response to perturbations, for instance) is much
smaller [13]. Indeed, a relatively small number of principles underlying the
emergence of self-sustained oscillations have been identified [14]. The most
fundamental of these principles is delayed negative feedback [15]. Simple
mathematical models show that when a signaling pathway inhibits itself with a
time lag (Fig. 1A), large enough delays trigger self-sustained oscillations,
as identified in pioneering models of the segmentation clock and other
recurrent pathways [16, 17].

Figure 1: Oscillations can arise from delayed negative feedback. (A) Schematic
representation of the circuit. The solid line represents a direct interaction
and the dashed line represent a delayed interaction with time lag $\tau$. (B)
Simulation of the system 1 for increasing values of the time lag $\tau$. The
values of the rest of parameters, together with the simulation code in Julia,
can be found in https://github.com/dsb-lab/SignalingOscillators.

Mathematically, this can be represented for instance by the following system
of delayed-differential equations:

$\displaystyle\frac{dX}{dt}$
$\displaystyle=\frac{\beta_{x}K_{r}^{n}}{K_{r}^{n}+Y^{n}(t-\tau)}-\delta_{x}X$
(1a) $\displaystyle\frac{dY}{dt}$
$\displaystyle=\frac{\beta_{y}X^{m}}{K_{a}^{m}+X^{m}}-\delta_{y}Y$ (1b)

Here the species $X$ activates $Y$ in a sigmoidal manner, with threshold
$K_{a}$, and $Y$ inhibits $X$ (also in a sigmoidal manner, with threshold
$K_{r}$) after a time delay $\tau$, represented by the explicit dependence of
$Y$ on $t-\tau$ in the first term on the right hand side of Eq. (1a) (all
other instances of $X$ and $Y$ in the equations above are computed at time
$t$, reflecting instantaneous processes). As shown in Fig. 1B, simulating the
system of equations (1) leads to self-sustained oscillations for large enough
$\tau$.

In model (1), $X$ could represent an mRNA species that is translated into its
own protein $Y$, although other molecular interpretations (such as $X$ and $Y$
being two different proteins in a recurrent pathway) are possible. The time
lag $\tau$ can also affect the other interaction in the circuit (from $X$ to
$Y$), or be distributed among the two [18]. The functional forms of the
various terms in Eqs. (1) can also be different, as long as the total
nonlinearity in the feedback loop is large enough [19]. There is thus
substantial freedom in how the delayed negative feedback can be implemented,
and indeed evolution has come across this design principle in many different
ways. In what follows we review three different instances of oscillatory
signaling circuits that implement negative feedback in various forms, with the
time delay arising in correspondingly distinct manners. We discuss the
molecular mechanisms underlying the different circuits, as well as the
biological functions that oscillations have in each case. All simulations
shown in this review have been generated with custom-made Julia code that can
be obtained from https://github.com/dsb-lab/SignalingOscillators.

##  II Notch oscillations in tissue maintenance

The Notch signaling system is a classical juxtacrine pathway involved in the
formation and maintenance of most tissues, among many other cellular functions
[20, 21]. Even though this pathway is involved in different lineage decisions,
its core circuitry is highly conserved [22]. The versatility of Notch
signaling relies on its multiple levels of control [23, 24, 25], its high
sensitivity to perturbations [1], and its ability to exhibit different
responses in the presence of different ligands [26]. Some of these properties
arise from, or are enabled by, the dynamical character of the pathway [1, 26].

The relation between the Notch pathway and signaling oscillations is
underpinned by the fact that one of the targets of Notch is Hes1, the core
component of the segmentation clock [17]. Additionally, the Notch ligand
Delta-like1 (Dll1) has been seen to oscillate in multiple tissues, including
the presomitic mesoderm, pancreatic and neural progenitors cells [27, 28] and,
more recently, in myogenic cells [29] (Fig. 2A). In the latter case, the
authors show that Dll1 is repressed by Hes1 directly, which suggests that the
Dll1 oscillations are a downstream effect of the well-known oscillatory
character of Hes1.

Figure 2: Oscillations in Notch signaling. (A) Dll1 oscillates in activated
muscular stem cells. (B) Schematic representation of the whole Hes, MyoD and
Dll1 circuit. (C) Time traces of Hes1, MyoD, Dll1 simulated with Eqs. (2) and
its downstream elements as presented in [29]. The vertical dashed line
represents the moment of the commitment and the end of the oscillations.
Adapted from [29]. The simulations in panel (C) were generated with the Julia
code posted in https://github.com/dsb-lab/SignalingOscillators, which includes
the parameter values used here.

Besides its direct repressing effect, Zhang et al also show [29] that Hes1
inhibits Dll1 indirectly by repressing the myogenic factor MyoD, which
activates Dll1 (Fig. 2B). Interestingly, the dynamics of MyoD determines
whether a myogenic cell self-renew or differentiates: while sustained
oscillations of MyoD in activated stem cells are necessary for self-renewal
and maintenance of the stem cell pool, sustained MyoD expression leads to
differentiation. This is shown in the simulations of Fig. 2C, in which Hes1
oscillations arise from the following circuit:

$\displaystyle\frac{dH}{dt}$ $\displaystyle=k_{1}h-k_{2}HF-k_{3}H$ (2a)
$\displaystyle\frac{dh}{dt}$ $\displaystyle=\frac{k_{4}}{1+H^{2}}-k_{5}h$ (2b)
$\displaystyle\frac{dF}{dt}$ $\displaystyle=\frac{k_{6}}{1+H^{2}}-k_{2}HF-
k_{7}F$ (2c)

where $H$ and $h$ represent Hes1 protein and mRNA, respectively, and $F$ is an
interacting factor introduced in early models of the circuit [30, 31]. The
model shows that the delay introduced by $F$, together with the negative
feedback acting through $h$, are enough to produce sustained oscillations of
Hes1. The oscillation is then propagated to the rest of the system by the
repression of $H$ on MyoD and Dll1 (Fig. 2B). Furthermore, since Dll1
activates Notch signaling (which in turn activates Hes1) in neighboring cells,
an additional inter-cell feedback arises that couples the oscillations in
neighboring cells, the time scale of which needs to be tuned to avoid
oscillation quenching, as Zhang et al verify experimentally using Dll1 mutants
[29]. This brings us back to the somitogenesis clock, where Notch signaling
was seen to synchronize the oscillations that arise in the presomitic mesoderm
[32].

##  III Intermittent ERK oscillations under FGF4 stimulation

Our second example involves the extracellular regulated kinase (ERK). ERK is a
classical mitogen-activated kinase (MAPK) module that, like Notch, is involved
in a large number of cellular processes including differentiation, growth,
proliferation, cell survival and apoptosis [33, 34, 35]. The ERK signaling
pathway is a paracrine activated cascade by which an extracellular growth
factor, such as the fibroblast growth factor (FGF), activates the Ras-Raf-MEK-
ERK cascade (Fig 3A).

Figure 3: Oscillations in ERK activity mediated by negative feeback and
ultrasensitivity. (A) Schematic representation of the circuit. The dashed line
representes the ultrasensitive step. (B) Experimental observations of ERK
oscillations in mouse ESCs, from [36]. Top: ERK-KTR expressing cells growing
in serum + LIF without (top filmstrip) and with MEKi (bottom filmstrip).
Dashed line indicates cell outlines. Bottom: time traces of the ERK activity
of cells treated with a MEK inhibitor (left) or with serum + LIF (right). (C)
Simulation of model (3). The values of the rest of parameters, together with
the simulation code, can be found in https://github.com/dsb-
lab/SignalingOscillators.

The cascade consists of a chain of serial phosphorylations that result in the
double phosphorylation of ERK and its shuttling to the nucleus, where it
affects the expression of more than 600 direct targets [37]. Even though the
pathway is relatively direct, the dynamics of ERK localization and activity
can be very complex: ERK is able to display a large variety of dynamic
responses in different cellular contexts, even when presented with similar
extracellular signals. In particular, under sustained growth-factor
stimulation, ERK can display sustained, pulsatile or oscillatory dynamics [38,
39, 40].

Recently, it was shown that treating mouse embryonic stem cells (mESCs) with
sustained concentrations of FGF4 results in oscillations of their ERK
activity, which disappear in the presence of a MEK inhibitor [36] (Fig 3B).
The oscillations became more regular as the concentration of FGF4 is
increased, although the shape of the pulses does not depend noticeably on the
FGF4 levels. Notably, ERK activity jumps between an oscillatory and non-
oscillatory state over a physiological range of ligand concentrations. The
role of these oscillations is still to be determined. It could merely be a
mechanism of adaptation to keep the ERK response within physiological ranges,
as proposed by the authors [36].

Similar oscillations in the ERK signaling pathway have been identified in
other studies. Stimulating mammalian epithelial cells with continuous
epidermal growth factor (EGF), for instance, leads to oscillations of ERK
nuclear translocation with a periodicity of 15 mins [38]. Increasing EGF
levels reduce the amplitude of the oscillations while the pulse duration
remains constant, similarly to the observations in mESCs described above. In
contrast, other studies have shown that pulsing can be more stochastic than
regular, and that growth-factor concentration can affect the frequency of the
oscillation [41, 42].

The molecular mechanism of the oscillations observed by Raina et al [36] is
still unclear. A natural candidate is negative feedback, which is commonly
found affecting ERK signaling [43]. Given that the time scale of the
oscillation is of the order of 5 to 10 minutes, it is possible that the
feedback is acting at the level of ERK phosphorilation/dephosphorilation, as
suggested previously [38, 39, 44]. The system depicted in figure 3A is a
simplification of the cascade, where we only consider the activated kinases
and the feedback affects directly the input of the cascade. Its dynamics can
be represented by a model such as:

$\displaystyle\frac{dR}{dt}$
$\displaystyle=\beta_{R1}\frac{K_{R1}^{n}}{K_{R1}^{n}+E^{n}}\frac{F}{K_{R2}+F}-\frac{\beta_{R2}R}{K_{R3}+R}$
(3a) $\displaystyle\frac{dM}{dt}$
$\displaystyle=\beta_{M1}\frac{R^{m}}{K_{M1}^{m}+R^{m}}-\frac{\beta_{M2}M}{K_{M2}+M}$
(3b) $\displaystyle\frac{dE}{dt}$
$\displaystyle=\beta_{E1}\frac{M^{p}}{K_{E1}^{p}+M^{p}}-\frac{\beta_{E2}E}{E_{E2}+E}$
(3c)

Simulations of this system (3) show that in the presence of FGF (represented
in the model by $F$) and large enough ultrasensitivity ($m\gg 1$) in the
activation of MEK ($M$) by RAF ($R$), ERK ($E$) displays sustained
oscillations (Fig. 3C). These results are not sufficient, however, to explain
all the characteristics that ERK activity displays in the presence of FGF4, in
particular their intermittent character [36]. Potential extensions of the
model (both conceptually and mathematically) could include the existence of
multiple feedback loops acting at different steps of the cascade and at
different timescales [43].

##  IV ERK mechanochemical wave

The negative feedbacks that we have considered so far have been purely
biochemical. In this last example, we are going to focus on signaling
oscillations mediated by a delayed negative feedback that involves an elegant
combination of biochemical and mechanical signals. Mechanical inputs are
important factors in many different biological processes, such as the
collective migration of the cells forming monolayers of epithelial tissue
(Fig. 4A,B) [45, 46].

Figure 4: Mechanochemical feedbacks mediated by ERK drive collective motion.
(A) Schematic representation of the mechanochemical system. Black arrows
represent direction of cell movement while brown arrow represents direction of
ERK wave. Time runs from top to bottom. (B) Immunofluorescence images of the
Golgi apparatus (GM130) and the nucleus (EKAREV-NLS) in MDCK cells at 21 h
after migration. Adapted from Hino et al [47].(C) Schematic representation of
the feedback circuit

Waves of cell motion appear in contexts such as wound healing. Both in vivo
and in vitro studies have shown that waves of ERK activation propagate counter
to the direction of cell migration and cell polarization [46]. ERK waves are
responsible for the directionality of the migration, by orienting the cells in
the direction opposite to their propagation. In the absence of an external cue
(e.g. a wound), there is no collective net movement of the tissue, even though
there might be stochastic cellular motion. In the context of wound healing, on
the other hand, the presence of stress, caused by the loss of congruency in
the tissue, serves as a continuous input of mechanical force into the
monolayer. Such continuous pulling produces a wave of ERK that propagates in
the opposite direction of net cell movement [47, 48].

What causes these ERK waves? During wound healing each cell following the
leader cell undergoes a series of activations and inactivations of ERK
activity (Fig. 4A). The mechanism by which cells transduce the mechanical
intercellular interaction into an intracellular biochemical response was
unknown until very recently. Hino and colleagues [47] have demonstrated that
the mechanical pulling of cells by their neighbors activates ERK. This
increase in ERK activity then triggers the contraction of the cell, which in
turns removes the original mechanical activation of ERK. The resulting
oscillation in cell activity is then propagated mechanically along the tissue.
The presence of such mechanochemical feedback (Fig. 4C) is an ideal example of
how the simple scheme of a delayed negative feedback can drive complex
functions like collective polarization of cells to migrate in a specific
direction. Boocock et al have [48] have proposed a viable model of collective
motion based on the above-mentioned scheme (Fig. 4C) of delayed feedback
between ERK activity and cellular contraction. Such simple but elegant
mechanism is enough to reproduce the intricacies of both stochastic and
directed collective movements in monolayers of epithelial tissue.

##  V Discussion

Cells are constantly receiving signals coming from their environment, and they
need to decide how to react appropriately to the information (commonly
dynamical, frequently unpredictable) contained in these signals. The signaling
pathways involved in cellular communication and decision are scarce compared
to the amount of possible environmental situations that a cell can face. To
solve this underactuation problem, cells have evolved sophisticated
communication codes [40] through which information is actively processed.

One way in which external signals can be “multiplexed” within a number of
signaling systems is by using dynamics [2]. In this approach, different
dynamical regimes of a regulatory factor (in the limiting case oscillations
versus stationary behavior) can code for different situations, and thereby for
different reactions that the cell must follow. Here we have seen an example of
this phenomenon in the case of tissue maintenance in muscle stem cells (where
oscillations of Notch pathway elements determine the fate of those cells). We
have also seen that not only adult stem cells, but embryonic stem cells as
well, exhibit signaling oscillations (in this case of the ERK pathway). The
biological function of these oscillations is however not yet clear.

Both of the above-mentioned instances of signaling oscillations seem to be
based on delayed negative feedback. While the feedback in those cases is
purely biochemical, the importance of mechanical interactions in tissues makes
it reasonable to expect that mechanics might also be involved in feedback
loops that could underlie oscillatory behavior. This is the case of the third
example considered here, involving collective cell motion in epithelial
tissue. This situation also involves ERK activation, but here mechanical
regulation is an essential part of the feedback. Furthermore, mechanics
naturally leads to the spatial propagation of the oscillations in the form of
waves of ERK activity, which regulate the cell motion.

These are just a few recent examples that show the relevance of oscillations
in cell signaling systems. More work is needed to identify further regulatory
processes driven by oscillatory dynamics, which will remain hidden unless
single-cell time-resolved measurements are systematically performed in cell-
biology studies.

##  VI Acknowledgments

We acknowledge financial support from the Spanish Ministry of Science and
Innovation and FEDER (grant PGC2018-101251-B-I00), by the “Maria de Maeztu”
Programme for Units of Excellence in R&D (grant CEX2018-000792-M), and by the
Generalitat de Catalunya (ICREA Academia programme). P.C.-G. is supported by a
FPI doctoral fellowship from the Spanish Ministry of Science and Innovation
(reference XXXXXX).

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# Combinatorics of $m=1$ Grasstopes

Yelena Mandelshtam  Dmitrii Pavlov  and Elizabeth Pratt

###### 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 the work of Karp and Williams [13]. Inspired by this
characterization, we also suggest a notion of a Grasstope arising from an
arbitrary oriented matroid.

##  1 Introduction

The (tree) amplituhedron, introduced by Arkani-Hamed and Trnka in [1], is a
geometric object playing an important role in calculations of scattering
amplitudes in planar $N=4$ Super-Yang-Mills theory. It is defined as the image
of the totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ under a
totally positive linear map
$\tilde{Z}:\text{Gr}(k,n)\dashrightarrow\text{Gr}(k,k+m)$ given by an
$n\times(k+m)$ matrix $Z$. While immediate physical relevance of the
amplituhedron becomes manifest at $m=4$, it is an object of independent
mathematical interest for any $m$. It is known that when $k=1$, it is a cyclic
polytope [21] and when $k+m=n$, it is isomorphic to the totally nonnegative
Grassmannian $\text{Gr}_{\geq 0}(k,n)$.

In recent years, the amplituhedron has been studied extensively from the point
of view of algebraic combinatorics for $m=1,2,4$ (see [4, 9, 12, 13, 16, 18]).
The structure of the amplituhedron in the $m=1$ case is particularly simple:
Karp and Williams [13] show that it is linearly homeomorphic to the complex of
bounded cells of an affine hyperplane arrangement and therefore is
homeomorphic to a closed ball.

One reason that the amplituhedron is so amenable to combinatorial study is
that the totally nonnegative Grassmannian itself has a rich combinatorial
structure. In particular, $\text{Gr}_{\geq 0}(k,n)$ admits a stratification by
positroid cells, which are all homeomorphic to open balls [19]. However,
amplituhedra are images of very special linear maps, just as cyclic polytopes
are very special polytopes. From this point of view, it makes sense to
consider images of the totally nonnegative Grassmannian under arbitrary linear
maps. In [14], Lam considers the images of positroid cells under arbitrary
linear maps, and calls them _Grassmann polytopes_. Images of the whole totally
nonnegative Grassmannian are referred to as _full Grassmann polytopes_.

While amplituhedra have attracted significant attention from the mathematical
community, many results in this area rely on the total positivity of the map
$\tilde{Z}$, and much less is known about more general Grassmann polytopes.
Most of the known results are assembled in [14, Part 2]. Many of them are
limited to the case when $\tilde{Z}$ does not have base points on the totally
nonnegative Grassmannian, i.e. is regular (well-defined) on $\text{Gr}_{\geq
0}(k,n)$.

In this paper, we initiate the study of general full Grassmann polytopes
(_Grasstopes_) and focus on the case of $m=1$, when the ambient Grassmannian
is the dual projective space $(\mathbb{P}^{k})^{\vee}$. We extend the results
of [13], showing that when $\tilde{Z}$ is regular on $\text{Gr}_{\geq
0}(k,n)$, the resulting Grasstope is a union of cells of a projective
hyperplane arrangement satisfying a certain sign variation condition. This, in
particular, implies that Grasstopes arising from such $\tilde{Z}$ are closed
and connected, in accordance with [14, Proposition 15.2]. When there are no
additional restrictions on $\tilde{Z}$, we show that the image of the totally
positive Grassmannian $\text{Gr}_{>0}(k,n)$ can still be characterized in
terms of sign changes, although the image of $\text{Gr}_{\geq 0}(k,n)$ might
have irregular boundary. We also show that, unlike amplituhedra, general $m=1$
Grasstopes are not necessarily homeomorphic to closed balls or even
contractible.

This paper is organized as follows. In Section 2, we define Grassmanians,
their totally nonnegative and totally positive parts, and Grasstopes. We
divide Grasstopes into three categories (tame, wild, and rational) based on
the properties of the map $\tilde{Z}$. We introduce the concepts necessary for
the sign variation characterization of $m=1$ Grasstopes and prove some
auxiliary results about general Grasstopes. In Section 3, we study the
combinatorics and geometry of Grasstopes for $m=1$ and prove the sign
variation characterization results for tame and wild Grasstopes, as well as
for open rational Grasstopes. Section 4 is devoted to examples. In Section 5,
we give background on oriented matroids, which is useful for Section 6.
Finally, in Section 6 we investigate how many regions of a hyperplane
arrangement can be in an $m=1$ Grasstope, and, based on the sign variation
characterization, suggest a definition of the Grasstope of a (not necessarily
realizable) oriented matroid.

##  2 Preliminaries

The _real Grassmannian_ $\text{Gr}(k,n)$ is the variety parameterizing
$k$-dimensional subspaces of an $n$-dimensional vector space $\mathbb{R}^{n}$.
This can be identified with the variety of $(k-1)$-dimensional subspaces of an
$(n-1)$-dimensional real projective space $\mathbb{P}^{n-1}$. A linear space,
considered as an element of $\text{Gr}(k,n)$, can be represented by a $k\times
n$ matrix $A$ whose rows span the space. The Grassmannian $\text{Gr}(k,n)$ can
be realized as a subvariety of the projective space
$\mathbb{P}^{\binom{n}{k}-1}$ via the _Plücker embedding_ , which sends a
matrix $A$ representing an element in $\text{Gr}(k,n)$ to the vector of its
maximal minors. Maximal minors of $A$ are called _Plücker coordinates_. This
embedding does not depend on the choice of matrix representatives. For more
details on the Grassmannian and the Plücker embedding, see [17, Chapter 5].

The _totally nonnegative Grassmannian_ $\text{Gr}_{\geq 0}(k,n)$ is the subset
of $\text{Gr}(k,n)$ consisting of the elements whose non-zero Plücker
coordinates all have the same sign [19, 22]. Each element in $\text{Gr}_{\geq
0}(k,n)$ can be represented by a _totally nonnegative_ $k\times n$ matrix,
that is, by a matrix with nonnegative maximal minors. The _totally positive
Grassmannian_ $\text{Gr}_{>0}(k,n)$ is a subset of $\text{Gr}(k,n)$ consisting
of the elements whose Plücker coordinates are all non-zero and have the same
sign. Elements of $\text{Gr}_{>0}(k,n)$ can be represented by $k\times n$
matrices with strictly positive maximal minors (such matrices are called
_totally positive_).

Let $Z$ be a real $n\times(k+m)$ matrix of full rank, where $k+m\leq n$. The
matrix $Z$ defines a rational map
$\tilde{Z}:\text{Gr}(k,n)\dashrightarrow\text{Gr}(k,k+m)$ by $[A]\mapsto[AZ]$,
where $[A]$ is the class in $\text{Gr}(k,n)$ of a matrix $A$.

######  Definition 2.1 (Grasstopes).

The image $\tilde{Z}(\text{Gr}_{\geq 0}(k,n))\subseteq\text{Gr}(k,k+m)$ is
called the $(n,k,m)$-_Grasstope_ of $Z$ and is denoted by
$\mathcal{G}_{n,k,m}(Z)$.

The totally nonnegative Grassmannian $\text{Gr}_{\geq 0}(k,n)$ is a
semialgebraic set (that is, it can be described by polynomial equations and
inequalities) and $\mathcal{G}_{n,k,m}(Z)$ is its image under a polynomial
map. Thus, it follows from the Tarski-Seidenberg theorem [3, Theorem A.49]
that $\mathcal{G}_{n,k,m}(Z)$ is also semialgebraic.

In [14] $\mathcal{G}_{n,k,m}(Z)$ are called _full Grassmann polytopes_. When
$Z$ is a totally positive matrix, one recovers the definition of the _(tree)
amplituhedron_ $\mathcal{A}_{n,k,m}(Z)$ [1], a geometric object of fundamental
importance to calculating scattering amplitudes in particle physics.

Note that the matrix $AZ$ defines an element in $\text{Gr}(k,k+m)$ if and only
if $AZ$ has full rank. It is a priori not guaranteed that the map $\tilde{Z}$
is _well-defined_ on $\text{Gr}_{\geq 0}(k,n)$, that is, that $AZ$ has full
rank for any totally nonnegative $k\times n$ matrix $A$. In general, the map
$\tilde{Z}$ has base locus

$\mathcal{B}(\tilde{Z}):=\\{V:\text{dim}(V\cap\ker Z^{T})\geq
1\\}\subset\text{Gr}(k,n),$

which may or may not intersect $\text{Gr}_{\geq 0}(k,n).$ Note that the
$\mathcal{B}(\tilde{Z})$ is a Schubert variety, so in particular, it is closed
in $\text{Gr}(k,n)$ [14, Section 17]. We will often view $\text{Gr}_{\geq
0}(k,n)$ as a parameter space of $(k-1)$-dimensional subspaces of
$\mathbb{P}^{n-1},$ in which case the base locus is all projective subspaces
which are not disjoint from $\mathbb{P}(\ker Z^{T})$.

Finding combinatorial conditions for $\tilde{Z}$ to be well-defined on
$\text{Gr}_{\geq 0}(k,n)$ had been an active area of research for several
years. In [14, Proposition 15.2] Lam proved that if $\tilde{Z}$ is well-
defined on $\text{Gr}_{\geq 0}(k,n)$, then $\mathcal{G}_{n,k,m}(Z)$ is closed
and connected. In the same proposition he showed that the following condition
is sufficient for $\tilde{Z}$ to be well-defined.

$\text{There exists a }(k+m)\times k\text{ matrix }M\text{ such that all
}k\times k\text{ minors of }ZM\text{ are positive}.$ (1)

Geometrically, condition (1) means that the element of $\text{Gr}(k+m,n)$
represented by $Z$ contains a totally positive $k$-dimensional subspace, that
is, an element of $\text{Gr}_{>0}(k,n)$. Lam also conjectured that this
condition is necessary for $\tilde{Z}$ to be well-defined on $\text{Gr}_{\geq
0}(k,n)$. This conjecture turned out to be false, with a counterexample given
by Galashin (see [13, Remark 9.3] and Example 4.2). A combinatorial criterion
was given in [11, Theorem 4.2]. Lam’s conjecture gives rise to the following
definition.

######  Definition 2.2 (Tame Grasstope).

The Grasstope $\mathcal{G}_{n,k,m}(Z)$ is called _tame_ if $Z$ satisfies (1).

Let $\phi:V\to W$ be a map of vector spaces. We define its $k^{\text{th}}$
exterior power $\wedge^{k}\phi:\bigwedge^{k}V\to\bigwedge^{k}W$ by
$v_{1}\wedge\ldots\wedge
v_{k}\mapsto\phi(v_{1})\wedge\ldots\wedge\phi(v_{k}).$ If $M$ is a matrix
representing $\phi$ in bases $\\{e_{i}\\}$ of $V$ and $\\{\tilde{e}_{j}\\}$ of
$W$, we denote the matrix representing $\wedge^{k}\phi$ in the induced bases
of $\bigwedge_{k}V$ and $\bigwedge_{k}W$ by $\wedge_{k}M$. Then the matrix of
$\tilde{Z}$ is $\wedge^{k}Z,$ where we use the standard embedding
$\text{Gr}(k,n)\to\bigwedge^{k}(\mathbb{R}^{n}),\
\text{Span}\\{v_{1},\ldots,v_{k}\\}\mapsto v_{1}\wedge\ldots\wedge v_{k}$.
Concretely, the entries of $\wedge^{k}M$ are the $k\times k$ minors of $M$.
Those minors are ordered by multi-indices in reverse lexicographic order
(according to Macaulay2 convention).

We now give a simple geometric criterion of tameness. Although well-known to
specialists in this area, this result seems to have not appeared in the
literature yet. In what follows, for the sake of simplicity, we slightly abuse
notation and write $\mathcal{G}_{n,k,m}(Z)$ both for the Grasstope as a subset
of $\text{Gr}(k,k+m)$ and its image under the Plücker embedding.

######  Proposition 2.3.

An $n\times(k+m)$ matrix $Z$ satisfies (1) (i.e. $\mathcal{G}_{n,k,m}(Z)$ is
tame) if and only if there exists a hyperplane in
$\mathbb{P}^{\binom{k+m}{k}-1}$ which does not intersect
$\mathcal{G}_{n,k,m}(Z)$.

###### Proof.

Any $(k+m)\times k$ matrix $M$ defines a hyperplane in
$\mathbb{P}^{\binom{k+m}{k}-1}$ by its Plücker coordinates: a point
$p\in\mathbb{P}^{\binom{k+m}{k}-1}$ lies on the hyperplane defined by $M$ if
and only if $(\wedge_{k}M)^{T}p=0$. Suppose that $Z$ satisfies (1) and $M$ is
a $(k+m)\times k$ matrix such that $ZM$ is totally positive. Suppose that a
point in $\mathcal{G}_{n,k,m}(Z)$ lies on the hyperplane defined by $M$. Then,
for some $A\in\text{Gr}_{\geq 0}(k,n)$, it holds that
$(\wedge_{k}M)^{T}(\wedge_{k}AZ)^{T}=0$, which implies that
$(\wedge_{k}(ZM))^{T}(\wedge_{k}A)^{T}=0$. Since all $k\times k$ minors of $A$
are non-negative, and at least one is nonzero, it is not possible for all
$k\times k$ minors of $ZM$ to have the same sign. Thus, if there exists $M$
such that $ZM$ has all positive (or negative) $k\times k$ minors, then the
image $\mathcal{G}_{n,k,m}(Z)$ does not intersect the hyperplane defined by
$M$.

Now suppose that $Z$ does not satisfy (1), that is, for any $M$ the matrix
$ZM$ has either a zero $k\times k$ minor or at least one positive and one
negative $k\times k$ minor. We will show that there exists a matrix
$A\in\text{Gr}_{\geq 0}(k,n)$ such that
$(\wedge_{k}(ZM))^{T}(\wedge_{k}A)^{T}=0$, so that the hyperplane defined by
$M$ intersects $\tilde{Z}(\text{Gr}_{\geq 0}(k,n))$.

In the first case, when $ZM$ has a zero minor in the $i^{\text{th}}$ Plücker
coordinate, one can find an element $A\in\text{Gr}_{\geq 0}(k,n)$ which has
all Plücker coordinates equal to zero except for the $i^{\text{th}}$ one. In
this case, $(\wedge_{k}(ZM))^{T}(\wedge_{k}A)^{T}=0$.

Now consider the second case, in which $ZM$ has at least one positive and one
negative $k\times k$ minor. By the pigeonhole principle, there exists a set of
column indices $I=\\{i_{1},\dots,i_{k-1}\\}$ such that two Plücker coordinates
involving $I$ (which we label $p_{I\cup\\{i\\}}$ and $p_{I\cup\\{j\\}}$) have
different signs. Then, take $(q_{1,\dots
k}:\dots:q_{n-k+1,\dots,n})\in\mathbb{P}^{\binom{n}{k}-1}$ such that all
coordinates except for $q_{I\cup\\{i\\}},q_{I\cup\\{j\\}}$ are zero, and
$q_{I\cup\\{i\\}}=|p_{I\cup\\{j\\}}|$ and
$q_{I\cup\\{j\\}}=|p_{I\cup\\{i\\}}|$. Then, all Plücker relations are
satisfied so this point represents an element $A\in\text{Gr}_{\geq 0}(k,n)$.
We have $(\wedge_{k}(ZM))^{T}(\wedge_{k}A)^{T}=0$, so the hyperplane given by
$M$ intersects the Grasstope $\mathcal{G}_{n,k,m}(Z)$. ∎

By choosing a hyperplane disjoint from $\mathcal{G}_{n,k,m}(Z)$ to be the
hyperplane at infinity, we arrive at the following result.

######  Corollary 2.4.

The Grasstope $\mathcal{G}_{n,k,m}(Z)$ is tame if and only if its image under
the Plücker embedding is contained in some affine chart of
$\mathbb{P}^{\binom{k+m}{k}-1}$.

Tame Grasstopes share many nice properties with amplituhedra. In particular,
for $m=1$ they are homeomorphic to closed balls and can be described as
complexes of bounded cells of affine hyperplane arrangements [13, Section 9].
The focus of this paper, however, is to take a step away from the tame case
and study Grasstopes that behave somewhat less regularly.

######  Definition 2.5 (Wild Grasstope).

If the map $\tilde{Z}$ is well-defined on $\text{Gr}(k,n)_{\geq 0}$ but $Z$
does not satisfy (1), then the Grasstope $\mathcal{G}_{n,k,m}(Z)$ is called
_wild_.

Even though $\tilde{Z}$ might not be well-defined on $\text{Gr}_{\geq
0}(k,n)$, it still makes sense to consider the image of $\text{Gr}_{\geq
0}(k,n)\setminus(\mathcal{B}(\tilde{Z})\cap\text{Gr}_{\geq 0}(k,n))$, where
$\mathcal{B}(\tilde{Z})$ is the base locus of $\tilde{Z}$. In this case we
will still write $\tilde{Z}(\text{Gr}_{\geq 0}(k,n))$ for this image.

######  Definition 2.6 (Rational Grasstope).

Suppose the map $\tilde{Z}$ is not well-defined on $\text{Gr}(k,n)_{\geq 0}$.
Then the image $\mathcal{G}_{n,k,m}(Z)=\tilde{Z}(\text{Gr}(k,n)_{\geq 0})$ is
called a _rational_ Grasstope. The image
$\mathcal{G}^{\circ}_{n,k,m}(Z)=\tilde{Z}(\text{Gr}(k,n)_{>0})$ of the totally
positive Grassmannian is called an _open rational_ Grasstope. This is indeed
an open set, as shown in Proposition 3.4.

We conclude this section with technical results that will prove useful in
characterization of $m=1$ Grasstopes in Section 3.

Given a point $u$ in $\mathbb{P}^{n},$ we associate a sign pattern
$\sigma=(\sigma_{0},\ldots,\sigma_{n})\in\\{+,-,0\\}^{n+1}$ to $u$ in the
following way. Pick $i$ such that $u_{i}\neq 0$ and set $\sigma_{i}:=+$. Then
$\sigma_{j}:=\operatorname{sign}(u_{i}u_{j})$, which is a well-defined
function of homogeneous coordinates of $u$. Since we associate sign labels to
points in projective space, we will identify sign labels $\sigma$ and
$-\sigma$. Each orthant of the $\mathbb{P}^{n}$ consists of the points with
the same sign pattern. For instance, the sign pattern $(+:+:-:+)$ for a point
in $\mathbb{P}^{3}$ represents the orthant defined by

$\\{u_{0}u_{1}>0,u_{0}u_{2}<0,u_{0}u_{3}>0,u_{1}u_{2}<0,u_{1}u_{3}>0,u_{2}u_{3}<0\\}.$

Given a point $x\in\mathbb{P}^{k}$ being the image of a hyperplane
$X=\text{Span}\\{w_{1},...,w_{k}\\}$ under the Plücker embedding of
$\text{Gr}(k,k+1),$ and any point $v\in\mathbb{P}^{k},$ one may consider the
bilinear map

$\displaystyle T:\mathbb{P}^{k}\times\mathbb{P}^{k}$
$\displaystyle\to\mathbb{R}$ $\displaystyle x,v$
$\displaystyle\mapsto\det\begin{bmatrix}|&&|&|\\\ w_{1}&...&w_{k}&v\\\
|&&|&|\end{bmatrix}=\sum_{j=1}^{k}(-1)^{j}p_{1...\hat{j}...k}v_{j}.$

Then $T(x,v)=0$ if and only if $v$ is contained in $X.$ If $v=Z_{i}$ for some
row $Z_{i}$ of $Z,$ then $T(x,Z_{i})$ is commonly called a _twistor coordinate
of $X$ with respect to $Z$_ [22, Definition 4.5].

######  Remark 2.7.

One may also fix either argument $x$ or $v$ of $T$ to get a linear form on
$\mathbb{P}^{k}.$ In this paper we will consider the twistor coordinates as
functions of $x$ by setting $v=Z_{i}$ and denote the resulting forms by
$l_{i}(x)$. We will see in Theorem 3.1 that the vanishing loci of the
$l_{i}$’s are exactly the hyperplanes which contain the boundaries of
$\mathcal{G}_{n,k,m}(Z)$.

We now recall the definition of sign variation from [13].

######  Definition 2.8 (Sign variation).

Given a sequence $v$ of $n$ real numbers, let $var(v)$ be the number of sign
changes in $v$ (zeros are ignored). Let
$\overline{var}(v):=\max\\{var(w):w\in\mathbb{R}^{n}\text{ such that
}w_{i}=v_{i}\text{ for all }i\in[n]\text{ with }v_{i}\neq 0\\}$, i.e.
$\overline{var}(v)$ is the maximum number of sign changes in $v$ after a sign
for each zero component is chosen. Note that both $var$ and $\overline{var}$
are well-defined functions of homogeneous coordinates of a point in
$\mathbb{P}^{n-1}$.

We call a $(k-1)$-dimensional subspace of $\mathbb{P}^{n-1}$ _positive_ if it
is a point in $\text{Gr}_{\geq 0}(k,n)$ and _totally positive_ if it is a
point in $\text{Gr}_{>0}(k,n)$. Given a point $u\in\mathbb{P}^{n-1},$ we
define the _hyperplane $H_{u}$ given by $u$_ to be the hyperplane orthogonal
to $u$ with respect to the standard dot product, i.e.
$H_{u}:=\\{v\in\mathbb{P}^{n-1}:u\cdot v=0\\}.$ Then we have the following
proposition.

######  Proposition 2.9.

Let $H_{u}$ be the hyperplane given by $u\in\mathbb{P}^{n-1}.$

  1. 1.

$H_{u}$ contains a positive subspace of dimension $k-1$ if and only if
$\overline{var}(u)\geq k$.

  2. 2.

$H_{u}$ contains a totally positive susbspace of dimension $k-1$ if and only
if $var(u)\geq k$.

###### Proof.

We start by proving the first statement. A hyperplane $H_{u}$ contains a
positive subspace of dimension $k$ if and only if $u$ is in the kernel of some
matrix of the form $\left(\begin{array}[]{ccc}&A&\\\
\hline\cr&B&\end{array}\right)$ where $A\in\text{Gr}_{\geq 0}(k,n)$, and
$B\in\text{Gr}(n-k-1,n)$. Here $A$ represents the positive subspace, and $B$
the additional points to define a hyperplane. Note that
$u\in\ker\left(\begin{array}[]{ccc}&A&\\\ \hline\cr&B&\end{array}\right)$ if
and only if $u\in\ker(A)$ and $u\in\ker(B)$. Then, the “only if” direction
follows directly from [10, Theorems V1 and V6] (also [13, Theorem 3.4(i)]).
The “if” direction also follows, with the note that one can always find
$n-k-1$ additional points to form the matrix $B$ such that $u\in\ker(B)$. The
second statement is proved analogously, with replacing $\text{Gr}_{\geq
0}(k,n)$ by $\text{Gr}_{>0}(k,n)$ and using [13, Theorem 3.4(ii)]. ∎

##  3 Grasstopes for $m=1$ Tame, Wild and Rational

We begin by stating our main theorem which describes any $m=1$ Grasstope
$\mathcal{G}_{n,k,1}(Z)$ arising from a well-defined map $\tilde{Z}$ as a
subset of $\mathbb{P}^{k}\cong\text{Gr}(k,k+1)$. This theorem recovers and
generalizes many of the results of Karp and Williams describing $m=1$
amplituhedra and tame Grasstopes ([13, Section 6]).

######  Theorem 3.1.

Suppose $\tilde{Z}:\text{Gr}(k,n)\dashrightarrow\text{Gr}(k,k+1)$ is well-
defined on $\text{Gr}_{\geq 0}(k,n)$. Then the Grasstope
$\mathcal{G}_{n,k,1}(Z)$ consists of the points $\mathbf{x}\in\mathbb{P}^{k}$
such that $\overline{var}(L(\mathbf{x}))\geq k$, where $L(\mathbf{x})$ is the
vector of twistor coordinates of $\mathbf{x}$ with respect to $Z.$

###### Proof.

Let $L(\mathbf{x})$ =
$(l_{1}(\mathbf{x}):...:l_{n}(\mathbf{x}))\in\mathbb{P}^{n-1}$ (see Remark
2.7). By Proposition 2.9, it suffices to show that $H_{L(\mathbf{x})}$
contains a positive subspace if and only if $X\in\mathcal{G}_{n,k,1}(Z)$.

For the “if” direction, suppose that $A\in\text{Gr}_{\geq 0}(k,n)$ and
$\mathbf{x}$ is the vector of Plücker coordinates of $[AZ].$ Then for each row
$A_{i}$ of $A,$ we have

$L(\mathbf{x})\cdot
A_{i}=\sum_{j=1}^{n}T(\mathbf{x},Z_{j})A_{ij}=T(\mathbf{x},\sum_{j=1}^{n}A_{ij}Z_{j})=0,$

since $\sum A_{ij}Z_{j}$ is a row of $AZ.$ Thus $H_{L(\mathbf{x})}$ contains
$A.$

For the “only if” direction, suppose that $H_{L(\mathbf{x})}$ contains a
positive subspace $A$, and let $v\in\ker Z^{T}.$ Then

$L(\mathbf{x})\cdot
v=\sum_{i=1}^{n}T(\mathbf{x},Z_{i})v_{i}=T(\mathbf{x},\sum_{i=1}^{n}Z_{i}v_{i})=T(\mathbf{x},0)=0.$

So $H_{L(\mathbf{x})}$ contains $\mathbb{P}(\ker Z^{T})$. Since
$\mathbb{P}(\ker Z^{T})\cap A=\emptyset$ by regularity of $\tilde{Z}$ on
$\text{Gr}_{\geq 0}(k,n),$ we have by dimension considerations that any two
hyperplanes containing $A$ and $\mathbb{P}(\ker Z^{T})$ must be equal.
However, by the “if” direction, if $\mathbf{a}$ is the vector of Plücker
coordinates of $[AZ],$ then $H_{L(\mathbf{a})}$ contains $A$. In addition,
$H_{L(\mathbf{a})}$ contains $\mathbb{P}(\ker Z^{T})$. Therefore
$H_{L(\mathbf{x})}=H_{L(\mathbf{a})}.$ Since $u\mapsto H_{u}$ is injective, we
obtain $\mathbf{x}=\mathbf{a}.$

∎

Note that in the proof of Theorem 3.1, the well-defined condition is
necessary, since otherwise there might be positive subspaces which intersect
$\mathbb{P}(\ker Z^{T})$. In this case, there is no guarantee that a
hyperplane containing a positive subspace also contains a positive subspace
disjoint from $\mathbb{P}(\ker Z^{T})$, which might present problems on the
boundary of the Grasstope.

However, as we show in the following results, we can still describe the open
rational Grasstope in the case that the map $\tilde{Z}$ is not well-defined on
$\text{Gr}_{\geq 0}(k,n)$. We first need the following lemma about totally
positive subspaces to show that intersection with $\mathbb{P}(\ker Z^{T})$
does not cause issues.

######  Lemma 3.2.

Let $H$ be a hyperplane in $\mathbb{P}^{n-1}$ containing an
$(n-k-1)$-dimensional subspace $P$. If $H$ contains a totally positive
$(k-1)$-dimensional subspace, then it contains a totally positive
$(k-1)$-dimensional subspace disjoint from $P$.

###### Proof.

Let $V\subset H$ be a totally positive $(k-1)$-dimensional subspace. Let
$l=\dim(V\cap P)$. Then $\dim(V+P)=n-l-2$ and $V+P\subset H$. In particular,
the codimension of $V+P$ inside $H$ is equal to $l$. Pick $l+1$ points
$q_{1},\ldots,q_{l+1}$ in general position in $H\setminus(V+P)$. Let $M$ be a
matrix representing $V+P\in\text{Gr}(n-l-1,n)$ such that the first $k+1$ rows
of $M$ represent $V\in\text{Gr}_{>0}(k,n)$ and there is a subset of rows
$r_{1},\ldots,r_{l+1}$ of $M$ representing $V\cap P\in\text{Gr}(l,n)$. Denote
by $B$ the submatrix of $M$ given by its first $k+1$ rows. Note that the $k-l$
rows $s_{1},\ldots,s_{k-l}$ of $B$ that are not $r_{1},\ldots,r_{l+1}$ span a
subspace $W$ disjoint from $P$. Consider a matrix $M^{\prime}$ obtained from
$M$ by replacing $r_{i}$ with $r_{i}+\varepsilon_{i}\tilde{q}_{i}$ for
$i=1,\ldots,l+1$, where $\varepsilon_{i}>0$ and $\tilde{q}_{i}$ is a vector of
homogeneous coordinates of the point $q_{i}$. Since the points
$q_{1},\ldots,q_{l+1}$ are in general position in $H\setminus(V+P)$, the
matrix $M^{\prime}$ has full rank and represents an $(n-l-2)$-dimensional
subspace of $H$. Denote the first $k+1$ rows of $M^{\prime}$ by $B^{\prime}$.
Then for $\varepsilon_{1},\ldots,\varepsilon_{l+1}$ small enough $B^{\prime}$
represents a subspace $V^{\prime}\in\text{Gr}_{>0}(k,n)$. We claim that
$V^{\prime}$ is disjoint from $P$. To show this, consider the matrix $D$
obtained by stacking $B^{\prime}$ below a matrix $C$ representing $P$. Note
that $D$ represents the projective subspace $V^{\prime}+P$. Since the points
in $\mathbb{P}^{n-1}$ defined by rows $r_{1},\ldots,r_{l+1}$ are in $P$, by
performing row operations on $D$, one can reduce it to the form
$D^{\prime}=\begin{pmatrix}C\\\ \hline\cr Q\end{pmatrix}$ , where $Q$ consists
of the rows $s_{1},\ldots,s_{k-l},\tilde{q}_{1},\ldots,\tilde{q}_{l+1}$. Due
to the choice of $q_{1},\ldots,q_{l+1}$ and the fact that $W$ is disjoint from
$P$, $D^{\prime}$ has full rank. This means that
$\dim(V^{\prime}+P)=\dim(V^{\prime})+\dim(P)$, so $V^{\prime}$ and $P$ are
disjoint. ∎

Following Lemma 3.2, we are ready to describe the open rational Grasstope
$\tilde{Z}(\text{Gr}_{>0}(k,n))$. Recall that $\mathcal{B}(\tilde{Z})$ denotes
the set of base points of $\tilde{Z}$.

######  Proposition 3.3.

For a map $\tilde{Z}:\text{Gr}_{\geqslant
0}(k,n)\dashrightarrow\text{Gr}(k,k+1)$ given by a matrix $Z$ the open
Grasstope $\tilde{Z}(\text{Gr}_{>0}(k,n))$ consists of the points
$\mathbf{x}\in\mathbb{P}^{k}$ such that $var(L(\mathbf{x}))\geq k$, where
$L(\mathbf{x})$ is the vector of twistor coordinates of $\mathbf{x}$ with
respect to $Z.$

###### Proof.

We will prove this statement by slightly modifying the proof of Theorem 3.1.
Fix a totally positive matrix
$A\in\text{Gr}_{>0}(k,n)\setminus\mathcal{B}(\tilde{Z})$ and let $\mathbf{x}$
be the vector of Plücker coordinates of $[AZ].$ Then the hyperplane
$H_{L(\mathbf{x})}$ contains $\mathbb{P}(\ker Z^{T})$ and the totally positive
subspace $A$. Conversely, if a hyperplane $H$ contains $\mathbb{P}(\ker
Z^{T})$ and a totally positive subspace $A$, then by Lemma 3.2 $H$ contains a
totally positive subspace $A^{\prime}$ disjoint from $\mathbb{P}(\ker Z^{T})$
(thus, $A^{\prime}\not\in\mathcal{B}(\tilde{Z})$). The hyperplane is then
uniquely determined by its containment of $\mathbb{P}(\ker Z^{T})$ and
$A^{\prime}$, so it must be $H_{L(\mathbf{x})},$ where $\mathbf{x}$ is the
vector of Plücker coordinates of $[A^{\prime}Z]$. We now use the second part
of Proposition 2.9 to conclude that the open Grasstope
$\tilde{Z}(\text{Gr}_{>0}(k,n))$ consists of the points in
$\mathbf{x}\in\mathbb{P}^{k}$ such that $var(L(\mathbf{x}))\geq k$.

∎

######  Proposition 3.4.

The open rational Grasstope of $Z$ is open and, if $Z$ has no zero rows, the
rational Grasstope of $Z$ is contained in the closure of the open rational
Grasstope of $Z.$

###### Proof.

Let $Z$ define a rational Grasstope and consider a point $\mathbf{x}$ with
$var(L(\mathbf{x}))\geq k$, where $L(\mathbf{x})$ is the vector of twistor
coordinates of $\mathbf{x}$ with respect to $Z.$ Then for all points
$\mathbf{x^{\prime}}$ in a sufficiently small neighborhood around it,
$L(\mathbf{x^{\prime}})$ has the same signs as $L(\mathbf{x})$ in all the
indices of the nonzero entries of $L(\mathbf{x})$. Since changing the zero
entries cannot decrease the sign variation, the neighborhood is contained in
the open rational Grasstope.

Similarly to the reasoning of Theorem 3.1, it follows from the second part of
Proposition 2.9 that the rational Grasstope of $Z$ must be contained in the
set $C:=\\{\mathbf{x}:\overline{var}(L(\mathbf{x}))\geq k$}. We show that when
$Z$ has no zero rows, $C$ is the closure of the open rational Grasstope of
$Z$. First, we show that the complement of $C$, the set
$\\{\mathbf{x}:\overline{var}(L(\mathbf{x}))<k\\},$ is open. Let $\mathbf{x}$
be such that $\overline{var}(L(\mathbf{x}))<k$. Then for all points
$\mathbf{x^{\prime}}$ in a sufficiently small neighborhood around
$\mathbf{x}$, $L(\mathbf{x^{\prime}})$ has the same signs as $L(\mathbf{x})$
in all the indices of the nonzero entries of $L(\mathbf{x})$. Since changing
the values of the zero entries cannot increase $\overline{var}$, we know
$\overline{var}(L(\mathbf{x^{\prime}}))<k$. Therefore $C$ is closed.

Now consider a point $\mathbf{x}\in C$ and an open neighborhood $N$ of points
around it. We will show that there is some $\mathbf{x^{\prime}}\in N$ with
$var(L(\mathbf{x}^{\prime}))\geq k$, which is sufficient to conclude that $C$
is the closure of the open rational Grasstope of $Z$. Since $Z$ has no zero
rows, each zero entry of $L(\mathbf{x})$ corresponds to containment of
$\mathbf{x}$ in a hyperplane. Any open neighborhood around $\mathbf{x}$
contains points on either side of the hyperplane. Similarly, if $\mathbf{x}$
lies in the intersection of several hyperplanes, any open neighborhood of
$\mathbf{x}$ contains points in each orthant defined by these hyperplanes, so
any sign pattern can be achieved in the entries which are zero in
$L(\mathbf{x})$. In particular, this means that there is a point
$\mathbf{x^{\prime}}$ with signs in the nonzero entries of $L(\mathbf{x})$
equal to the signs of the corresponding entries of $L(\mathbf{x^{\prime}})$
(since we can restrict $N$ to be small enough such that the signs in the
nonzero entries of $L(\mathbf{x})$ are unchanged), and signs of the zero
entries replaced by the signs which ensure that
$var(L(\mathbf{x^{\prime}}))\geq k$.

∎

In Section 4, we show that the rational Grasstope may be equal to the closure
of the open rational Grasstope. However, we do not know if this holds in
general, since as far as we can tell, it might be possible for points in the
boundary to fall out. Thus the question of fully describing which parts of the
boundary are contained in $m=1$ rational Grasstopes remains open.

##  4 Examples

In this section we provide examples of the families of Grasstopes we
considered. We begin with an example of a tame Grasstope.

######  Example 4.1 (A tame Grasstope).

Let

$Z=\begin{bmatrix}1&0&-1&-3&-2\\\ 0&1&1&2&1\\\ 0&0&1&-1&-2\end{bmatrix}^{T}.$

This matrix is not totally positive, since $p_{123}=1$ and $p_{124}=-1$.
However, the first two rows of $Z^{T}$ span a totally positive line, so $Z$
satisfies (1) with the matrix $M$ being

$\begin{bmatrix}1&0&0\\\ 0&1&0\end{bmatrix}^{T}.$

Therefore the resulting Grasstope is tame but is not an amplituhedron. The
rows of $Z$ define $5$ linear forms, as noted in Remark 2.7:

$l_{1}=z,\quad l_{2}=-y,\quad l_{3}=x-y-z,\quad l_{4}=-x-2y-3z,\quad
l_{5}=-2x-y-2z.$

Not every affine chart that we choose results in a bounded picture. For
instance, if we map $(x:y:z)\mapsto(x+y:y:z)$ and dehomogenize with respect to
the first coordinate, the resulting picture is unbounded. However, as
predicted by Corollary 2.4, there are lines disjoint from the Grasstope. One
of them is $\\{-4x+z=0\\}$ in $\mathbb{P}^{2}$. By picking it to be the line
at infinity (that is, by mapping $(x:y:z)\mapsto(-4x+z:y:z)$ and
dehomogenizing with respect to the first coordinate), we obtain affine lines
given by the linear forms

$\tilde{l}_{1}=\tilde{y},\quad\tilde{l}_{2}=-\tilde{x},\quad\tilde{l}_{3}=\frac{-1-4\tilde{x}-3\tilde{y}}{4},\quad\tilde{l}_{4}=\frac{-1-8\tilde{x}-13\tilde{y}}{4},\quad\tilde{l}_{5}=\frac{1-2\tilde{x}-5\tilde{y}}{2}.$

Each line has an orientation, with the positive half-space given by the points
$(\tilde{x},\tilde{y})$ for which $\tilde{l}(\tilde{x},\tilde{y})\geq 0$. The
lines divide the affine plane into regions, each of which has a corresponding
sign vector with $i^{th}$ coordinate being $+$ if
$\tilde{l}_{i}(\tilde{x},\tilde{y})>0$ for all $(\tilde{x},\tilde{y})$ in the
region, and $-$ if $\tilde{l}_{i}(\tilde{x},\tilde{y})<0$. The Grasstope of
$Z$ consists exactly of those points in the regions for which
$\overline{var}(u)\geq 2$, as can be seen in Figure 1.

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.

######  Example 4.2 (A wild Grasstope).

Let

$Z=\begin{bmatrix}2&2&0&-1&1&0\\\ 2&3&1&0&2&0\\\
2&2&0&0&2&1\end{bmatrix}^{T}.$

This is the example found by Galashin (and communicated to us by Lam [15]) to
show that wild Grasstopes exist. Indeed, suppose there exists a $3\times 2$
matrix $M$ such that $ZM$ has positive $2\times 2$ minors. The $2\times 2$
minors of $ZM$ are the entries of
$\wedge_{2}(ZM)=\wedge_{2}(Z)\times\wedge_{2}(M)=$

$=\setcounter{MaxMatrixCols}{15}\begin{bmatrix}2&2&2&2&3&1&2&1&-1&-2&0&0&0&0&0\\\
0&0&0&2&2&0&2&2&0&-2&2&2&0&-1&1\\\
-2&-2&-2&0&0&0&0&2&2&0&2&3&1&0&2\end{bmatrix}^{T}\begin{bmatrix}p_{12}\\\
p_{13}\\\ p_{23}\end{bmatrix},$

where $p_{12},p_{13},p_{23}$ are the minors of $M$. Then, column 4 of
$(\wedge_{2}Z)^{T}$ tells us that $p_{12}+p_{13}>0$ but column 10 tells us
that $p_{12}+p_{13}<0$, so no such matrix $M$ can exist.

The six rows of $Z$ correspond to the six linear forms

$\displaystyle l_{1}$ $\displaystyle=2x-2y+2z,$ $\displaystyle l_{2}$
$\displaystyle=2x-3y+2z,$ $\displaystyle l_{3}$ $\displaystyle=-y$
$\displaystyle l_{4}$ $\displaystyle=-z,$ $\displaystyle l_{5}$
$\displaystyle=2x-2y+z,$ $\displaystyle l_{6}$ $\displaystyle=x.$

Mapping $(x:y:z)\mapsto(x+y:y:z)$ and dehomogenizing with respect to the first
coordinate, we obtain affine lines given by the linear forms

$\displaystyle\tilde{l}_{1}$ $\displaystyle=2\tilde{y}-4\tilde{x}+2,$
$\displaystyle\tilde{l}_{2}$ $\displaystyle=2\tilde{y}-5\tilde{x}+2,$
$\displaystyle\tilde{l}_{3}$ $\displaystyle=-\tilde{x},$
$\displaystyle\tilde{l}_{4}$ $\displaystyle=-\tilde{y},$
$\displaystyle\tilde{l}_{5}$ $\displaystyle=\tilde{y}-4\tilde{x}+2,$
$\displaystyle\tilde{l}_{6}$ $\displaystyle=-\tilde{x}+1.$

We draw these in the affine plane and color them (in order) red, orange,
yellow, green, blue, and purple. We also give the lines orientations with the
positive half-space given by the points $(\tilde{x},\tilde{y})$ for which
$\tilde{l}(\tilde{x},\tilde{y})>0$. Then the Grasstope of $Z$ consists exactly
of those points in the regions between the lines for which
$\overline{var}(u)\geq 2$, as can be seen in Figure 2.

Figure 2: 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.

######  Example 4.3 (A rational Grasstope with closed boundary and Möbius
strip topology).

Let

$Z=\begin{bmatrix}1&0&0&-1&0&0\\\ 0&1&0&-1&1&-1\\\
0&0&3&0&-2&-1\end{bmatrix}^{T}.$

Note that $(1,1,1,1,1,1)\in\ker(Z^{T})$, so the map $\tilde{Z}$ has base
points on $\text{Gr}_{\geq 0}(2,6)$. Following Proposition 3.3 we can still
describe its open Grasstope. As in the previous examples, we find 6
dehomogenized linear forms corresponding to six affine lines

$\displaystyle\tilde{l}_{1}$ $\displaystyle=\tilde{y},$
$\displaystyle\tilde{l}_{2}$ $\displaystyle=-\tilde{x},$
$\displaystyle\tilde{l}_{3}$ $\displaystyle=-3\tilde{x}+3,$
$\displaystyle\tilde{l}_{4}$ $\displaystyle=\tilde{x}-\tilde{y},$
$\displaystyle\tilde{l}_{5}$ $\displaystyle=\tilde{x}-2,$
$\displaystyle\tilde{l}_{6}$ $\displaystyle=2\tilde{x}-1.$

Then we can find the open rational Grasstope of $Z$ as in Figure 3.

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.

We claim that the rational Grasstope of $Z$ is the closure of the open
rational Grasstope. One can check this by directly finding Plücker coordinates
for some $M\in\text{Gr}_{\geq 0}(2,6)$ which map to the points that lie on the
boundary. For example, to find such $M$ for any point of the form $(0,a)$ with
$a\geq 0$, we solve $\wedge_{2}(Z)\times\wedge_{2}(M)=(1:0:a)$. Then
$\wedge_{2}(M)$ must have nonnegative entries which satisfy the Plücker
relations. This process is made easier if one recalls that the Plücker
relations are trivially satisfied if all entries are zero except for ones
which correspond to pairwise overlapping submatrices, that is any two nonzero
minors come from submatrices which share a column. In this case, since

$\wedge_{2}(Z)=\setcounter{MaxMatrixCols}{15}\begin{bmatrix}1&0&0&-1&1&0&1&0&0&-1&-1&0&0&1&0\\\
0&3&0&0&0&3&-2&0&0&2&-1&0&0&1&0\\\
0&0&3&0&0&3&0&-2&-3&2&0&-1&3&1&-3\end{bmatrix}^{T},$

we find that the point $(1:0:a)$ is given by
$\wedge_{2}(M)=(0:0:a/3:0:1:0:0:0:0:0:0:0:0:0:0)$. Since the only nonzero
entries correspond to the minors $p_{23}$ and $p_{24}$, the Plücker relations
are satisfied. Thus the portion of the boundary line $x=0$ with $y\geq 0$ is
part of the rational Grasstope of $Z$. One can similarly check that all other
parts of all six lines are included. Therefore the rational Grasstope of $Z$
is closed. Furthermore, topologically, as one can see from Figure 3, it is a
Möbius strip.

##  5 Background on Oriented Matroids

Much of the material on hyperplane arrangements can be naturally generalized
to oriented matroids [2]. In this section, we review the basics of oriented
matroid theory and recall a dictionary between hyperplane arrangements and
oriented matroids. This will prove useful in algorithmically counting the
number of regions in a given Grasstope (see Section 6). We begin by discussing
signed circuits, which we will often abbreviate as circuits if the meaning is
clear from context.

######  Definition 5.1 (Signed circuit axioms).

An oriented matroid consists of a ground set $\mathcal{E}$ and a collection
$\mathcal{C}$ of tuples of the form $X=(X^{+},X^{-})$ called _circuits_ ,
where $X^{+},X^{-}$ are disjoint subsets of $\mathcal{E}$ satisfying

  1. 1.

$\emptyset$ is not a circuit.

  2. 2.

If $X$ is a circuit, then so is $-X=(X^{-},X^{+})$.

  3. 3.

No proper subset of a circuit is a circuit.

  4. 4.

(Elimination). If $X_{0}$ and $X_{1}$ are two signed circuits with $X_{0}\neq
X_{1}$ and $e\in X_{0}^{+}\cap X_{1}^{-},$ then there is a third circuit
$X\in\mathcal{C}$ with $X^{+}\subset(X_{0}^{+}\cup X_{1}^{+})\setminus\\{e\\}$
and $X^{-}\subset(X_{0}^{-}\cup X_{1}^{-})\setminus\\{e\\}.$

We can obtain the oriented matroid of a matrix as follows.

######  Definition 5.2 (Oriented matroid of a matrix).

Fix a matrix $A$ and let $\sum_{i}\lambda_{i}v_{i}$ be a minimal linear
dependency among its rows. Associate to this dependency the signed set
$X=(X^{-},X^{+}),$ where

$\displaystyle X^{-}=\\{i:\lambda_{i}<0\\}$ $\displaystyle
X^{+}=\\{i:\lambda_{i}>0\\}.$

Then the _oriented matroid $\mathcal{M}_{A}$_ associated to $A$ has as its
signed circuits the signed sets coming from minimal linear dependencies.

One can check that oriented matroids of matrices satisfy the signed circuit
axioms.

######  Example 5.3 (Matroid of a matrix $A$).

Consider the matrix

$A=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 2&-3&4\end{bmatrix}.$

Then the only linear dependency up to scaling is
$v_{4}-4v_{3}+3v_{2}-v_{1}=0.$ Thus the only circuit (when only one of $X,-X$
is considered) is $24\overline{13},$ where we use a more compact notation in
which the bar indicates being in the negative part of the signed set.

######  Remark 5.4.

One can also define $\mathcal{M}_{A}$ by signed bases; the matroid is the map
which assigns to each size $k$ subset $I\subset[n]$ the sign of the
determinant of $A_{I}.$ This definition is called the _chirotope definition_
[2, page 6] and satisfies the _chirotope axioms,_ which we do not describe
here. In particular, the database [5] used in Section 6 indexes matroids by
chirotope.

We recall a few definitions we will need to explain how a hyperplane
arrangement can be viewed as an oriented matroid.

######  Definition 5.5 (Composition).

Let $X=(X^{+},X^{-})$ and $Y=(Y^{+},Y^{-})$ be signed sets. Then their
_composition_ $X\circ Y$ is $(X^{+}\cup(Y^{+}\setminus
X^{-}),X^{-}\cup(Y^{-}\setminus X^{+})).$

######  Definition 5.6 (Orthgonality).

Let $X=(X^{+},X^{-})$ and $Y=(Y^{+},Y^{-})$ be signed sets. Define
$S(X,Y)=(X^{+}\cap Y^{-})\cup(X^{-}\cap Y^{+}).$ We say $X$ and $Y$ are
_orthogonal_ if $S(X,Y)$ and $S(X,-Y)$ are both empty or both non-empty.

We define _vectors_ as compositions of circuits, and _cocircuits_ and
_covectors_ as the circuits and vectors of the dual matroid [2, page 4],
respectively. An equivalent definition which is easier for computing is that
the covectors of $\mathcal{M}$ are the signed sets which are orthogonal to all
vectors of $\mathcal{M}$. For more detail, see [2, Chapter 1]. There is yet
another definition in the case of vector configurations.

######  Definition 5.7 (Oriented matroid of a vector configuration).

One can also view the rows $v_{i}$ of the matrix $A$ as vectors in
$\mathbb{R}^{k}.$ For such a vector configuration, the covectors can be
defined as the set of tuples $Y_{H}=(Y_{H}^{+},Y_{H}^{-})$ as $H$ runs over
oriented hyperplanes, where $Y_{H}^{+}$ is the set of vectors in the positive
halfspace defined by $H$, and $Y_{H}^{-}$ is the set of vectors in the
negative halfspace. The _cocircuits_ are the minimal covectors. They arise
from hyperplanes that are spanned by subsets of $\\{v_{1},...,v_{n}\\}.$

######  Definition 5.8 (Oriented matroid of a hyperplane arrangment).

Let $H_{i}$ be the hyperplane given by the vanishing of $l_{i}(x)=a_{i}\cdot
x.$ Then the oriented matroid of the arrangement $\\{H_{i}\\}_{i=1}^{n}$ is
the matroid of the vector configuration given by $a_{1},...,a_{n},$ or
equivalently, the oriented matroid of the matrix with rows $a_{1},...,a_{n}.$

######  Remark 5.9.

Faces of a hyperplane arrangement correspond to covectors of its oriented
matroid, and regions correspond to maximal covectors. The _rank_ (denoted by
$r$) of the oriented matroid is $k+1,$ where $k$ is the dimension of the
ambient space [2, Chapter 1].

######  Example 5.10 (Matroid of a matrix $A$).

Let $A$ be the matrix

$A=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 2&-3&4\end{bmatrix}$

as in Example 5.3. The cocircuits are given considering hyperplanes $H$
spanned by pairs of rows. For example, if
$H=\text{Span}\\{a_{1},a_{3}\\}=\\{y=0\\},$ then $a_{2}$ is in the positive
halfspace (since the $y$-coordinate is $1$) and $a_{4}$ is in the negative
halfspace (since the $y$ coordinate is $-3$). Thus we obtain the cocircuit
$2\bar{4}$.

The total set of cocircuits (not including negations) is
$\\{2\bar{4},34,23,14,3\bar{1},12\\}$. In the previous sections we have used
sign vector notation; that is, to each signed set we associate a length $n$
vector with $\pm$ at index $i$ if $i$ is in $X^{\pm}$ and $0$ otherwise.
Applying this convention, one can check that the covectors are exactly the
sign vectors with fewer than $3$ sign changes. For example, $2\bar{4}\circ
3\bar{1}=23\overline{14},$ which corresponds to $(-++-).$

######  Definition 5.11.

A matroid is _realizable_ if it arises from a hyperplane arrangement over
$\mathbb{R}$.

##  6 Extremal Counts and Oriented Matroid Grasstopes

Given an $n\times(k+1)$ matrix $Z$, one may ask questions about the topology
of the resulting $m=1$ Grasstope. For instance, is it closed, connected,
contractible? How many regions of the hyperplane arrangement does it contain?
For the $m=1$ amplituhedron, the answer to the first three questions is “yes”
[13, Corollary 6.18]. As for the latter, the $m$ = 1 amplituhedron contains as
few regions as possible, that is, all possible sign vectors with $var<k$
appear as labels of regions in the corresponding arrangement [13, Proposition
6.14]. In this section, we investigate this last question for more general
$m=1$ Grasstopes.

######  Definition 6.1.

A _region_ of a hyperplane arrangement is a connected component of the
complement of the union of hyperplanes in the arrangement.

In [23], Zaslavsky gives the following formulae for the number of total
regions $t(\mathcal{A})$ and bounded regions $b(\mathcal{A})$ of an affine
arrangement $\mathcal{A}$ of $n$ generic hyperplanes in $\mathbb{A}^{k}$:

$\displaystyle t(\mathcal{A})$
$\displaystyle=1+n+\binom{n}{2}+...+\binom{n}{k},$ $\displaystyle
b(\mathcal{A})$ $\displaystyle=\binom{n-1}{k}.$

Note that since intersections of two hyperplanes in any projective arrangement
$\mathcal{P}$ have codimension two in $\mathbb{P}^{k}$ and there are a finite
number of them, we can always find some hyperplane avoiding them, and hence an
an affine chart which contains all of them (see Figure 4). A projective
arrangement $\mathcal{P}$ naturally induces an affine arrangement
$\mathcal{A}$ in this affine chart. If a region of $\mathcal{P}$ intersects
the chosen hyperplane at infinity, it induces two unbounded regions of
$\mathcal{A}$. Otherwise it induces a single bounded region. Thus, the total
number of regions in $\mathcal{P}$ is

$r(\mathcal{P})=b(\mathcal{A})+\frac{t(\mathcal{A})-b(\mathcal{A})}{2}.$ (2)

$---$$--+$$-++$$-+-$$++-$$+--$$+++$$1$${\color[rgb]{0.29,0.56,0.89}\definecolor[named]{pgfstrokecolor}{rgb}{0.29,0.56,0.89}2}$${\color[rgb]{0.49,0.83,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.49,0.83,0.13}3}$
Figure 4: An oriented hyperplane arrangement and its sign labels.

######  Definition 6.2.

Let $\mathcal{P}$ be an arrangement of $n$ hyperplanes in $\mathbb{P}^{k}$.
Then $\mathcal{P}$ is called _simple_ if the intersection of any $i$
hyperplanes in $\mathcal{P}$ has codimension $i$ in $\mathbb{P}^{k}$ for all
$i\leq k$, and is empty for $i>k$.

We write $\beta(k,n)$ for the number of possible sign patterns of length $n$
with sign variation less than $k$ and $\gamma(k,n)$ for the number of sign
patterns with variation greater or equal than $k$ (we identify sign patterns
$\sigma$ and $-\sigma$). Note that
$\beta(k,n)=1+\binom{n-1}{2}+...+\binom{n-1}{k},$ and
$\gamma(k,n)=2^{n-1}-\beta(k,n).$ Theorem 3.1 and Proposition 3.4 then give
the lower bound $r(\mathcal{P})-\beta(k,n)$ for the number of regions in
$\mathcal{G}_{n,k,1}(Z)$, where $\mathcal{P}$ is the hyperplane arragnement
defined by $Z$. An upper bound is given by the minimum of $\gamma(k,n)$ and
$r(\mathcal{P})$.

The database [5] contains a catalog of isomorphism classes of oriented
matroids [6, Section 6]. Each matroid is indexed by a vector of signs of its
bases and each hyperplane arrangement corresponds to a realizable matroid, as
explained in the previous section. An arrangement is simple if the vector of
signs of bases of its matroid does not contain zeros, that is, the matroid is
uniform.

We iterate over all uniform oriented matroid isomorphism classes in this
catalog for small values of $k$ and $n$. Note that, for all the values of $k$
and $n$ which we consider, all uniform matroids are realizable [8], that is,
arise from hyperplane arrangements. Within each isomorphism class, we iterate
over all possible reorderings of the ground set and reorientations. If the
matroid is realizable, at the level of the matrix $Z$ defining the arrangement
as described in Section 2, reorderings correspond to permuting the rows and
reorientations to negating certain rows. For each isomorphism class, ordering
of the hyperplanes, and a choice of orientation, we compute the number of
regions in the corresponding Grasstope (that is, the number of maximal
covectors with sign variation greater or equal than $k$; see Remark 5.9). It
turns out that for many values of $k$ and $n$ the minimal and maximal number
of regions in the Grasstope when iterating over reorderings and reorientations
does not depend on the oriented matroid isomorophism class. The minimal and
maximal number of regions in the Grasstope for these values of $k$ and $n$ are
presented in Table 1. The Python code used to extract this data is available
at https://mathrepo.mis.mpg.de/Grasstopes. We therefore have a computational
proof of the following statement.

######  Proposition 6.3.

For each pair of values of $k$ and $n$ in Table 1 the minimal and maximal
possible number of regions in a Grasstope arising from a simple arrangement of
$n$ hyperplanes in $\mathbb{P}^{k}$ do not depend on the choice of
arrangement.

Out of the entries in Table 1, note that for $k=2$ and the pairs $(3,5)$ and
$(4,6),$ there is only one oriented matroid up to isomorphism [5]. For $k=2$,
this is because any matrix may be turned into a totally positive matrix by
permuting the rows. This can be done by viewing rows as vectors in the plane
and arranging them in counterclockwise position.

$k$, $n$ |  Minimal |  Maximal |  $r(\mathcal{P})$ |  $\beta(k,n)$ |  $\gamma(k,n)$
---|---|---|---|---|---
$2$, $6$ |  $10$ |  $16$ |  $16$ |  $6$ |  $26$
$2$, $7$ |  $15$ |  $22$ |  $22$ |  $7$ |  $57$
$3$, $5$ |  $4$ |  $5$ |  $15$ |  $11$ |  $5$
$3$, $6$ |  $10$ |  $16$ |  $26$ |  $16$ |  $16$
$4$, $6$ |  $5$ |  $6$ |  $31$ |  $26$ |  $6$
$4$, $7$ |  $15$ |  $22$ |  $57$ |  $42$ |  $22$
Table 1: Minimal and maximal possible number of regions in a Grasstope.

The pair $(3,7)$ does not appear in Table 1. This is the first time we see
variation depending on which simple arrangement we choose, with the maximal
number of regions ranging from $38$ to $42$. The reorientations and
reorderings of a totally positive matrix (i.e. the amplituhedron case) give at
most $42$ regions, while all other oriented matroid classes achieve fewer. We
can see the maximal numbers of regions for other small $k,n$ in Table 2. It
would be interesting to see whether the maximal number of regions attained by
reorienting and reordering a totally positive matrix attains the upper bound
in general.

$k$, $n$ |  Maximal |  $r(\mathcal{P})$ |  $\gamma(k,n)$
---|---|---|---
$3$, $7$ |  $42$ |  $42$ |  $42$
$3$, $8$ |  $64$ |  $64$ |  $99$
$4$, $8$ |  $64$ |  $99$ |  $64$
$5$, $8$ |  $29$ |  $120$ |  $29$
$2$, $9$ |  $37$ |  $37$ |  $247$
$3$, $9$ |  $93$ |  $93$ |  $219$
$4$, $9$ |  $163$ |  $163$ |  $163$
Table 2: Maximal number of regions from reorienting and reordering a positive
matrix.

######  Example 6.4.

Any totally positive $6\times 3$ matrix with the second and fourth rows
negated yields a Grasstope which includes all 16 regions counted by Equation
(2). The resulting hyperplane arrangement is cyclic, with just two
orientations flipped. See Figure 5 to see all of the regions labelled with
sign patterns.

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
$\mathbb{P}^{2}$.

######  Example 6.5.

An example of a $6\times 3$ matrix whose Grasstope has $16$ regions is any
totally positive matrix with the $2$nd and $4$th rows swapped. For examples of
totally positive matrices, one can take the Vandermonde matrix

$\begin{bmatrix}1&1&1&1&1&1\\\ d_{1}&d_{2}&d_{3}&d_{4}&d_{5}&d_{6}\\\
d_{1}^{2}&d_{2}^{2}&d_{3}^{2}&d_{4}^{2}&d_{5}^{2}&d_{6}^{2}\end{bmatrix}^{T}.$

with $0<d_{1}<...<d_{6}.$

Note that the lower bound $r(\mathcal{P})-\beta(k,n)$ for the number of
regions in a Grasstope is actually attained for all $k$ and $n$, by the $m=1$
amplituhedron [13, Proposition 6.14, Theorem 6.16]. The upper bound is also
attained by the $m=1$ amplituhedron for the values in Tables 1 and 2. One
interesting question to study is to determine whether this holds in general,
and to describe all oriented matroids achieving this upper bound.

The dictionary between hyperplane arrangements and oriented matroids (Remark
5.9) guides us to generalize our definition of a Grasstope to oriented
matroids that are not necessarily realizable, such that the definitions agree
when $\mathcal{M}=\mathcal{M}_{Z}$.

######  Definition 6.6 (Grasstope of an oriented matroid).

Let $\mathcal{M}$ be an oriented matroid of rank $r$, and $<$ be a total order
on the ground set $\mathcal{E}$ of $\mathcal{M}.$ Then we define the
_Grasstope $\mathcal{G}(\mathcal{M},<)$_ to be the subset of covectors
$\\{v:\overline{var}(v)\geq r-1\\},$ where $r$ is the rank of $\mathcal{M}$
and the variation is with respect to $<$.

Note that by Topological Representation Theorem [7, Theorem 20], every
oriented matroid arises from a pseudoline arrangement, with covectors
labelling the cells of this arrangement. In particular, the Grasstope
$\mathcal{G}(\mathcal{M},<)$ can be identified with the union of cells of a
pseudoline arrangement that satisfy the sign variation condition from
Definition 6.6. Therefore, Grasstopes of oriented matroids are meaningful
geometric objects, and topological concepts such as connectedness and
contractibility generalize naturally to them. Studying their topological
properties is an interesting topic for future research.

Finally, we use our code to analyze a non-realizable example, which does not
attain the upper bound.

######  Example 6.7.

Consider the non-realizable matroid FMR(8) of rank $4$ on $8$ elements, whose
signed cocircuits are given in [20, Table 1]. Reorientations and reorderings
give at least $34$ and at most $63$ regions. Thus, unlike the amplituhedron,
FMR(8) does not achieve the upper bound of $64$.

## Acknowledgements

The authors would like to thank Bernd Sturmfels for suggesting the topic and
for many helpful conversations. The authors would also like to thank Matteo
Parisi and Thomas Lam for interesting discussions. Y.M. was partially
supported by NSF grant DGE2146752.

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Authors’ addresses:

Yelena Mandelshtam, UC Berkeley<EMAIL_ADDRESS>
Dmitrii Pavlov, MPI-MiS Leipzig<EMAIL_ADDRESS>
Elizabeth Pratt, UC Berkeley<EMAIL_ADDRESS>