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\title{The $m=2$ Amplituhedron and the Hypersimplex}


%\author{Matteo Parisi}
%\author{Melissa Sherman-Bennett}
%\author{Lauren K.\ Williams}
%\address{}
%\email{\href{mailto:matteo.parisi@maths.ox.ac.uk}{matteo.parisi@maths.ox.ac.uk}}
%\email{\href{mailto:msherben@umich.edu}{msherben@umich.edu}}
%\email{\href{mailto:williams@math.harvard.edu}{williams@math.harvard.edu}}



%% define your title in the usual way
%\title[Optional shorter title]{Something to do with math}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[M.~Parisi, M.~Sherman-Bennet, and L.~Williams]{Matteo Parisi\thanks{\href{mailto:mparisi@cmsa.fas.harvard.edu}{mparisi@cmsa.fas.harvard.edu}. Supported by the Sachs Scholarship at Princeton and the ERC grant number 724638.}\addressmark{1,2}, Melissa Sherman-Bennett\thanks{\href{mailto:msherben@umich.edu}{msherben@umich.edu}. Supported by the NSF Graduate Research Fellowship, grant number DGE-1752814.}\addressmark{3}, \and Lauren Williams\thanks{\href{mailto:williams@math.harvard.edu}{williams@math.harvard.edu}. Supported by the NSF grants DMS-1854316 and DMS-1854512, as well as the Harvard--Radcliffe Institute for Advanced Study.}\addressmark{4}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Center for Mathematical Sciences and Applications, Harvard University, Cambridge MA \\ \addressmark{2}School of Natural Sciences, Institute for Advanced Study, Princeton NJ \\
 \addressmark{3}Department of Mathematics, University of Michigan, Ann Arbor MI\\ \addressmark{4}Department of Mathematics, Harvard University, Cambridge MA}


%% put the date of submission here
\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{The hypersimplex $\Delta_{k+1,n}$ is the image of the positive
	Grassmannian $Gr^{\geq 0}_{k+1,n}$ under the moment map.  
	It is a polytope of dimension $n-1$ in $\RR^n$.  
	Meanwhile, the amplituhedron
	$\A$ is the image of $Gr^{\geq 0}_{k,n}$ under an amplituhedron map $\widetilde{Z}$ induced by 
	a positive matrix $Z$.
	Introduced in the context of \emph{scattering amplitudes}, it is not a polytope, and 
	is a full dimensional subset of $Gr_{k,k+2}$.  
	Nevertheless, there seem to be remarkable connections between these two objects,
	as conjectured by Lukowski--Parisi--Williams (LPW).
	We use ideas from oriented matroid theory, total positivity, 
	and the geometry of the hypersimplex and positroid polytopes
	to obtain a deeper understanding of the amplituhedron.
	We show that the inequalities cutting out \emph{positroid polytopes} --- moment map images of positroid cells --- translate into sign conditions cutting out \emph{Grasstopes} --- amplituhedron map images of positroid cells.
	Moreover, we subdivide the amplituhedron into \emph{chambers}, just as the hypersimplex can be subdivided into simplices --- with both chambers and 
	simplices enumerated by the Eulerian numbers.
	We use these properties to prove the main conjecture of (LPW): 
	a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron $\A$ for all $Z$.
	We also prove 
	Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of $\A$.%,
%	and Lukowski--Parisi--Spradlin--Volovich's conjectures on 
%	\emph{$m=2$ cluster adjacency} and on \emph{positroid tiles} 
%	(images of $2k$-dimensional positroid cells which map injectively into $\mathcal{A}_{n,k,2}$). 
%	Finally, we introduce new cluster structures in the amplituhedron.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{oriented matroid, Eulerian number, amplituhedron, matroid polytope, positroid, hypersimplex}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
The \emph{positive Grassmannian}\footnote{More formally, 
	the \emph{totally nonnegative
		Grassmannian}.} $\Grk$ 
is the subset of the real Grassmannian $\Gr_{k,n}$ 
where all Pl\"ucker coordinates
are nonnegative \cite{lusztig,postnikov, rietsch}.  
This is a remarkable space 
with connections to cluster algebras, integrable systems,
and high energy physics \cite{abcgpt,ca1, KodamaWilliams, Scott},
and it has a beautiful CW decomposition into \emph{positroid cells} $S_{\pi}$,
which are indexed by various 
combinatorial objects including 
\emph{decorated permutations}
$\pi$ \cite{postnikov}.

There are several interesting maps which one can apply 
to the positive Grassmannian $\Grk$
and its cells.  The first map is the \emph{moment map} $\mu$, 
initially studied by 
Gelfand--Goresky--MacPherson--Serganova \cite{GGMS} in the context of the Grassmannian
and its torus orbits, who showed that the image of the Grassmannian is the
\emph{hypersimplex} $\Delta_{k,n} \subset \R^{n}$, a polytope of dimension
$n-1$.
When one restricts $\mu$ to $\Grk$,
the image is still the hypersimplex \cite{tsukerman_williams}.


The second map is the \emph{amplituhedron map}, introduced by 
Arkani-Hamed and Trnka~\cite{arkani-hamed_trnka}
in the context of 
\emph{scattering amplitudes} in $\mathcal{N}=4$ SYM.  In particular,
any $n \times (k+m)$ matrix $Z$ with maximal minors positive induces
a map $\widetilde{Z}$ from $\Grk$ to the Grassmannian $\Gr_{k,k+m}$, whose
image has full dimension $mk$ and is called the \emph{amplituhedron}
$\Am$. %\MP{I would use the notation $\mathcal{A}_{n,k,2}(Z)$. In any case, the notation has to be consistent thorough the text.} \MSB{Using the superscript for the moment to save space---we can change it back if we hate it.} \MP{Yes, it's ok for me, now it seems the notation is uniform!}


Given a surjective map $\phi \colon \Grk \to X$,
it is natural to try to decompose $X$ using images
of positroid cells under $\phi$.   
This leads to the following definition.\footnote{There are many reasonable
	variations of this definition.  One might want to relax the injectivity
	assumption, or to impose further restrictions on how boundaries of the 
	images of cells should overlap.}
\begin{definition}\label{def:tri}
	Let $\phi \colon \Grk \to X$ be 
	a continuous surjective map onto a cell complex or subset thereof,
	where $\dim X=d$. A \emph{positroid tile} $\overline{\phi(S_\pi)}$ is (the closure of) the image of a $d$-dimensional positroid cell $S_{\pi}$ on which $\phi$ is injective.
	A \emph{positroid tiling} of $X$ (with respect to $\phi$) 
	is a collection $\{\overline{\phi(S_{\pi})}\}$ of positroid tiles such that pairs of distinct images $\phi(S_{\pi})$
	and $\phi(S_{\pi'})$ are disjoint and $\cup \overline{\phi(S_{\pi})} = X$.
\end{definition}

When $\phi$ is the moment map, the 
(closures of) the
images of the positroid cells $S_{\pi}$ 
are the \emph{positroid polytopes} $\Gamma_{\pi}$
\cite{tsukerman_williams}, so a positroid tiling of the hypersimplex
is a decomposition into positroid polytopes.
When $\phi$ is the amplituhedron map $\widetilde{Z}$,
the (closures of) the images 
of the positroid cells $S_{\pi}$
are \emph{Grasstopes} $Z_{\pi}$, 
which were first studied in~\cite{arkani-hamed_trnka}
as the building blocks of conjectural positroid tilings of the amplituhedron.
Note that neither the amplituhedron nor the Grasstopes are polytopes.



At first glance, the $(n-1)$-dimensional hypersimplex $\Delta_{k+1,n} \subset \R^n$ does not seem to have any relation to the $2k$-dimensional
amplituhedron $\A \subset \Gr_{k,k+2}$.  Nevertheless,~\cite{LPW} showed that there are 
surprising parallels between them.  
In particular, \cite{LPW} showed that \emph{T-duality} gives a bijection between
loopless cells $S_{\pi}$ of $\Gr_{k+1,n}^{\geq 0}$ and coloopless cells 
$S_{\widehat{\pi}}$ of 
$\Grk$, and conjectured that T-duality gives a bijection between
positroid 
tilings %$\{\Gamma_{\pi}\}$ 
of the hypersimplex $\Delta_{k+1,n}$ 
and positroid tilings %$\{Z_{\widehat{\pi}}\}$ 
of the
amplituhedron $\A$. They verified this conjecture for the recursively-defined tilings of the amplituhedron found by~\cite{BaoHe}.

Here, we use \emph{twistor coordinates}
and the geometry of the hypersimplex and positroid
polytopes 
to obtain a deeper understanding of the amplituhedron.
We prove the conjecture of Lukowski--Parisi--Spradlin--Volovich 
\cite{Lukowski:2019sxw} 
classifying the positroid tiles of $\A$. We then give a new characterization of positroid tiles as regions where certain twistor coordinates have specified sign.
We use this result to prove a conjecture  of Arkani-Hamed--Thomas--Trnka~\cite{ATT}:
$\A$ can be characterized using sign flips of twistor coordinates.
%And we prove two results relating the amplituhedron to cluster algebras.
%First, we prove the \emph{cluster adjacency} conjecture
%\cite{Lukowski:2019sxw} for $\A$,
%which says that the 
%Pl\"ucker coordinates labeling facets of a given generalized 
%triangle
%consist of pairwise compatible cluster variables. We also state and prove a generalization of this conjecture by showing that twistor coordinates of a generalized triangle associated to Pl\"ucker coordinates compatible with the ones labelling its facets have constant sign. 
%Second,
%we associate a cluster variety to each
%generalized triangle in $\A \subset \Gr_{k, k+2}$,
%and show that the generalized triangle
%is the totally positive part of that cluster variety.
%We then have the strange phenomenon that 
%the $2k$-dimensional amplituhedron
%$\A$ can be triangulated into ${n-2 \choose k}$
%$2k$-dimensional generalized triangles,
%each of which is the totally positive part of a cluster variety.
%(Moreover, there are many such triangulations.)


Additionally, we draw striking parallels between
$\Delta_{k+1,n}$ and $\A$.
We show that positroid tiles for $\Delta_{k+1, n}$ and $\A$ are in bijection and establish a close connection between the inequalities cutting out corresponding positroid tiles.
Further, we subdivide the amplituhedron into \emph{sign chambers} according to the signs of twistor coordinates. We prove that the \emph{realizable} chambers are exactly enumerated by the Eulerian numbers $E_{k,n-1}$, so the chamber decomposition of the amplituhedron is analogous to the subdivision of the hypersimplex
into simplices enumerated by $E_{k,n-1}$.
We then  prove the main conjecture of \cite{LPW}:
a collection of positroid polytopes is a positroid tiling of 
$\Delta_{k+1,n}$ if and only if
the collection of T-dual Grasstopes is a positroid tiling of 
$\A$ for all $Z$.

%\begin{landscape}
%	\thispagestyle{empty}
%\begin{table}
%	\begin{adjustwidth}{+0.4in}{-1in}
%		\caption{Table of Correspondences via T-duality: the Hypersimplex VS the Amplituhedron.} % title name of the table
%		%	\caption{Table of Correspondences via T-duality $\pi \mapsto \widehat{\pi}$}
%		\centering
%		\begin{tabular}{l c r} % creating 10 columns
%			\hline\hline % inserting double-line
%			\rule{0pt}{3ex}  \large The Hypersimplex $\Delta_{k+1,n}$ & \large VS & \large The Amplituhedron $\AA_{n,k,2}$ \\ [1ex]
%			\hline % inserts single-line
%			% Entering 1st row
%			\rule{0pt}{3ex} $\Delta_{k+1,n}=\mu(\Gr_{k+1,n}^{\geq 0})$ (moment map) & {} & (Amplituhedron map) $\AA_{n,k,2}=\widetilde{Z}(\Gr_{k,n}^{\geq 0})$\\[1ex]
%			\rule{0pt}{3ex} $\mbox{dim}(\Delta_{k+1,n})=n-1$ in $\mathbb{R}^n$  & {} &  $\mbox{dim}(\AA_{n,k,2})=2k$ in $\Gr_{k,k+2}$\\[1ex]
%			\hline
%			\rule{0pt}{2ex} {}& GENERALIZED TRIANGLES (GTs)& {}\\
%			\hline
%			\rule{0pt}{3ex} $\Gamma_{G(\T)}$ (Positroid Polytope) & unpunctured plabic tiling $\T$ in an $n$-gon & (Grasstope) $Z_{\widehat{G}(\T)}$\\[1ex]
%			% Entering 2nd row
%			$x_{[h,j-1]} \geq \area_{\T}(h \rightarrow j)$  & compatible arc $h \rightarrow j$ & $\sgn \langle Y hj\rangle=(-1)^{\area_{\T}(h \rightarrow j)}$\\[1ex]
%			% Entering 3rd row
%			$x_{[h,j-1]} = \area_{\T}(h \rightarrow j)$  & facet defining arc $h \rightarrow j$ &  $\langle Y hj\rangle=0$\\[1ex]
%			\hline
%			\rule{0pt}{2ex} {}&$w$-SIMPLICES and $w$-CHAMBERS& {}\\
%			\hline
%			\rule{0pt}{3ex} $w$-simplex $\Delta_w \subset \Delta_{k+1,n}$ such that: & $w \in S_n$: $w_n=n, \# \des(w)=k$ & $w$-chamber $\widehat{\Delta}_w(Z)\subset \AA_{n,k,2}$ such that:  \\[1ex]
%			$\Delta_w = \conv\{e_{I_1},\ldots,e_{I_n}\}$ & 
%			$I_a=I_a(w):=\cdes(w^{(a-1)})$ &$ \flip(
%			\langle Y a \widehat{1} \rangle , \langle Y a \widehat{2} \rangle, \dots, \langle Y a n \rangle)=I_a \setminus \{a\} $\\[1ex]
%			Hypersimplex $\Delta_{k+1,n}= \bigcup_w {\Delta_w}$ & {} &  Amplituhedron $\AA_{n, k, 2}(Z)= \bigcup_w {\asimp{w}}$\\[1ex]
%			GT $\Gamma_{\pi}= \bigcup_{\Delta_{w} \cap \Gamma^{\circ}_{\pi} \neq \emptyset} {\Delta_{w}}$ & & GT $\gt{\widehat{\pi}}= \bigcup_{\asimp{w} \cap \gto{\widehat{\pi}} \neq \emptyset} {\asimp{w}}$\\[1ex]
%			$\simp{w} \subset \Gamma_{\pi}$ & $\Leftrightarrow$ &$\asimp{w} \subset \gt{\td{\pi}}$\\[1ex]
%			Hypercube $\mbox{\mancube}_{n-1}=\bigcup_k \Delta_{k+1,n}=\bigcup_w \Delta_w$ & $w \in S_n: w_n=n$ & Total Amplituhedron $\mathcal{G}_n^{(2)}=\bigcup_k \mathcal{G}_{n,k,2}=\bigcup_w \widehat{\Delta}_w(\mathcal{G})$ \\[1ex]
%			\hline
%			\rule{0pt}{2ex} {}&TRIANGULATIONS& {}\\
%			\hline
%			\rule{0pt}{3ex}  $\{\Gamma_{\pi}\}$ triangulates $\Delta_{k+1, n}$& $\Leftrightarrow$ & $\{\gt{\td{\pi}}\}$ triangulates $\AA_{n, k, 2}(Z),$ for all $Z$\\[1ex]
%			BCFW triangulation $\{\Gamma_\pi\}$ & BCFW recurrences & BCFW triangulation $\{\gt{\td{\pi}}\}$  \\[1ex]
%			Regular triangulation $\{\Gamma_\pi\}$ & max cones of $\mbox{Trop}^+\Gr_{k+1,n}$ & Regular triangulation $\{\gt{\td{\pi}}\}$  \\[1ex]
%			Catalan triangulation $\{\Gamma_\pi\}$ & Coloring vertices of a fixed planar 
%			tree & Catalan triangulation $\{\gt{\td{\pi}}\}$  \\[1ex]
%			Descent triangulation $\{\Gamma_\pi\}$ & positions of descents/sign flips & Sign-flip triangulation $\{\gt{\td{\pi}}\}$  \\[1ex]
%			% [1ex] adds vertical space
%			\hline % inserts single-line
%			
%		\end{tabular}
%		\label{tab:PPer}
%	\end{adjustwidth}
%\end{table}
%\end{landscape}
\section{Background}\label{sec:background}

\subsection{The Grassmannian and positive Grassmannian}
Fix integers $0<k\leq n$, let $[n]$ denote $\{1,\dots,n\}$, and $\binom{[n]}{k}$ the set of all $k$-element subsets of $[n]$.
The {\itshape (real) Grassmannian} $\Gr_{k,n}$ is the space of all
$k$-dimensional subspaces of $\R^n$.  An element of
$\Gr_{k,n}$ can be viewed as a full rank $k\times n$ matrix, modulo left
multiplication by invertible $k \times k$ matrices. We often abuse notation and identify a full-rank matrix with its rowspan.
%That is, two
%$k\times n$ matrices of rank $k$ represent the same point in $\Gr_{k,n}$ if and only if they
%can be obtained from each other by invertible row operations. For $C$ a full-rank $k \times n$ matrix, we will often abuse notation and write $C \in \Gr_{k, n}$, identifying $C$ with its rowspan.
We embed $\Gr_{k,n}$ into $\PP(\wedge^k \RR^n)$ via the Pl\"ucker embedding: the subspace $V$ represented by matrix $C$ with rows $C_1, \dots, C_k$ is mapped to the line spanned by $C_1 \wedge \cdots \wedge C_k$. We also frequently abuse notation and identify a vector with its span. The \emph{Pl\"ucker coordinates} of $V$, denoted $p_I(V)$ for $I \in \binom{[n]}{k}$, are the homogeneous coordinates of $C_1 \wedge \cdots \wedge C_k$ in $\PP(\wedge^k \RR^n)$, using the standard basis on $\RR^n$ and $\wedge^k \RR^n$. Alternately, $p_I(V)$ is the maximal minor of $C$ using column set $I$.
%Let $[n]$ denote $\{1,\dots,n\}$, and $\binom{[n]}{k}$ the set of all $k$-element subsets of $[n]$. We embed $\Gr_{k,n}$ into projective space $\PP(\wedge^k \RR^n)$ in the usual way. That is, choose $V \in \Gr_{k,n}$ and any representative matrix $C$ with rows $C_1, \dots, C_k$. We map $V$ to the equivalence class of $C_1 \wedge \dots \wedge C_k$ in $\PP(\wedge^k \RR^n)$. This equivalence class depends only on $V$, not on the choice of $C$.

%The embedding $V \mapsto C_1 \wedge \dots \wedge C_k$ gives a natural choice of coordinates for the Grassmannian.
%Let $\{e_1,\dots,e_n\}$ be the standard basis of $\R^n$, and for 
%$I= \{i_1 < i_2 < \dots < i_k\} \subset {[n] \choose k}$, let 
%$E_I:= e_{i_1} \wedge \dots \wedge e_{i_k}.$ 
%Writing $C_1\wedge \dots\wedge C_k$ in terms of the $E_I$, we obtain
%\begin{equation}
%	C_1 \wedge \dots \wedge C_k = \sum_{I\in {[n] \choose k}} p_I(V) E_I \in \wedge^k(\R^n),
%\end{equation}
%where $p_I(V)$ is the maximal minor of $C$ located in column set $I$. The $p_I(V)$ are the {\itshape Pl\"{u}cker coordinates} of $V$, and are independent of $C$ (up to simultaneous rescaling by a  constant).

We will also use the notation $\lrangle{C_1, \dots, C_k}$ for $C_1 \wedge \cdots \wedge C_k$.

\begin{defn}[{\cite[Section~3]{postnikov}}]\label{def:positroid}
	We say that $V\in\Gr_{k,n}$ is {\itshape totally nonnegative} (resp. \emph{totally positive}) if it has a representative $C$ so that $p_I(C)\ge 0$ (resp. $p_I(C)>0$) for all $I\in\binom{[n]}{k}$. In an abuse of notation, we identify $V \in \Gr_{k,n}^{\ge 0}$ with its totally nonnegative representative $C$. The set of all totally nonnegative $V\in\Gr_{k,n}$ is the {\it totally nonnegative Grassmannian} 
	$\Gr_{k,n}^{\ge 0}$. %, 
	%and the set of all totally positive $C$ is the {\itshape totally positive Grassmannian} $\Gr_{k,n}^{>0}$. 
%	\LW{do we need the notion of totally positive part?}\MP{I don't think so}
	 For $\M\subseteq \binom{[n]}{k}$,
	the {\it positroid cell} $S_{\M}$ is
	the set of $C\in\Gr_{k,n}^{\geq 0}$ such that 
	$p_I(C)>0$ if and only if $I\in \M$. We call $\M$ a \emph{positroid} if $S_\M$ is nonempty. 
%	We let $Q_{k,n}$ denote the poset on the cells of $\Gr_{k,n}^{\ge 0}$ defined by
%	$S_{\M} \leq S_{\M'}$ if and only if
%	$S_\M \subseteq \overline{S_{\M'}}$.
\end{defn}


\begin{remark}
	Lusztig \cite{lusztig} earlier defined the positive and totally nonnegative part of a flag variety $G/P$, which Rietsch showed had a cell decomposition \cite{rietsch}. The two definitions, and the two cell complexes, agree \cite{rietsch_private,talaska_williams}.
\end{remark}


%\begin{remark}\label{rem:PL}
%	The positive and nonnegative part of a flag variety $G/P$ was 
%	first introduced by 
%	Lusztig \cite{lusztig} (who gave a Lie-theoretic definition
%	of $(G/P)_{>0}$ and defined $(G/P)_{\geq 0} :=\overline{(G/P)_{>0}}$),
%	and proved to have a cell decomposition by Rietsch \cite{rietsch}.  
%	Postnikov \cite{postnikov} subsequently defined the nonnegative part of the 
%	Grassmannian as in \cref{def:positroid}, and independently gave the above
%	decomposition into cells.
%	From the beginning it was believed by experts that 
%	Postnikov's definition of $\Grk$ should agree
%	with Lusztig's (in the case $G/P$ is the Grassmannian); this was first 
%	proved by Rietsch \cite{rietsch_private}, and reproved  
%	in \cite[Corollary 1.2]{talaska_williams}, where the authors additionally proved
%	that the two cell decompositions
%	coincide.  Two subsequent proofs 
%	that the two definitions of $\Grk$ coincide were given in 
%	\cite{lam, Lusztig3}.
%\end{remark}



There are many ways to index the positroid cells of $\Grk$
\cite{postnikov}, including
\emph{decorated permutations} $\pi$
and \emph{plabic graphs} $G$. Accordingly, we also use the notation $S_\pi$ and $S_G$.

\begin{defn}
	Let $\M$ be a positroid and choose $C \in S_\M$. The \emph{decorated permutation} $\pi$ of $\M$ is defined by $\pi_i:=j$, where $j$ is the label of the first column $c_j$ of $C$ such that $c_i$ lies in the span of $c_{i+1}, c_{i+2}, \dots, c_j$ (where indices are modulo $n$). If $\pi_i=i$, then $i$ is a \emph{loop} if $c_i$ is the zero vector and is a \emph{coloop} otherwise.
\end{defn}

\begin{defn}
	Let $G$ be a \emph{plabic graph}, \textit{i.e.}\ a planar bipartite graph embedded in a disk, with black vertices $1, 2, \dots, n$ on the boundary of the disk. An \emph{almost perfect matching} $M$ of $G$ is a collection of edges which covers each internal vertex of $G$ exactly once. The \emph{boundary} of $M$, denoted $\partial M$, is the set of boundary vertices covered by $M$. The positroid associated to $G$ is the collection $\M=\M(G):=\{\partial M: M \text{ an almost perfect matching of }G \}$.
\end{defn}

See \cref{fig:unpuncTduality} for examples of plabic graphs. We will always assume that plabic graphs are \emph{reduced} \cite[Definition 12.5]{postnikov}, a technical condition meaning roughly that among all plabic graphs with the same positroid, we consider only those with the fewest possible faces.

\subsection{The amplituhedron}

In this section we discuss the \emph{(tree) amplituhedron}, which was introduced by
Arkani-Hamed and Trnka~\cite{arkani-hamed_trnka}.
In what follows, 
we fix positive integers $k, m, n$ with $k+m \leq n$.
We let $\Mat_{n,p}^{>0}$ denote the set of $n\times p$ matrices whose maximal minors
are positive. 

\begin{defn}\label{defn_amplituhedron}
	Let $Z\in \Mat_{n,k+m}^{>0}$.
	Then $Z$ induces a map
$	\widetilde{Z} \colon \Gr_{k,n}^{\ge 0} \to \Gr_{k,k+m}$ which maps
		$C \mapsto CZ.$
%	$$\widetilde{Z}(\langle C_1,\dots, C_k \rangle) := \langle Z(C_1),\dots, Z(C_k) \rangle.$$
%	Equivalently, if $C$ is a matrix representing an element of 
%	$\Gr_{k,n}^{\ge 0}$, then $\widetilde{Z}(C)$ is defined to be 
%	the element of $\Gr_{k,k+m}$
%	represented by 
%	the matrix $CZ$.  
	The \emph{(tree) amplituhedron} $\Am$ is  the image
	$\widetilde{Z}(\Gr_{k,n}^{\ge 0})$ inside $\Gr_{k,k+m}$.
\end{defn}

The fact that $Z$ has positive maximal minors ensures that $\widetilde{Z}$
is well defined~\cite{arkani-hamed_trnka}. 
%See \cite[Theorem 4.2]{karp} for a necessary and sufficient condition 
%(in terms of sign-variation)
%for a matrix $Z$ 
%to give rise to a well-defined map $\widetilde{Z}$.
The amplituhedron $\Am$  has full dimension $km$ inside $\Gr_{k,k+m}$. In special cases the amplituhedron recovers familiar objects: $\mathcal{A}_{k+m,k,m}^{Z}$ is isomorphic to $\Gr_{k,n}^{\ge 0}$, as in this case $Z$ is a square matrix; $\mathcal{A}_{n,1,m}^{Z}$ is a {\itshape cyclic polytope} in projective space $\PP^m$ \cite{Sturmfels}; and  $\mathcal{A}_{n,k,1}^{Z}$ can be identified with the 
complex of bounded faces of a cyclic hyperplane arrangement 
\cite{karpwilliams}.
% If $Z$ is a square matrix, \textit{i.e.}
%$k+m=n$, then $\Am$ is isomorphic to
%$\Gr_{k,n}^{\ge 0}$.  $k=1$,  $\mathcal{A}_{n,1,m}^{Z}$ is a {\itshape cyclic polytope} in projective space $\PP^m$ \cite{Sturmfels}.
%If $m=1$, then $\mathcal{A}_{n,k,1}(Z)$ can be identified with the 
%complex of bounded faces of a cyclic hyperplane arrangement 
%\cite{karpwilliams}.

We will consider the restriction of the $\widetilde{Z}$-map to positroid 
cells in 
$\Gr_{k,n}^{\ge 0}$. Recall the definition of positroid tile\footnote{In the literature, \textit{e.g.}~\cite{Lukowski:2019sxw}, these are sometimes called ``generalized triangles''.} from \cref{def:tri}, with $\phi$ taken to be the $\widetilde{Z}$ map.

\begin{definition}
	Let
	$Z\in \Mat_{n,k+m}^{>0}$.  
	Given a positroid cell $S_{\pi}$ of 
	$\Gr_{k,n}^{\ge 0}$, we define the \emph{open Grasstope}
	$\gto{\pi} := \widetilde{Z}(S_{\pi})$ and the \emph{Grasstope}\footnote{This is short for ``Grassmann polytope'', language introduced in \cite{lam}.}
	$\gt{\pi} := \widetilde{Z}(\overline{S_{\pi}})$.
%	If $\gt{\pi}$ is a positroid tile of $\A$, we call
%$\gto{\pi}$
%an \emph{open positroid tile}
%	of $\Am$.
\end{definition}
%\MP{Do we use somewhere later ``open positroid tile''? If not, maybe remove that?} \MSB{I think we don't use it--thanks!}

By \cite[Proposition 15.2]{lam},  $\widetilde{Z}(\overline{S}_\pi)=\overline{
	\widetilde{Z}(S_{\pi})}$, so the Grasstope $\gt{\pi}$ is the closure of the open Grasstope $\gto{\pi}$. If $k=1$ and $m=2$, the amplituhedron $\mathcal{A}_{n,1,2}^{Z}$ is a convex $n$-gon in $\PP^2$. Its positroid tiles are exactly the triangles using 3 vertices of the polygon.


%\begin{defn}\label{def:facetG} \MSB{omit if unneeded}
%	Let $\gt{\pi}$ be a Grasstope of 
%	$\Am$. We say that $\gt{\pi'}$ is a \emph{facet} of $\gt{\pi}$ if it is maximal by inclusion among the Grasstopes $\gt{\mu}$ such that $S_\mu$ is contained in $\overline{S_\pi}$, $\gt{\mu}$ is contained in the boundary $\partial \gt{\pi}$, and $\gt{\mu}$ is codimension 1 in $\gt{\pi}$.
%\end{defn}



%\begin{rmk}\label{rmk:closure} By \cite[Proposition 15.2]{lam},  $\widetilde{Z}(\overline{S}_\pi)=\overline{
%		\widetilde{Z}(S_{\pi})}$.
%\end{rmk}


Images of positroid cells under the map $\widetilde{Z}$ 
have been studied since~\cite{arkani-hamed_trnka}, which conjectured that the images
of certain \emph{BCFW} collections of $4k$-dimensional cells in 
$\Gr_{k,n}^{\geq 0}$ give a positroid tiling of the amplituhedron 
$\mathcal{A}_{n,k,4}^{Z}$.

\begin{remark}
	It is believed that many 
	combinatorial properties of $\Am$ are independent of $Z$, 
	for example, whether or not 
	$\gt{\pi}$ is a positroid tile.
	We will see that this 
	is true in \cref{cor:GTsInBijection} for $m=2$.
	It is also believed that the collections of positroids $\{\M\}$ indexing a positroid tiling 
	of 
	$\Am$ should be independent of $Z$. 
\end{remark}


\subsubsection{Twistor coordinates and the sign stratification of \texorpdfstring{$\Am$}{Am}}
Our results on  $\A$ will use functions on $\Gr_{k,k+2}$ called \emph{twistor coordinates}~\cite{ATT,arkani-hamed_trnka}.

\begin{definition}\label{def:twistor}
	Choose
	$Z \in \Mat_{n,k+m}^{>0}$ and denote its rows by $Z_1,\dots, Z_n \in \R^{k+m}$.
	Given a matrix $Y$ with rows $Y_1,\dots,Y_k$ representing an element of 
	$Gr_{k,k+m}$, 
	and $i_1,\dots, i_m$  a sequence of elements of $[n]$,
	the \emph{twistor coordinate} is defined to be
	$$\langle Y i_1 i_2 \cdots i_m \rangle:=
	\langle Y_1, \dots,Y_k, Z_{i_1}, \dots, Z_{i_m} \rangle,$$ 
	the determinant of the 
	$(k+m) \times (k+m)$ matrix with rows 
	$Y_1, \dots,Y_k, Z_{i_1}, \dots, Z_{i_m}$. 
	
\end{definition}

Note that the twistor coordinates of $Y \in \Gr_{k, k+m}$ are defined only up to a common scalar multiple. It follows from \cite[Lemma 3.10, Proposition 3.12]{karpwilliams} that an element of $\Gr_{k, k+m}$ is uniquely determined by its twistor coordinates. In fact, \cite{karpwilliams} shows that $\Gr_{k,k+m}$ can be embedded into $\Gr_{m,n}$ so that the twistor coordinate $\lrangle{Y i_1 \cdots i_m}$ is the pullback of the Pl\"ucker coordinate $p_{i_1, \dots, i_m}$ in $\Gr{m,n}$.

Since $Y\in \Gr_{k,k+m}$ is uniquely determined by its twistor coordinates, 
we can 
 stratify $\Am \subset \Gr_{k,k+m}$ by the signs of the twistor coordinates.
This was done in \cite{karpwilliams} when $m=1$.
Moreover, this sign stratification is closely related to
the \emph{oriented matroid stratification} of the Grassmannian $\Gr_{m,n}$ using the embedding $\Gr_{k,k+m} \hookrightarrow \Gr_{m,n}$ mentioned above.

%, which partitions 
%elements of the real Grassmannian into strata based on the signs of the Pl\"ucker coordinates.
%By \cref{prop:twistor}, the twistor coordinates of $Y\in \Am$ are 
%Pl\"ucker coordinates 
%on the corresponding element of the \emph{B-amplituhedron} \cite{karpwilliams} or \emph{amplituhedron in momentum twistor space} \cite{ATT},
%so this sign stratification reduces to the oriented matroid stratification 
%in momentum twistor space.


\begin{definition}[Amplituhedron chambers]\label{def:chamber}
	Fix positive $k<n$ and $m$ such that $k+m \leq n$.
	Let $\sigma= (\sigma_{i_1,\dots,i_m})\in \{0, +,-\}^{n \choose m}$ be a nonzero sign vector, considered\footnote{Pl\"ucker and twistor coordinates are 
		defined only up to multiplication by a common scalar.} 
modulo multiplication by 
		$\pm 1$.
	Set $$\mathcal{A}^{Z,\sigma}_{n,k,m}:=\{Y\in \Am \ \vert \ 
	\sign \lrangle{Y i_1 \cdots i_m} = \sigma_{i_1,\dots,i_m}\}.$$
	We call 
	$\mathcal{A}^{Z,\sigma}_{n,k,m}$ an \emph{(amplituhedron) sign stratum.}
	Clearly $$\Am = \bigsqcup_{\sigma} \mathcal{A}^{Z,\sigma}_{n,k,m}.$$
	If $\sigma\in \{+,-\}^{n \choose m}$,
	we call 
	$\mathcal{A}^{Z,\sigma}_{n,k,m}$ an open 
	\emph{(amplituhedron) chamber}.\footnote{We borrow the word ``chamber''
		from the theory of hyperplane arrangements.}
\end{definition}
%\MP{Do we want to call them ``sign chambers'' instead?}

For $m=1$, all strata are nonempty \cite[Definition 5.2]{karpwilliams},
but this is not true for $m>1$.  Moreover, whether or not 
$\mathcal{A}^{\sigma}_{n,k,m}(Z)$ is empty depends on  $Z$.
\begin{definition} \label{def:realizable}
	We say that a sign vector $\sigma$ (or sign stratum 
	$\mathcal{A}^{\sigma}_{n,k,m}$) is \emph{realizable} for $\mathcal{A}^{\sigma}_{n,k,m}$
	if 
	$\mathcal{A}^{Z,\sigma}_{n,k,m}$ is nonempty for some $Z\in \Mat^{>0}_{n,k+m}$. %\MP{Shall we add: $Z \in \mbox{Mat}^{>0}_{n,k+2}$?}
\end{definition}

%\subsubsection{Sign variation and sign flips}
%
%
%
%Signs and sign flips will be important to our description of the amplituhedron, so we 
%introduce some useful terminology here.
%
%\begin{definition}\label{def:var}
%	Given $v\in \R^n$, let
%	$\var(v)$ be the number of times $v$ changes sign 
%	when we read the components from left to right and ignore any
%	zeros.  We also define
%	$$\overline{\var}(v):=\max\{\var(w) \ \vert \ 
%	w\in \R^n \text{ such that }w_i=v_i\text{ for all }
%	i\in [n] \text{ with }v_i \neq 0\}.$$
%	If $v\in \{0,+,-\}^n$, we define $\var(v)$ and 
%	$\overline{\var}(v)$ in the obvious way.
%\end{definition}
%
%For example, if $v:=(4,-1,0,-2)\in \R^4$ then $\var(v)=1$
%and $\overline{\var}(v)=3$.

%\begin{definition}\label{def:flip}
%	If $v\in \R^n$, or $v\in \{+,-, 0\}^n$, we say that 
%	$v$ has a \emph{sign flip in position $i$} if $v_i, v_{i+1} \neq 0$
%	and they have different signs, where indices are considered modulo $n$.
%	We define
%	$$\flip(v) = \flip(v_1,\dots,v_n):= \{i \ \vert \ v \text{ has a sign flip in position }i\} \subseteq [n].$$
%\end{definition}
%
%\begin{remark} We caution the reader that $|\flip(v)|$ may not equal $\var(v)$. For example, the sequence $(+,0,-,0,+,+,-)\in \{+,-, 0\}^7$ has sign flips in positions
%	$\{6,7\}$ , but $\var(+,0,-,0,+,+,-)$ is 3.
%\end{remark}

%\begin{remark}\label{rem:twisted}
%	Matrices in $\Mat^{>0}_{n,p}$ have a twisted cyclic symmetry.
%	If $Z\in \Mat^{>0}_{n,p}$ with $n \geq p$ has rows $Z_1,Z_2, \dots, Z_n$, and if we let $\widehat{Z}_i$ denote
%	$(-1)^{p-1} Z_i$, then the matrix with rows 
%	$Z_2,\dots, Z_n, \widehat{Z}_1$ also lies in $\Mat^{>0}_{n,p}$.  
%	Similarly for the matrix with rows $Z_3,\dots, Z_n, \widehat{Z}_1, \widehat{Z}_2$, etc\footnote{We will use the `hat' notation $\widehat{}$ also in the context of T-duality with a different meaning. It will be always clear from context which one we mean.}.
%\end{remark}

\noindent \textbf{Previous work on the $m=2$ amplituhedron.}
The original amplituhedron paper~\cite{arkani-hamed_trnka}
gave a conjectural
positroid tiling of $\A$.
A BCFW-style recursion for producing tilings of 
$\A$ was conjectured in \cite{Karp:2017ouj} and proved in \cite{BaoHe}.  
A conjectural classification of $m=2$ positroid tiles was 
given in \cite{Lukowski:2019sxw}.
A conjectural description of the boundaries of the $m=2$
amplituhedron was given in \cite{Lukowski:2019kqi}. In another direction, \cite{ATT} gave a conjectural description of $\A$ in terms of sign flips; \cite{ATT} and independently \cite{Karp:2017ouj} proved one direction.


\subsection{The hypersimplex and positroid polytopes}



%In \cite{LPW}, a surprising parallel was found between the 
%amplituhedron map $\widetilde{Z}$ on $\Gr^{\geq 0}_{k, n}$ and the moment map $\mu$ on $\Gr^{\geq 0}_{k+1, n}$. A correspondence called \emph{T-duality} was used to relate Grasstopes in the amplituhedron $\AA_{n, k, 2}$ to positroid polytopes in the hypersimplex $\Delta_{k+1, n}$.We present relevant background here\footnote{We
%	use `$k+1$' instead of `$k$' here in order to match conventions of later sections.}.
%



In this section, we review the necessary background material on positroid polytopes. Throughout, for $x \in \R^n$ and $I \subset [n]$, we use the notation $x_I:=\sum_{i \in I} x_i$. 

\begin{definition}
	Let $e_I := \sum_{i \in I} e_i \in \R^n$, where
	$\{e_1, \dotsc, e_n\}$ is the standard basis of $\RR^n$.
	The \emph{$(k+1,n)$-hypersimplex} is 
	$\Delta_{k+1,n}:=\convex(e_I : I \in \binom{[n]}{k})$.
\end{definition}

%\begin{remark}\label{rem:hypercube} The hypersimplex $\Delta_{k+1,n}$ is obtained by intersecting the unit hypercube $\mbox{\mancube}_{n}$ with the hyperplane $x_{[n]}=k+1$.
%	Alternatively, under the projection $P:(x_1,\dots,x_n) \mapsto (x_1,\dots,x_{n-1})$,
%	$\Delta_{k+1,n}$ is linearly equivalent to
%	$$\widetilde{\Delta}_{k+1,n} := \{(x_1,\dots,x_{n-1}) \ \vert \
%	0 \leq x_i \leq 1; k \leq x_{[n-1]} \leq k+1\} \subset \RR^{n-1}.$$
%	That is,
%	$\widetilde{\Delta}_{k+1,n}$ is the slice of $\mbox{\mancube}_{n-1}$ between the hyperplanes
%	$x_{[n-1]} = k$ and $x_{[n-1]}= k+1$.
%\end{remark}

The torus $T = \R^n \setminus \{0\}$ acts on $k \times n$ matrices by rescaling columns. This descends to an $(n-1)$-dimensional torus action on $Gr(k,n)$.
We let $TA$ denote the orbit of $A$ under the action of $T$, and 
$\overline{TA}$ its closure.  

The \emph{moment map} from the Grassmannian $\Gr_{k+1,n}$ to $\R^n$
is defined as follows.
\begin{definition}[The moment map]
	Let $A$ be a $(k+1) \times n$ matrix representing a point of 
	$\Gr_{k+1,n}$.
	The \emph{moment map} 
	$\mu \colon \Gr_{k+1,n} \to \R^n$ is defined by 
	$$\mu(A) = \frac{ \sum_{I \in \binom{[n]}{k+1}} |p_I(A)|^2 e_I}
	{\sum_{I \in \binom{[n]}{k+1}} |p_I(A)|^2}.$$
\end{definition}
It is well-known that the image of the Grassmannian $\Gr_{k+1,n}$ under the moment
map is the hypersimplex
$\Delta_{k+1,n}$. 
If one restricts the moment map to $\Gr_{k+1,n}^{\geq 0}$ then the image is again the hypersimplex
$\Delta_{k+1,n}$ 
\cite[Proposition 7.10]{tsukerman_williams}.



In general, it follows from classical work of 
Atiyah \cite{A:82} and Guillemin--Sternberg \cite{GS} that the image
$\mu(\overline{TA})$ is a convex polytope, whose vertices
are the images of the torus-fixed points, \textit{i.e.}\ the vertices are the
points $e_I$ such that $p_I(A) \neq 0$. This motivates the notion of \emph{matroid polytope}.  
Recall that any full
rank $(k+1)\times n$ matrix $A$ gives rise to a matroid 
$\M(A)=([n],\B)$, where $\B = \{I \in \binom{[n]}{k+1} \ \vert \ p_I(A) \neq 0\}$.

\begin{definition}\label{def:mpolytope}
	Given a matroid $\M=([n],\B)$, the (basis) \emph{matroid polytope} of $\M$ is
	$
	\Gamma_\M := \convex\{e_B : B \in \B\} \subset \RR^n.
	$
\end{definition}



%Matroid polytopes also have a straightforward description in terms of inequalities.
%
%\begin{proposition}[\cite{welsh}]\label{prop:inequalitiesmatroids}
%	Let $\M = ([n], \B)$ be any matroid of rank $k$, and let $r_\M:2^{[n]} \to \Z_{\geq 0}$ be its rank function. Then the matroid polytope $\Gamma_\M$ can be described as
%	\begin{align*}
%		\Gamma_\M &= \{ {\bf x} \in \RR^n : x_{[n] }= k, \, x_A \leq r_\M(A) \, \text{ for all $A \subset [n]$} \}\\
%		\Gamma_\M &= \{ {\bf x} \in \RR^n :x_{[n] }= k, \, x_A \geq k- r_\M([n]\setminus A) \, \text{ for all $A \subset [n]$} \}.
%	\end{align*}
%\end{proposition}


Here, we are interested in \emph{positroid polytopes}, that is, matroid polytopes $\Gamma_\M$ where $\M$ is a positroid. They arise as $\mu(\overline{TA})$ where $A$ is a totally nonnegative matrix. Of more interest to us, they are also moment map images of positroid cells. 

\begin{proposition}[\protect{\cite{tsukerman_williams}}]\label{prop:moment}
	Let $\M$ be a positroid.  Then 
	$\Gamma_{\M} = \mu(\overline{S_{\M}}) = \overline{\mu(S_{\M})}$.
\end{proposition}

The interior $\Gamma_{\M}^\circ$ of a positroid polytope is the moment map image of $S_\M$; we call $\Gamma_{\M}^\circ$ the \emph{open} positroid polytope. As in the amplituhedron case, we will focus on those positroid polytopes which are positroid tiles (\textit{i.e.}\ $(n-1)$-dimensional positroid polytopes $\Gamma_\M$ where $\mu$ is injective on $S_\M$).


%\begin{definition}[Positroid Polytopes]\label{def:Gamma}
%	Given a positroid cell $S_{\pi}$ of 
%	$\Gr_{k+1,n}^{\ge 0}$, we let 
%	$\Gamma^{\circ}_{\pi} = \mu(S_{\pi})$ and 
%	$\Gamma_{\pi} = \overline{\mu(S_{\pi})}$, and we refer to 
%	$\Gamma^{\circ}$ and $\Gamma_{\pi}$ as \emph{open positroid polytopes} and 
%	\emph{positroid polytopes}, respectively.
%	We call 
%	$\Gamma_{\pi}$ a 
%	\emph{generalized triangle} 
%	for $\Delta_{k+1,n}$ 
%	if $\dim(S_\pi) =n-1$, and 
%	$\mu$ is injective on $S_\pi$.
%\end{definition}

\begin{theorem}[\protect{\cite{LPW}}]\label{prop:homeo2}
%	Consider a positroid cell $S_{G} \subset \Gr_{k+1,n}^{\geq 0}$, with $G$ a reduced plabic graph.
%	Then the moment map is injective on $S_G$ if and only if $G$ is a forest.  When $G$ is a forest, $\mu$ is moreover a stratification-preserving
%	homeomorphism from $\overline{S_{G}}$ to  the polytope
%	$\Gamma_G \subset \mathbb{R}^n$.
%	We have $\dim S_{G}
%	=\dim \Gamma_{G} = n-c,$ where $c$ is the number of connected components of $G$. 
A positroid polytope $\Gamma_G$ is a tile for $\Delta_{k+1,n}$ if and only if $G$ is a tree.
\end{theorem}
%\MP{Maybe we can mention this as a Remark after Th 3.1?}.

\subsection{T-duality}

In \cite{LPW}, a curious parallel was conjectured between positroid tilings of $\A$ and $\Delta_{k+1,n}$ involving \emph{T-duality} \cite{Karp:2017ouj}.

\begin{definition}[T-duality]\label{hatmap}
	Let  $\pi=a_1 a_2 \cdots a_n$ be a loopless decorated permutation (written
	in one-line notation). The \emph{T-dual} decorated permutation is
	$\widehat{\pi} = a_n a_1 a_2 \cdots a_{n-1}$, where any fixed points in $\widehat{\pi}$ are declared to be loops.
\end{definition}

T-duality induces a bijection from loopless positroid cells $S_\pi \in \Gr^{\ge 0}_{k+1,n}$ to coloopless positroid cells $S_{\widehat{\pi}} \in \Gr^{\ge 0}_{k,n}$ \cite[Lemma 5.2]{LPW}. It thus also induces a bijection between loop\-less positroid polytopes $\Gamma_{\pi} \in \Delta_{k+1, n}$ and coloopless Grasstopes $\gt{\widehat{\pi}} \in \Am$; we say $\Gamma_{\pi}$ and $\gt{\widehat{\pi}}$ are \emph{T-dual}.

\begin{conj}
	A positroid polytope $\Gamma_{\pi}$ is a positroid tile for $\Delta_{k+1, n}$ if and only if the T-dual Grasstope $\gt{\widehat{\pi}}$ is a tile for $\A$. Further, a collection $\{\Gamma_{\pi}\}$ of positroid polytopes is a positroid tiling of $\Delta_{k+1, n}$ if and only if the collection $\{\gt{\widehat{\pi}}\}$ of T-dual Grasstopes is a positroid tiling of $\A$ for all $Z \in \Mat_{n,k+2}^{>0}$. 
\end{conj}

We prove this conjecture in \cref{cor:GTsInBijection} and \cref{thm:simpTriangGiveAmp}.


%\subsection{T-duality and positroid triangulations}
%
%Recall the definition of positroid triangulation from \cref{def:tri}.  Specializing to 
%$\A$, we get the following.
%\begin{definition}[Positroid triangulations of $\mathcal{A}_{n,k,2}$]\label{def:dissectionAmp}
%	Let $\mathcal{C} = \{\gt{\pi}\}$ be a collection of Grasstopes, with $\{S_{\pi} \}$ positroid cells of $\Gr^{\geq 0}_{k,n}$.  
%	We say that 
%	$\mathcal{C}$ is 
%	a \emph{positroid triangulation} of $\A$
%	if:
%	\begin{itemize}
%		\item each Grasstope $\gt{\pi} $ is a generalized triangle (\textit{i.e.} $\widetilde{Z}$ is injective on $S_{\pi}$ and $\dim \gt{\pi} = 2k$);
%		\item pairs of distinct open Grasstopes $\gto{\pi}$ and $\gto{\pi'}$ in the collection are disjoint; 
%		\item $\cup_{\pi} \gt{\pi} = \A$. 
%	\end{itemize}
%\end{definition}
%
%
%
%
%\begin{remark}
%	Alternatively, one could define a positroid triangulation as coming
%	from a collection $\{S_{\pi}\}$ of cells
%	such that $\{Z_{\pi}\}$ is a triangulation (as above)
%	for \emph{all} choices of $Z$.
%	We use \cref{def:dissectionAmp} here
%	since some objects we define will be sensitive to the choice of $Z$. 
%\end{remark}
%
%
%In the case of the hypersimplex, a positroid triangulation is as follows.
%
%\begin{definition}[Positroid triangulations of $\Delta_{k+1,n}$]\label{def:dissectionSimp}
%	Let $\mathcal{C} = \{\Gamma_{\pi}\}$ be a collection 
%	of positroid polytopes, with $\{S_{\pi}\}$ positroid cells of $\Gr^{\geq 0}_{k+1,n}$. We say that 
%	$\mathcal{C}$ is a \emph{positroid triangulation} of $\Delta_{k+1,n}$ if:
%	\begin{itemize}
%		\item each $\Gamma_\pi$ is a generalized triangle ($\mu$ is injective on $S_{\pi}$, and $\dim S_{\pi} = n-1$);
%		\item pairs of distinct open positroid polytopes $\Gamma^{\circ}_{\pi}$ and $\Gamma^{\circ}_{\pi'}$ in the collection are disjoint;
%		\item $\cup_{\pi} \Gamma_{\pi} = \Delta_{k+1,n}$.
%	\end{itemize}
%\end{definition}
%
%
%\begin{remark}
%	In this work, the definitions of ``positroid triangulations'' are slightly different from that of \cite{LPW}.
%\end{remark}
%
%
%By \cref{prop:homeo2},  the generalized triangles of $\Delta_{k+1, n}$ are the
%positroid polytopes $\Gamma_G$ where $G$ is a plabic tree. 
%And by \cref{thm:allGTs},  the generalized triangles of $\AA_{n, k, 2}(Z)$ are 
%the Grasstopes $\gt{\widehatG(\T)}$ for $\T$ a $(k, n)$-unpunctured plabic tiling.
%In \cite{LPW}, it was conjectured that generalized triangles 
%and the two notions of positroid triangulation are 
%related by a very simple correspondence, called \emph{T-duality}.
%
%\begin{definition}[T-duality on decorated permutations]\label{hatmap}
%	Let  $\pi=a_1 a_2\dots a_n$ be a loopless decorated permutation (written
%	in one-line notation). The \emph{T-dual} decorated permutation is
%	$\widehat{\pi} : i \mapsto \pi(i-1)$, so that
%	$\widehat{\pi} = a_n a_1 a_2\dots a_{n-1}$.
%	Any fixed points in $\widehat{\pi}$ are declared to be loops.
%\end{definition}
%
%\begin{rmk}
%	This map was previously defined in \cite[Definition 4.5]{Karp:2017ouj}) and 
%	was studied in \cite{LPW}, where it was used to 
%	draw parallels between the hypersimplex $\Delta_{k+1,n}$ 
%	and the 
%	$m=2$ amplituhedron $\A$.  
%	The T-duality map was 
%	also studied in \cite{posquotients,
%		GalCritVar}.
%	The map $\pi \to \widehat{\pi}$ is an $m=2$ version 
%	of a map that appeared in \cite{abcgpt} for the case $m=4$.
%\end{rmk}
%
%
%
%\begin{lem}[\protect{\cite[Lemma 5.2]{LPW}}]\label{lem:tDualityBiject}
%	The T-duality map $\pi \mapsto \td{\pi}$ is a bijection from loopless decorated permutations of type $(k+1, n)$ to coloopless decorated permutations of type $(k, n)$. That is, the map $S_\pi \mapsto S_{\td{\pi}}$ is a bijection from the set of loopless cells in $\Gr_{k+1,n}^{\geq 0}$ to the set of coloopless cells in $\Grk$.
%\end{lem}
%
%
%The philosophy of \cite{LPW} is that if the moment map behaves well on $S_\pi$, then the $\widetilde{Z}$-map behaves well on $S_{\td{\pi}}$. For example, if the image of $S_\pi$ is a generalized triangle for $\Delta_{k+1,n}$, then the image of $S_{\td{\pi}}$ is a generalized triangle for $\A$ 
%\cite[Proposition 6.6.]{LPW}. Moreover, there is a main conjecture involving positroid triangulations:
%
%\begin{conj}[\protect{\cite[Conjecture 6.9]{LPW}}]\label{conj:triangCorresp}
%	A collection $\{\Gamma_\pi\}$ of positroid polytopes in $\Delta_{k+1, n}$ gives a positroid triangulation of $\Delta_{k+1, n}$ if and only if for all $Z \in \Mat_{n,k+2}^{>0}$, the collection $\{\gt{\td{\pi}}\}$ of Grasstopes gives a positroid triangulation of $\AA_{n, k, 2}(Z)$. 
%\end{conj}
%
%In \cref{sec:Tplabic}, we will prove a number of additional results on T-duality, 
%upgrading it to a map on plabic graphs.
%We will also prove  \cref{conj:triangCorresp} in \cref{thm:simpTriangGiveAmp}.

\section{Results}

Our first main result is a characterization of the positroid tiles for $\A$.\footnote{The positroid tiles for $\Delta_{k+1, n}$ were characterized in \cite{LPW} using \emph{plabic trees}.}
 A \emph{bicolored subdivision} of the $n$-gon $P_n$ is a choice of noncrossing diagonals decomposing $P_n$ into subpolygons. Each subpolygon is colored grey or white so that no two subpolygons of the same color share an edge;
 we view the middle diagram of \cref{fig:unpuncTduality} as a decomposition of 
 $P_9$ into a grey quadrilateral, a grey pentagon, and two white triangles.  The \emph{area} of a subdivision is the total number of grey triangles in any triangulation of the grey subpolygons.

\begin{thm}\label{cor:GTsInBijection}
Positroid tiles of $\Delta_{k+1, n}$ and $\A$ are in bijection with bicolored subdivisions of $P_n$ with area $k$.
We read $\Gamma_G$ and $Z_{\widehat{G}}$ off of the same bicolored subdivision $\sigma$ of the $n$-gon by first choosing any triangulation $\tau$ of the grey polygons in $\sigma$. Then $G:= G(\tau)$ is the dual plabic tree\footnote{technically, one should insert degree 2 white vertices to get a bipartite tree} of $\sigma$ and $\widehat{G}:=\widehat{G}(\tau)$ is obtained by placing a ``tripod'' in each grey triangle (see \cref{fig:unpuncTduality}). %\MP{placing black vertices on each vertex of $P_n$ and connecting them to boundary vertices of $\widehat{G}$.}
\end{thm}

\begin{figure}[h]
	\centering
	\includegraphics[width=0.8\textwidth]{unpuncTduality}
	\caption{A triangulation of a bicolored subdivision $\sigma$ with area $5$, with the dual graph $G(\tau)$ to its left, and the T-dual graph $\widehatG(\tau)$ to 
		its right.}
	\label{fig:unpuncTduality}
\end{figure}

Using \cref{cor:GTsInBijection}, we can enumerate the positroid tiles.

\begin{prop}
	Positroid tiles of $\Delta_{k+1}$ and $\A$ are in bijection with separable permutations on $[n-1]$ with $k$ descents. They are enumerated by $R_{k,n-2}$ from~\cite[A175124]{OEIS}.  
\end{prop}


From a bicolored subdivision $\sigma$, we also 
obtain inequality descriptions of the corresponding T-dual positroid tiles. Interestingly, the inequality descriptions use exactly the same combinatorial information, the area statistic. Given two positive numbers $a,b\in [n]$, the 
\emph{cyclic interval} $[a,b]$ is defined to be $\{a,a+1,\dots,b-1,b\}$ if $a \leq b$ and $\{a,a+1,\dots,n,1,\dots,b\}$ otherwise.


\begin{thm}[T-dual inequalities]\label{thm:inequalitiesMatch}
	Let $\sigma, \tau, G(\tau), \widehatG(\tau)$ be as in \cref{cor:GTsInBijection}. Let $h \to j$ be an arc in a grey triangle of $\tau$, with $h<j$. Then
	\begin{enumerate}
		\item $\area(h \to j)+1 > x_{[h, j-1]} > \area(h \to j)$ for $x \in \Gamma^{\circ}_{G(\tau)}$,
		\item $(-1)^{\area(h \to j)} \langle Y h j\rangle>0$ for $Y \in \gto{\widehatG(\tau)}$,
	\end{enumerate}
	where $\area(h \to j)$ is the number of grey triangles to the left of $h \to j$. Letting $h \to j$ vary over all arcs of grey triangles of $\tau$, these inequalities cut out the positroid tiles $\Gamma^{\circ}_{G(\tau)}$ and $\gto{\widehatG(\tau)}$.
\end{thm} 

For example, for the triangulation in \cref{fig:unpuncTduality}, $\area(3 \to 7)=2$, so the inequality $3> x_3 + x_4 +x_5+x_6 > 2$ holds for $x \in \Gamma_{G(\tau)}^{\circ}$ and the inequality $\lrangle{Y37}>0$ holds for $Y \in \gt{\widehat{G}(\tau)}$. \cref{thm:inequalitiesMatch} can be extended to give a correspondence between \emph{facets} of T-dual positroid tiles lying in the interior of $\Delta_{k+1, n}$ or $\A$, which correspond to arcs of $\sigma$ separating grey and white subpolygons \cite[Theorem 9.10]{ampm2}.

Using \cref{thm:inequalitiesMatch}, we also prove a description of $\A$ conjectured by \cite{ATT}.

%\MP{I would keep it if people of FPSAC are interested in enumerative combinatorics.}

%\begin{thm}[Facets via T-duality] \label{th:facetsGT} \MSB{Can omit or shorten if there's not space.}
%	Let $\T$ be a triangulated unpunctured plabic tiling and let $h \to j$ be a facet-defining arc of $\T$. Let $G:=G(\T)$ be the plabic tree dual to $\T$ and let $G'$ be the plabic forest obtained from $G$ by deleting the edge dual to $h \to j$. Let $\widehatG$ and $\widehatG'$ denote their T-duals.
%	\begin{enumerate}
%		\item  The positroid polytope $\Gamma_{G'}$ is a facet of $\Gamma_G$, and lies on the hyperplane
%		$$x_{[h, j-1]}= \area(h \to j). $$
%		\vspace{-0.5cm} \item The Grasstope $\gt{\widehatG'}$ is a facet of $\gt{\widehatG}$, and lies on the hypersurface
%		$$ \lrangle{Y h j}=0. $$
%	\end{enumerate}
%	Moreover, if we let $h \to j$ range over the facet-defining arcs of $\T$ which are not on the boundary of $\textbf{P}_n$, we obtain all facets of $\Gamma_G$ and $\gt{\widehatG}$ in the interior of $\Delta_{k+1, n}$ and $\AA_{n, k, 2}(Z)$.
%\end{thm}
%\MP{If we keep it, I would put the equations on the same line of the text to save space.}

\begin{theorem}[Sign-flip characterization of $\mathcal{A}_{n,k,2}$]\label{thm:main1}
	
	Fix $k<n$  and $Z\in \Mat^{>0}_{n,k+2}$.
	Let 
	\begin{align*}
		\mathcal{F}^\circ_{n,k,2}(Z):=	&\{Y\in Gr_{k,k+2} \ \vert \  
		\langle Y i ~ i+1 \rangle>0 \text{ for }1 \leq i \leq n-1,
		\text{ and }   (-1)^k \langle Y 1 ~n \rangle >0,\\
		& \; \text{ and the sequence } 
	{\langle Y 1 ~2\rangle, 
			\langle Y 1 ~3\rangle,  \dotsc,
			\langle Y 1 ~n\rangle} \text{ has exactly $k$ sign flips}\}.
	\end{align*}
	Then $\A = \overline{\mathcal{F}^\circ_{n,k,2}(Z)}.$
\end{theorem}
%\MP{Shall we define ``var'' and $\widehat{Z}_i$?}

We now turn to our results on tilings. In order to show the bijection between positroid tilings of $\Delta_{k+1, n}$ and $\A$, we consider a refinement of all positroid tilings on both sides. This decomposes $\Delta_{k+1, n}$ and $\A$ into pieces that are smaller than positroid tiles; we will give a correspondence between these smaller pieces.

On the hypersimplex side, we consider a triangulation of $\Delta_{k+1, n}$ \cite{LamPost,StanleyTriangulation,SturmfelsGrobner}, \textit{i.e.}\ a decomposition into simplices, which refines all positroid tilings. The maximal simplices $\Delta_w$ of this triangulation are indexed by permutations $w$ of $[n]$ with $k$ descents and $w_n=n$. Such permutations are enumerated by the \emph{Eulerian number} $E_{k, n-1}$ \cite{stanley_ec1}. We will describe the simplices $\Delta_w$ using the following permutation statistic.

\begin{defn}\label{defn:wsimplexHSimplex}
	Let $w \in S_n$. We call a letter $i$ in $w$ a \emph{cyclic descent}\footnote{These are closely related to \emph{left descents} of $w$, also called \emph{recoils} in the literature.} if either $i \geq 2$ and
	$w^{-1}(i) < w^{-1}(i-1)$ or if $i=1$ and $w^{-1}(1) < w^{-1}(n)$.
	We let $\cdes(w)$ denote the set of cyclic descents of $w$ and let  $D_{k+1, n}$ denote the set of permutations $w \in S_n$ with $k+1$ cyclic descents and $w_n=n$. Note that $|D_{k+1, n}|=E_{k, n-1}$.
	
	For $w \in D_{k+1, n}$, let $w^{(a)}$ denote the cyclic rotation of $w$ ending at $a$.
	We define %$I_1:=\{j \in [n]: j \text{ is a cyclic descent of } w\}$ and for $r=2, \dots, n$,
	$I_r:=\cdes(w^{(r-1)}).$
	The \emph{w-simplex} $\simp{w} \subseteq \Delta_{k+1, n}$ is the simplex with vertices $e_{I_1}, \dots, e_{I_n}$.
	%	\MP{Maybe we could define $w^{(r)}$ to be the rotation of $w$ ending at $r-1$ and define $I_r=\cdes(w^{(r)})$.}
\end{defn}

\begin{example}\label{ex:descents}
	Let $w=324156$ in one-line notation. Then $w$ has cyclic descents $\{1, 2, 3\}=I_1$. The rotation of $w$ ending at $1$ is $563241$, which has cyclic descents $I_2=\{ 2, 3, 5\}$. The rotation of $w$ ending at $2$ is $415632$, which has cyclic descents $I_3=\{1,3,4\}$.
\end{example}

Now, we turn to the amplituhedron side. We will consider the decomposition of $\A$ into (closures of) amplituhedron chambers. This is a refinement of all positroid tilings because the tiles for $\A$ are cut out by hypersurfaces $\lrangle{Yij}=0$. We define one amplituhedron chamber for each $w \in D_{k+1, n}$.



\begin{defn}[$w$-chambers]\label{def:ampchamber}
	Let $w \in D_{k+1,n}$ and let the vertices of $\simp{w}$ be $e_{I_1}, \dots, e_{I_n}$, as in Definition \ref{defn:wsimplexHSimplex}. Then the \emph{open $w$-chamber} $\AA_{n, k, 2}^{Z,w}$ consists of $Y \in \Gr_{k,k+2}$ such that
	\begin{align*} \sgn \langle Y a j \rangle& = (-1)^{|I_a \cap [a, j-1]|-1}
		&\hspace{-50pt}&	\text{for }j>a\\
		%and for $a>j$, 
		\sgn \langle Y a j \rangle&= (-1)^{|I_a \cap [a, j-1]|-k-1}
		&\hspace{-50pt}&	\text{for }j<a.
	\end{align*}
	The \emph{(closed) $w$-chamber} is the closure $\asimp{w}:= \overline{\AA_{n, k, 2}^{Z,w}}$. % \MSB{I added $Z$ to $w$-chamber notation-- is it ok or too cluttered? Also, should we call these ``$w$-chambers'' or ``amplituhedron $w$-chambers'' in the rest of the paper?}. 
	%\LW{I think $Z$ is important for $w$-chamber. And I think we can mostly call
	%the closed ones just $w$-chambers.}
\end{defn}

Continuing \cref{ex:descents}, $\lrangle{Y13}<0$ on $\asimp{w}$, since $|\{1, 2, 3\} \cap [1,2]|-1$ is odd. 

The $w$-chambers are a distinguished subset of the amplituhedron chambers.
We show that they are precisely the \emph{realizable} chambers,
in the sense of 
%\emph{A priori}, there could be many more  non-empty amplituhedron chambers than the $w$-chambers. However, this is not the case. Recall the definition of realizable from 
\cref{def:realizable}.

\begin{theorem}\label{thm:nonempty}
An amplituhedron chamber is realizable if and only if it is an open $w$-chamber. In particular, the amplituhedron $\A$ is the union of the $w$-chambers $\asimp{w}$, just as $\Delta_{k+1, n}$ is the union of the $w$-simplices $\Delta_w$.
\end{theorem}

Further, a $w$-simplex $\Delta_w$ and the corresponding $w$-chamber $\asimp{w}$ behave the same way with respect to containment in positroid tiles.

\begin{thm}\label{thm:tDualSimplex}
	Let $\asimp{w} \neq \emptyset$ and let $\gt{\widehat{\pi}}$ be a positroid tile for $\A$. Then $\asimp{w}$ is contained in $\gt{\widehat{\pi}}$ if and only if the $w$-simplex $\Delta_w$ is contained in the T-dual positroid tile $\Gamma_\pi$.
\end{thm}

Together, \cref{thm:nonempty,thm:tDualSimplex} give our main result on tilings.

\begin{thm}[Tilings of $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}$ are T-dual] \label{thm:simpTriangGiveAmp}
	The collection $\mathcal{C}=\{\Gamma_{\pi}\}$ is a positroid tiling of $\Delta_{k+1, n}$ if and only if for all $Z \in \Mat_{n,k+2}^{>0}$, the collection of T-dual Grasstopes $\td{\mathcal{C}}=\{\gt{\td{\pi}}\}$ is a positroid tiling of $\A$.
\end{thm}

\begin{table}[h!]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c}
\hline
{$(k,n)$}{}& $(1,n)$ & $(2,4)$ & $(2,5)$ & $(2,6)$ & $(2,7)$ & $(2,8)$ & $(3,5)$ & $(3,6)$ & $(3,7)$\\
\hline
$\#$ pos. tilings &$C_{n-2}$ & $1$ &$5$ &$120$ &$3073$ &$6 \, 443 \, 460$ &$1$ &$14$ &$3073$\\
\hline
\end{tabular}
\end{center}
	\vspace*{-0.5cm} \caption{  Number of tilings of $\Delta_{k+1,n}$ and $\A$ in bijection by \cref{thm:simpTriangGiveAmp} in the known cases. The \emph{Catalan number} $C_{n-2}$ is the number of triangulations of an $n$-gon. }
	\label{tab:numbtilings}
\end{table}

\acknowledgements{M.P. thanks Fatemeh Mohammadi, Leonid Monin, and Lionel Mason for useful discussions. 
	M.P. and L.W. thank Tomek Lukowski for the previous paper \cite{LPW} which inspired some of this work.  The authors thank
	Pasha Galashin, Tomek Lukowski, and David Speyer for helpful comments.

\medskip
\noindent
    Any opinions,
	findings and conclusions or recommendations expressed in this material
	are those of the authors and do not necessarily reflect the
	views of the NSF.}


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