%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2024}
\articlenumber{85}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

% \setlist[itemize]{leftmargin=*}
% \setlist[enumerate]{leftmargin=*}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem*{theorem*}{Theorem}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{observation}[theorem]{Observation}
%\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{claim}[theorem]{Claim}

%% Commands from paper
\DeclareMathOperator{\tree}{tree}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\co}{co}
\DeclareMathOperator{\tab}{tab}
\DeclareMathOperator{\rtop}{{\it k}} % root
\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\arr}{arr}
\DeclareMathOperator{\rt}{{\it k}} % root
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\inn}{in}
\DeclareMathOperator{\out}{out}
\DeclareMathOperator{\sort}{sort}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\tw}{tw}
\DeclareMathOperator{\Facet}{GeomFacet}
\DeclareMathOperator{\AFacet}{Froz}
\DeclareMathOperator{\Froz}{Froz}
\DeclareMathOperator{\refl}{refl}
\DeclareMathOperator{\cyc}{cyc}
\DeclareMathOperator{\op}{op}
\DeclareMathOperator{\pre}{pre}
\DeclareMathOperator{\inc}{inc}
\DeclareMathOperator{\Irr}{\xx}
\DeclareMathOperator{\cl}{cl}



\newcommand{\galp}{{\alpha}}
\newcommand{\gbet}{{\beta}}
\newcommand{\ggam}{{\gamma}}
\newcommand{\gdel}{{\delta}}
\newcommand{\geps}{{\varepsilon}}
\newcommand{\ga}{{a}}
\newcommand{\gb}{{b}}
\newcommand{\gc}{{c}}
\newcommand{\gd}{{d}}
\newcommand{\gn}{{n}}
\newcommand{\Ampl}{\mathcal{A}}
\def\CD{\mathcal{CD}_{n,k}}
\def\coord{\mathcal{C}}

 %% This is for Grasstopes $\tilde{Z}(\overline{S_\pi)}$

\newcommand{\Grk}{\Gr_{k,n}^{\scriptscriptstyle \ge 0}}
\newcommand{\GrDhat}{\widehat{\Gr_{4,n}({D})}}
\newcommand{\GrD}{{\Gr_{4,n}({D})}}
%\newcommand{\lrangle}[1]{\langle #1 \rangle}
\def\AA{\mathcal{A}^Z_{n,k,m}}
\newcommand{\Ank}{\mathcal{A}_{n, k, 4}({Z})}
\def\butterfly{BCFW }
\def\tZ{\tilde{Z}}
\def\bcfw{\bowtie}
\DeclareMathOperator{\bcfwp}{\widetilde{\bowtie}}
\def\mbcfw{\iota_{\bowtie}}
\def\mbcfwp{\iota_{\bowtie'}}
\def\4biddenprop{4-coindependent}
\newcommand{\lr}[1]{\langle #1 \rangle}
\newcommand{\llrr}[1]{\langle\!\langle #1 \rangle\!\rangle}
\newcommand{\gz}{{\color{lightgray}0}}
\newcommand{\pos}{\mathcal{P}}
\newcommand{\br}{\,|\,}
\newcommand{\gt}[1]{Z_{#1}}
\newcommand{\gto}[1]{Z_{#1}^\circ}
\def\ctop{D_{\rtop}}
\def\rchn{\nearrow}
\def\lchn{\nwarrow}
%arrow primed
%\newcommand{\rightarrowp}{\nearrow'}
%\newcommand{\leftarrowp}{\nwarrow'}
\DeclareMathOperator{\rightarrowp}{\widetilde{\rchn}}
\DeclareMathOperator{\leftarrowp}{\widetilde{\lchn}}

%Notation for cluster variables (irreducible factors of domino entries)
\newcommand{\rzeta}{\bar{\zeta}}
\newcommand{\ralpha}{\bar{\alpha}}
\newcommand{\rbeta}{\bar{\beta}}
\newcommand{\rgamma}{\bar{\gamma}}
\newcommand{\rdelta}{\bar{\delta}}
\newcommand{\repsilon}{\bar{\varepsilon}}
\DeclareMathOperator{\Mut}{Mut}
\DeclareMathOperator{\behind}{after}
\DeclareMathOperator{\after}{after}
\DeclareMathOperator{\below}{below}
\DeclareMathOperator{\sticky}{sticky}
\DeclareMathOperator{\nonsticky}{nonsticky}

% Notation for domino entries
\newcommand{\dzeta}{\hat{\zeta}}
\newcommand{\dalpha}{\hat{\alpha}}
\newcommand{\dbeta}{\hat{\beta}}
\newcommand{\dgamma}{\hat{\gamma}}
\newcommand{\ddelta}{\hat{\delta}}
\newcommand{\depsilon}{\hat{\varepsilon}}
\newcommand{\deta}{\hat{\eta}}
\newcommand{\dtheta}{\hat{\theta}}

% Notation for BCFW coordinates
\newcommand{\czeta}{{\zeta}}
\newcommand{\calpha}{{\alpha}}
\newcommand{\cbeta}{{\beta}}
\newcommand{\cgamma}{{\gamma}}
\newcommand{\cdelta}{{\delta}}
\newcommand{\cepsilon}{{\varepsilon}}


\newcommand{\Many}{{\star}}

\newcommand{\OPsi}{\overline{\Psi}}
\newcommand{\rPsi}{\overline{\Psi}}

\newcommand{\chR}{\Gr_{1,5}^{\scriptscriptstyle >0}}
\newcommand{\ZZ}{{L}}
\newcommand{\YY}{{M}}

\newcommand{\Sym}{\mathfrak{S}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\T}{\mathcal{T}}
\newcommand{\swingright}{\mathbin{{\rotatebox{180}{$\curvearrowleft$}}}}
\newcommand{\swingleft}{\mathbin{{\rotatebox{180}{$\curvearrowright$}}}}
\def\mtx{M}
\def\twmt{M^{\tw}}
\def\quord{quord}
\def\quords{quords}
\def\qdiag{quord diagram}
\def\qdiags{quord diagrams }

\def\O{\mathcal{O}}
\def\Xcal{\mathcal{X}}
\def\Acal{\mathcal{A}}
\def\Rcal{\mathcal{R}}
\def\Fcal{\mathcal{F}}
\def\Q{\mathbb{Q}}
\def\QQ{\mathbb{Q}}
\def\R{\mathbb{R}}
\def\F{\mathbb{F}}
%\def\ZZ{\mathbb{Z}}
\def\NN{\mathbb{N}}
\def\TT{\mathbb{T}}
\newcommand{\xx}{\mathbf{x}}
\newcommand{\txx}{\widetilde{\mathbf{x}}}
\newcommand{\bxx}{\overline{\mathbf{x}}}
\newcommand{\tSigma}{\tilde{\Sigma}}
\newcommand{\tQ}{\widetilde{Q}}
\newcommand{\ttxx}{\widetilde{\widetilde{\mathbf{x}}}}
\newcommand{\overunder}[2]{
\!\begin{array}{c}
\scriptstyle{#1}\\[-.1in]
-\!\!\!-\!\!\!-\\[-.1in]
\scriptstyle{#2}
\end{array}
\!
}
\def\rcp{\mathfrak{r}}
\def\rcpp{\mathfrak{p}}
\def\st{\text{FStep}}
\def\twcyc{\widetilde{\cyc}}
\def\twrefl{\widetilde{\refl}}

%Notation for bullet
\newcommand{\bulR}{\triangleleft}
\newcommand{\bulL}{\triangleright}

\usepackage{lipsum}
\usepackage{wrapfig}
\usepackage[rightcaption]{sidecap}

%%To delete
\def\MP#1{\textcolor{violet}{[MP: #1]}}


%% define your title in the usual way
\title[Cluster algebras and tilings for the $m=4$ amplituhedron]{Cluster algebras and tilings \\ for the $m=4$ amplituhedron}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[C. Zohar, T. Lakrec, M. Parisi, M. Sherman-Bennett, R. Tessler, and L. Williams]{  Chaim Even-Zohar\addressmark{1}, Tsviqa Lakrec\addressmark{2}, Matteo Parisi\thanks{\href{mailto:mparisi@cmsa.fas.harvard.edu}{mparisi@cmsa.fas.harvard.edu}}\addressmark{3}, Melissa Sherman-Bennett\addressmark{4}, Ran Tessler\addressmark{5}, \and Lauren Williams\addressmark{6}}


%% then use \addressmark to match authors to institutions here
\address{ \addressmark{1}Faculty of Mathematics, Technion, Haifa, Israel \\ \addressmark{2}Institute of Mathematics, University of Zurich, Switzerland \\ 
\addressmark{3} Institute for Advanced Study, Princeton, NJ; CMSA, Harvard University, Cambridge, MA\\
\addressmark{4} Department of Mathematics, MIT, Cambridge, MA \\ \addressmark{5} Department of Mathematics, Weizmann Institute of Science, Israel\\ \addressmark{6} Department of Mathematics, Harvard University, Cambridge, MA}

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

%% leave this blank until submitting a revised version
%\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 \emph{amplituhedron} $\mathcal{A}^Z_{n,k,m}$ is the
image of the
positive Grassmannian $\mbox{Gr}_{k,n}^{\geq 0}$ under the
 map $\tilde{Z}: \mbox{Gr}_{k,n}^{\geq 0} \to \mbox{Gr}_{k,k+m}$ induced by a positive linear map 
$Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$.
 It was originally introduced in physics in order to give a geometric interpretation of scattering amplitudes. More specifically, one can compute  scattering amplitudes in $\mathcal{N}=4$ SYM by
`tiling' the $m=4$ amplituhedron $\mathcal{A}^Z_{n,k,4}$
--- that is, decomposing $\mathcal{A}^Z_{n,k,4}$ into 
`tiles' (closures of images of $4k$-dimensional cells of $\mbox{Gr}_{k,n}^{\geq 0}$
on which $\tilde{Z}$ is injective).
In this article we deepen both our understanding of tiles and tilings of the $m=4$ amplituhedron and the connection with cluster algebras.
Firstly, we prove 
the \emph{cluster adjacency conjecture} for BCFW tiles of $\mathcal{A}^Z_{n,k,4}$, which says that
 facets of tiles are cut out by collections of compatible cluster variables 
for $\mbox{Gr}_{4,n}$.  Secondly, we describe each BCFW tile as the semialgebraic set in $\mbox{Gr}_{k,k+4}$ where certain cluster variables have particular signs.
Finally, we prove 
the \emph{BCFW tiling conjecture}, which says that any way of iterating the BCFW recurrence
gives rise to a tiling of the amplituhedron~$\mathcal{A}^Z_{n,k,4}$.
Along the way, we introduce a method to construct seeds for $\mbox{Gr}_{4,n}$ comprised of 
high-degree cluster variables, which may be of independent interest in the study of 
cluster algebras.}

%% put your French abstract here, or comment this out if you don't have one
% \resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{positroid, amplituhedron, cluster algebras, tile, tiling, BCFW}

%% 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; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\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 (tree) \emph{amplituhedron} $\mathcal{A}^Z_{n,k,m}$ is the image of the 
positive Grassmannian $\Gr_{k,n}^{\scriptscriptstyle\geq 0}$ under the 
\emph{amplituhedron map} $\tilde{Z}: \Gr_{k,n}^{\scriptscriptstyle\geq 0} \to \Gr_{k,k+m}$. 
It was introduced by 
Arkani-Hamed and Trnka \cite{arkani-hamed_trnka} in order  to give a 
geometric  interpretation of
\emph{scattering amplitudes} in $\mathcal{N}=4$ super Yang Mills theory (SYM):
in particular, one can 
 compute $\mathcal{N}=4$ SYM scattering amplitudes  by
`tiling' the $m=4$ amplituhedron $\mathcal{A}^Z_{n,k,4}$
--- that is, by decomposing the amplituhedron into smaller 
`tiles' --- and summing the `volumes' of the tiles.
While the case $m=4$ is most important for physics, the amplituhedron 
is defined for any positive $n,k,m$ with $k+m \leq n$, and 
has a very rich geometric and combinatorial
structure.  It generalizes cyclic polytopes (when $k=1$), 
cyclic hyperplane arrangements \cite{karpwilliams}
(when $m=1$), and the positive Grassmannian (when $k=n-m$), and it 
is connected to 
the hypersimplex and the positive tropical Grassmanian \cite{LPW,PSW} (when $m=2$). The amplituhedron is also an example of a \emph{Grassmann polytope} (`Grasstope') and conjectured to be a \emph{positive geometry} \cite{Arkani-Hamed:2017tmz,Lam:2022yly}. 
The followings are two of the guiding problems about the amplituhedron.

The first is the \emph{cluster adjacency conjecture}, which says that 
facets of tiles are cut out by collections of compatible cluster variables. 
This was motivated by physics where cluster algebras were shown to describe singularities of scattering 
amplitudes in $\mathcal{N}=4$ SYM \cite{Golden:2013xva}. In particular, \cite{Drummond:2017ssj, Drummond:2018dfd} conjectured that 
the terms in tree-level amplitudes 
coming from the BCFW recursions are rational functions 
whose poles correspond to compatible cluster variables of the Grassmannian $\Gr_{4,n}$, see also \cite{Mago:2019waa}.  The cluster adjacency
conjecture, formulated for the $m=2$ and 
$m=4$ amplituhedron in 
\cite{Lukowski:2019sxw} and 
 \cite{Gurdougan2020ClusterPI}, was proved for all tiles of
the $m=2$ amplituhedron in \cite{PSW}. 


The second is the \emph{BCFW tiling conjecture}, 
which says that any way of iterating the
BCFW recurrence gives rise to a collection of cells whose images  tile the
$m=4$ amplituhedron $\mathcal{A}^Z_{n,k,4}$. This arose alongside the 
definition of the amplituhedron
\cite{arkani-hamed_trnka} in order to give a geometric interpretation of the recurrence Britto--Cachazo--Feng--Witten \cite{BCFW} introduced to
compute scattering amplitudes.
BCFW-like tilings of the $m=1$ and $m=2$ amplituhedron 
were proved in \cite{karpwilliams}
and \cite{BaoHe}, building on \cite{AHTT}
and \cite{karp2020decompositions}.
Finally, extending the work of \cite{karp2020decompositions}, it was proved in
\cite{even2021amplituhedron} that the `standard' way of performing the BCFW recursion gives a tiling for the $m=4$ amplituhedron.


\textbf{Main results.}
In this paper we build on \cite{PSW} and 
 \cite{even2021amplituhedron} 
 to give a very 
complete picture of the $m=4$
amplituhedron. We show that arbitrary \emph{BCFW cells} give tiles (\cref{thm:bcfw_tiles}) and that they satisfy the cluster adjacency conjecture (\cref{thm:clusteradjacency}). 
We strengthen the connection with cluster algebras by associating to each BCFW tile a collection of compatible cluster variables for $\Gr_{4,n}$ (\cref{def:coord_cluster_vars}), which we use to describe the tile as a semialgebraic set in $\mbox{Gr}_{k,k+4}$ (\cref{th:sign_description}).
For `standard' BCFW tiles, one can also give a non-recursive description of these cluster variables and the underlying quiver, and define an associated cluster algebra \cite[Sections 8,9]{Even-Zohar:2023del}.  
Finally, we use these results to prove the BCFW tiling conjecture for the $m=4$ amplituhedron (\cref{thm:BCFWtiling}).

\textbf{Further motivation.}
From the point of view of cluster algebras, 
the study of tiles for the amplituhedron $\mathcal{A}_{n,k,m}$ is useful
because it is closely related to the 
cluster structure on the Grassmannian $\Gr_{m,n}$, as was shown for 
$m=2$ in 
\cite{PSW} and as this paper demonstrates for $m=4$.  In particular, for $m=4$,
the \emph{BCFW product} (\cref{def:butterfly}) used to recursively build tiles 
(\cref{def:BCFW_cell}) has a cluster quasi-homomorphism 
counterpart called \emph{product promotion} (\Cref{def:product_promotion}),
that can be used to recursively construct 
cluster variables and seeds in $\Gr_{4,n}$ (\cref{thm:promotion2}).

In the closely related field of \emph{total positivity}, one prototypical 
problem is to give an efficient characterization of
the `positive part' of a space as the subset
where a certain minimal collection of functions take on positive values \cite{fz-intel} (`positivity test').
For example,  for any cluster $\xx$ for $\Gr_{k,n}$ \cite{scott}, the \emph{positive Grassmannian} $\Gr_{k,n}^{>0}$ can be described as the region in  $\Gr_{k,n}$ where all the cluster variables of $\xx$ are positive.

We think of \cref{th:sign_description}
as a `positivity test'
for membership in a BCFW tile of the amplituhedron. 
See \cite[Theorem 6.8]{PSW} for an analogous result for $m=2$, 
and \cite[Conjecture 7.17]{Even-Zohar:2023del} for some conjectures for general $m$.


From the point of view of discrete geometry, it is interesting to study tiles 
and more generally
Grasstopes because
one can think of them as a generalization of polytopes in the Grassmannian.  
In particular, the positivity tests for the positive Grassmannian and BCFW tiles
can be thought of as 
analogues of the hyperplane description of polytopes. 
  Finally, it would be interesting to show that 
   tiles are positive geometries.


\section{Background}

\subsection{The (positive) Grassmannian}\label{sec:posGrass}

The \emph{Grassmannian} $\Gr_{k,n}(\F)$
is the space of all $k$-dimensional subspaces of
an $n$-dimensional vector space $\F^n$.
Let $[n]$ denote $\{1,\dots,n\}$, and $\binom{[n]}{k}$ denote the set of all $k$-element 
subsets of $[n]$.
We can
 represent a point $V \in
\Gr_{k,n}(\F)$  as the row-span 
of
a full-rank $k\times n$ matrix $C$ with entries in
$\F$.  
Then for $I=\{i_1 < \dots < i_k\} \in \binom{[n]}{k}$, we let $\lr{I}_V=\lr{i_1\,i_2\,\dots\,i_k}_V$ be the $k\times k$ minor of $C$ using the columns $I$. 
The $\lr{I}_V$ 
are called the {\itshape Pl\"{u}cker coordinates} of $V$, and are independent of the choice of matrix
representative $C$ (up to common rescaling). The \emph{Pl\"ucker embedding} 
$V \mapsto \{\lr{I}_V\}_{I\in \binom{[n]}{k}}$
embeds  $\Gr_{k,n}(\F)$ into
projective space\footnote{
We will sometimes abuse notation and identify $C$ with its row-span;
we will also drop the subscript $V$ on Pl\"ucker coordinates when it does not cause confusion.} . 
If $C$ has columns $v_1, \dots, v_n$, we may also identify $\lr{i_1\,i_2\,\dots\,i_k}$ with $v_{i_1} \wedge v_{i_2} \wedge \dots \wedge v_{i_k}$, hence e.g. $\lr{i_1\,i_2\,\dots\,i_k}=- \lr{i_2\,i_1\,\dots\,i_k}$. 
In this paper we will often be working with the 
\emph{real} Grassmannian $\Gr_{k,n}=
\Gr_{k,n}(\R)$.
We will also denote by $\Gr_{k,N}$ the Grassmannians of $k$-planes in a vector space with basis indexed by a set $N\subset [n]$.

\begin{definition}[Positive Grassmannian]\label{def:positroid}\cite{lusztig, postnikov}
We say that $V\in \Gr_{k,n}$ is \emph{totally nonnegative}
     if (up to a global change of sign)
       $\lr{I}_V \geq 0$ for all $I \in \binom{[n]}{k}$.
Similarly, $V$ is \emph{totally positive} if $\lr{I}_V >0$ for all $I
      \in \binom{[n]}{k}$.
We let $\Grk$ and $\Gr_{k,n}^{>0}$ denote the set of
totally nonnegative and totally positive elements of $\Gr_{k,n}$, respectively.  
$\Grk$ is called the \emph{totally nonnegative}  \emph{Grassmannian}, or
       sometimes just the \emph{positive Grassmannian}.
\end{definition}

If we partition $\Grk$ into strata based on which Pl\"ucker coordinates are strictly
positive and which are $0$, we obtain a cell decomposition of $\Grk$
into \emph{positroid cells} \cite{postnikov}.
Each positroid cell $S$ gives rise to a matroid $\mathcal{M}$, whose bases are precisely
the $k$-element subsets $I$ such that the Pl\"ucker coordinate
$\lr{I}$ does not vanish on $S$; $\mathcal{M}$ 
is called a \emph{positroid}.



There are many ways to index positroid cells in $\Grk$ \cite{postnikov}, such as \emph{plabic graphs}:

\begin{definition}
	Let $G$ be a \emph{plabic graph}, i.e. a planar bipartite graph\footnote{We will always assume that plabic graphs are \emph{reduced} \cite[Definition 12.5]{postnikov}.} 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 $\mathcal{M}=\mathcal{M}(G):=\{\partial M: M \text{ an almost perfect matching of }G \}$.
\end{definition}

Both $\Gr_{k,n}$ and $\Grk$ admit the following set of operations, 
which will be useful to us.

\begin{definition}[Operations on the Grassmannian]\label{def:opGr} We define the following maps on $\Mat_{k,n}$, which descends to maps on  $\Gr_{k,n}$ and $\Grk$, which we denote in the same way:
\begin{itemize}
    \item (cyclic shift) We define the \emph{cyclic shift}
 as the map $\cyc: \Mat_{k, n} \to \Mat_{k,n}$ which sends  $v_1 \mapsto (-1)^{k-1}v_{n}$ and $v_i \mapsto v_{i-1}, 2 \leq i \leq n$, and in terms of Pl\"uckers: $\lr{I} \mapsto \lr{I-1}$.
 \item  (reflection) We define \emph{reflection} as the map $\refl:\Mat_{k, n} \to \Mat_{k,n}$ which sends $v_i \mapsto v_{n+1-i}$ and rescales a row by $(-1)^{{k\choose 2}}$,  and in terms of Pl\"uckers: $\lr{I} \mapsto \lr{n+1-I}$.
 \item (zero column) We define the map $\pre_i:\Mat_{k, [n] \setminus \{i\}} \to \Mat_{k,n}$ which adds a zero column at $i$, and in terms of Pl\"uckers: $\lr{I} \mapsto \lr{I}$.
\end{itemize}
Here, $I-1$ is obtained from $I \in {[n] \choose k}$ by subtracting $1$ (mod $n$) from each element of $I$ and $n+1-I$ is obtained from $I$ by subtracting each element of $I$ from $n+1$.
\end{definition}


\subsection{The amplituhedron}

Building on \cite{abcgpt,hodges},
Arkani-Hamed and Trnka
\cite{arkani-hamed_trnka}
introduced
the \emph{(tree) amplituhedron}, which they defined as 
the image of the positive Grassmannian under a positive linear map.
Let $\Mat_{n,p}^{>0}$ denote the set of $n\times p$ matrices whose maximal minors
are positive.

\begin{definition}[Amplituhedron]\label{defn_amplituhedron}
Let $Z\in \Mat_{n,k+m}^{>0}$, where $k+m \leq n$. %, which we also regard
    The \emph{amplituhedron map}
$\tilde{Z}:\Gr_{k,n}^{\ge 0} \to \Gr_{k,k+m}$
        is defined by
        $\tilde{Z}(C):=CZ$,
    where
 $C$ is a $k \times n$ matrix representing an element of
        $\Gr_{k,n}^{\ge 0}$,  and  $CZ$ is a $k \times (k+m)$ matrix representing an
         element of $\Gr_{k,k+m}$.
        The \emph{amplituhedron} $\mathcal{A}^Z_{n,k,m} \subset \Gr_{k,k+m}$ is the image
$\tilde{Z}(\Gr_{k,n}^{\ge 0})$.
\end{definition}

In this article we will be concerned with the case where $m=4$. 

\begin{definition}[Tiles]
\label{defn_tile}
	Fix $k, n, m$ with $k+m \leq n$ and choose
	$Z\in \Mat_{n,k+m}^{>0}$.  
	Given a positroid cell $S$ of 
	$\Gr_{k,n}^{\ge 0}$, we let 
	$\gto{S} := \tilde{Z}(S)$ and 
	$\gt{S}: = \overline{
		\tilde{Z}(S)} = \tilde{Z}(\overline{S})$.
	We call
	$\gt{S}$  and $\gto{S}$
	a \emph{tile}  and an \emph{open tile}
	for $\AA$ 
	if $\dim(S) =km$ and 
	$\tilde{Z}$ is injective on $S$. 
\end{definition}


\begin{definition}[Tilings]\label{def:tiling}
	A \emph{tiling} 
                of $\AA$ is
                a collection
		$\{Z_{S} \ \vert \ S\in \mathcal{C}\}$
                 of tiles, such that their union equals $\AA$ and the open tiles $\gto{S},\gto{S'}$ are pairwise disjoint.
\end{definition}

There is a natural notion of  \emph{facet} of a tile, 
generalizing the notion of facet of a polytope. 

\begin{definition}[Facet of a cell and a tile]\label{def:facet2} 
Given two positroid cells $S'$ and $S$, we say that 
$S'$ is a \emph{facet} of $S$ if 
$S' \subset \partial{S}$ and $S'$ has codimension $1$ in $\overline{S}$.
If $S'$ is a facet of $S$ and $Z_S$ is a tile of $\AA$, we say that $Z_{S'}$ is a
	\emph{facet} 
 of $Z_{S}$ if 
         $\gt{S'} \subset \partial \gt{S}$ and has codimension 1 in $\gt{S}$.
\end{definition}


\begin{definition}[Twistor coordinates]\label{def:tw_coords}
Fix
  $Z \in \Mat_{n,k+m}^{>0}$
   with rows  $Z_1,\dots, Z_n \in \R^{k+m}$.
   Given  $Y \in \Gr_{k,k+m}$
   with rows $y_1,\dots,y_k$,
   and $\{i_1,\dots, i_m\} \subset [n]$,
   we define the \emph{twistor coordinate} $
	\llrr{{i_1} {i_2} \cdots {i_m}}$
        to be  the determinant of the
        matrix with rows
        $y_1, \dots,y_k, Z_{i_1}, \dots, Z_{i_m}$.
\end{definition}

Note that the twistor coordinates are defined only up to a common scalar multiple. An element of $\mbox{Gr}_{k, k+m}$ is uniquely determined by its twistor coordinates \cite{karpwilliams}. Moreover, $\mbox{Gr}_{k,k+m}$ can be embedded into $\mbox{Gr}_{m,n}$ so that the twistor coordinate $\llrr{i_1 \dots i_m}$ is the pullback of the Pl\"ucker coordinate $\lr{i_1, \dots, i_m}$ in $\mbox{Gr}_{m,n}$.
\begin{definition}\label{def:functionary}
We refer to a homogeneous polynomial in twistor coordinates as a \emph{functionary}. For $S \subseteq \Grk$, we say a functionary $F$ has a definite sign $s \in \{\pm 1\}$ (or vanishes) on $\gto{S}$ if for all $Z\in \Mat_{n,k+4}^{>0}$ and for all $Y \in \gto{S}$, $F(Y)$ has sign $s$ (or $0$, respectively).  
\end{definition}

Functionaries will be crucial to describe tiles of the amplituhedron, to prove the main theorems about cluster adjacency and BCFW tilings, and to connect with cluster algebras.



\subsection{Cluster Algebras}

Cluster algebras were introduced by Fomin and Zelevinsky in \cite{FZ1}, motivated by the study of total positivity; see \cite{FWZ} for 
an introduction.
We give a quick definition of cluster algebras from quivers. All cluster algebras here will be of \emph{geometric type}.

A \emph{quiver} $Q$ is an oriented graph given by a finite set of vertices. For a quiver without oriented cycles of length $1$ and $2$, one can define a \emph{quiver mutation} $\mu_k(Q)$ at each vertex $k$ of $Q$. This operation, described in \cite{FZ1}, is an involution: $\mu_k^2(Q) = Q$.

\begin{definition}
\label{def:seed0}
Choose $s\geq r$ positive integers.
Let $\Fcal$ be an \emph{ambient field}
of rational functions
in $r$ independent
variables
over
$\CC(x_{r+1},\dots,x_s)$.
A \emph{labeled seed} in~$\Fcal$ is
a pair $(\xx, Q)$, where $\xx = (x_1, \dots, x_s)$ forms a free generating
set for $\Fcal$ and $Q$ is a quiver with vertices $1,2, \dots, r$ called
\emph{mutable}, and vertices $r+1,\dots, s$ called \emph{frozen}.

We call~$\xx$ a \emph{cluster} and its elements $\{x_1, \dots, x_s\}$ \emph{cluster variables}.
The variables $x_1,\dots,x_r$ are called \emph{mutable}, and the variables $c=\{x_{r+1},\dots,x_s\}$ are called
	\emph{frozen}.
\end{definition}

\begin{definition}
[\emph{Seed mutations}]
\label{def:seed-mutation0}
Let $(\xx, Q)$ be a labeled seed in $\Fcal$,
and let $k \in \{1,\dots,r\}$.
The \emph{seed mutation} $\mu_k$ in direction~$k$ transforms
$(\xx, Q)$ into the labeled seed
$\mu_k(\xx,  Q)=(\xx', \mu_k(Q))$, where the cluster
$\xx'=(x'_1,\dots,x'_s)$ is defined as follows:
$x_j'=x_j$ for $j\neq k$,
whereas $x'_k \in \Fcal$ is determined
by the \emph{exchange relation}
\begin{equation}
\label{exchange relation0}
x'_k\ x_k =
 \ \prod_{i: \, i \rightarrow k} x_{i}
+ \ \prod_{i: \,  i \leftarrow k} x_{i} \, .
\end{equation}
\end{definition}

Where $i \rightarrow k$ (or $i \leftarrow k$) denotes an edge oriented from vertex $i$ to $k$ (or $k$ to $i$).
Note that one omits arrows between two frozen vertices as they do not
affect seed mutation. 

\begin{definition}
\label{def:patterns0}
Let $\TT_r$ be an \emph{$r$-regular tree}
whose edges are labeled by $1, \dots, r$,
so that edges emanating from each vertex receive
different labels.
A \emph{cluster pattern}  is an assignment
of a labeled seed $\Sigma_t=(\xx_t, Q_t)$
to every vertex $t \in \TT_n$, such that the seed assigned to the
endpoint of an edge $k$ emanated from $t$ is obtained by mutating $\Sigma_t$ in direction~$k$.
\end{definition}

\begin{definition}
[\emph{Cluster algebra}]
\label{def:cluster-algebra0}
Given a cluster pattern, we denote as $\Xcal$
the union of all mutable variables of all the seeds in the pattern.
Let $\CC[c^{\pm 1}]$ be the \emph{ground ring} consisting
of Laurent polynomials in the frozen variables. The
\emph{cluster algebra} $\Acal$ associated with a
given pattern is the $\CC[c^{\pm 1}]$-subalgebra of the ambient field $\Fcal$
generated by all mutable variables, 
with coefficients which are Laurent polynomials
in the frozen variables: $\Acal = \CC[c^{\pm 1}] [\Xcal]$.
We denote $\Acal = \Acal(\xx,  Q)$, where
$(\xx,Q)$
is any seed in the underlying cluster pattern.
We say that $\Acal$ has \emph{rank $r$} because each cluster contains
$r$ mutable variables. Cluster variables that belong to a common cluster are said to be \emph{compatible}.
\end{definition}



The Grassmannian $\mbox{Gr}_{k,n}(\mathbb{C})$
has a cluster structure \cite{scott}, defined starting from particularly nice seeds called \emph{rectangles seed} $\Sigma_{k,n}$, see \cref{fig:gr_quiver} and the exposition of \cite{FW6}. 

\begin{SCfigure}[50]
\includegraphics[width=0.5\textwidth]{gr_quiver.pdf}
  \caption{The rectangle seed $\Sigma_{4,7}$. Mutable variables are in the colored box.
}
\label{fig:gr_quiver}
\end{SCfigure}




\begin{theorem}[\cite{scott}]\label{thm:scott}
Let ${\Gr}_{k,n}^\circ$ be the open subset of the Grassmannian where the frozen variables
don't vanish. Then the coordinate ring $\C[\widehat{\Gr}_{k,n}^\circ]$ of the affine cone over ${\Gr}_{k,n}^\circ$ is the cluster algebra
 $\A(\Sigma_{k,n})$.
\end{theorem}



Moreover, the operations on the Grassmannian $\cyc, \refl, \pre$ in \cref{def:opGr} induce maps on $\CC[\widehat{\Gr}^\circ_{k,N}]$ which are compatible
with the cluster structure of \cref{thm:scott}:


\begin{proposition}\label{prop:pre_cluster}
       The maps $\cyc, \refl: \CC[\widehat{\Gr}^\circ_{k,n}] \to \CC[\widehat{\Gr}^\circ_{k,n}], \pre_i: \C[\widehat{\Gr}^\circ_{k, [n] \setminus \{i\}}] \to \C[\widehat{\Gr}^\circ_{k,n}]$ take cluster variables to cluster variables and preserve compatibility and exchange relations.
\end{proposition}








\section{Results}
Our first main result is proving that a class of cells called \emph{BCFW cells} give tiles for $\mathcal{A}^Z_{n,k,4}$. We will build BCFW cells recursively using the \emph{BCFW product}. Let us first introduce some notation we will use throughout this section. 

\begin{notation}\label{not:LR_cluster}
Choose integers $1\leq a<b<c<d<n$ with $a,b$ and $c,d,n$ consecutive.
	Let\footnote{
Note that we will overload the notation and let $n$
	index an element of a vector space basis for different
	vector spaces; however, in what follows, the meaning should
	be clear from context.} $N_L = \{n,1,2,\dots,a,b\}, N_R = \{b, \dots, c, d, n\}$ and $D=(a,b,c,d,n)$. Also fix $k \leq n$ and two nonnegative integers $k_L \leq |N_L|$ and $ k_R\leq |N_R|$ such that $k_L + k_R +1=k$. 
 Note that, for any set of indices $N \subset [n]$, our results hold with $N$ instead of $[n]$, by replacing $1$ and $n$ in the definition with the smallest and largest elements of $N$, respectively.
\end{notation}



\begin{definition}[BCFW product]\label{def:butterfly}
Let $S_L \subseteq \mbox{Gr}_{k_L,N_L}, S_R \subseteq \mbox{Gr}^{\geq 0}_{k_R,N_R}$ as in \cref{not:LR_cluster} and $G_L, G_R$ be the respective plabic graphs. The \emph{BCFW product} of $S_L$ and $S_R$ is the positroid cell $S_L \bcfw S_R \subseteq \mbox{Gr}_{k,n}$ corresponding the plabic graph in the right-hand side of \Cref{fig:butterfly}. When it is not clear from the context, we will say $\bcfw$ is performed `with indices $D$'.


\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{promotion_plabic.pdf}
\caption{The BCFW product $S_L \bcfw S_R$ of $S_L$ and $S_R$ in terms of their plabic graphs.
}
\label{fig:butterfly}
\end{figure}
\end{definition}

We now introduce the family of \emph{BCFW cells} to be the set of positroid cells which is closed under the operations in \Cref{def:opGr,def:butterfly}:

\begin{definition}[BCFW cells]\label{def:BCFW_cell}
The set of \emph{BCFW cells} is defined recursively. For $k=0$, let the trivial cell $\Gr^{\scriptscriptstyle>0}_{0,n}$ be a BCFW cell. If $S$ is a BCFW cell, so is the cell obtained by applying $\mbox{cyc}, \mbox{refl}, \mbox{pre}$ to $S$. If $S_L,S_R$ are BCFW cells, so is their BCFW product $S_L \bcfw S_R$.
\end{definition}



\begin{example}\label{ex:bcw_tile}
For $k=1$, the BCFW cells in $\Gr_{1,n}^{\geq 0}$ are as in \cref{fig:bcfw_tile} (left). They have Pl\"ucker coordinates $\lr{a},\lr{b},\lr{c},\lr{d},\lr{e}>0$ and all others zero. 
    In \cref{fig:bcfw_tile} (right), $S_{ex} \subset \mbox{Gr}^{\geq 0}_{2,7}$ is obtained as $S_L \bowtie S_R$, with $S_L, S_R$ BCFW cells in $\mbox{Gr}^{\geq 0}_{1,N_L}, \mbox{Gr}^{\geq 0}_{0,N_R}$ respectively, with  $N_L=\{7,1,2,3,4\}, N_R=\{4,5,6,7\}$ and $D=(3,4,5,6,7)$.
\end{example}



\begin{SCfigure}[85]
\includegraphics[width=0.4\textwidth]{bcfw_tile_ex.pdf}
  \caption{ \label{fig:bcfwcellsk1} Plabic graphs of:
   a BCFW cell in $\Gr^{\geq 0}_{1,n}$ (left);  a BCFW cell $S_{ex} \subset \mbox{Gr}^{\geq 0}_{2,7}$ (right).} \label{fig:bcfw_tile}
\end{SCfigure}





\begin{theorem}[BCFW tiles]\label{thm:bcfw_tiles}
The amplituhedron map is injective on each BCFW cell.
That is,
	the closure 
	${\gt{S} :=
	\overline{\tilde{Z}(S)}}$ of 
	the image of a BCFW cell $S$
 is a \emph{tile}, which we
	refer to as a \emph{BCFW tile}. 
\end{theorem}

A key ingredient to prove \cref{thm:bcfw_tiles} is inverting the amplituhedron map on BCFW tiles \cite[Theorem 7.7]{Even-Zohar:2023del} by using \emph{product promotion} -- an operation which interacts nicely both with the cluster structure on the Grassmannian and with the BCFW product.



  

\begin{definition}\label{def:product_promotion}


Using \cref{not:LR_cluster}, \emph{product promotion} is the homomorphism 
	$$\Psi_{D} = \Psi: \C(\widehat{\Gr}_{4,N_L})\times \C(\widehat{\Gr}_{4,N_R}) \to \C(\widehat{\Gr}_{4,n}),$$ 
induced 
by the following substitution:
\begin{equation*}
\text{on $\widehat{\Gr}_{4,N_L}$: } b \;\mapsto\; \frac{(ba)\cap (cdn)}{\lr{a\,c\,d\,n}}, \quad \text{on $\widehat{\Gr}_{4,N_R}$: } n \;\mapsto\;
\frac{(ba)\cap (cdn)}{\lr{a\,b\,c\,d}} 
\label{eq:promotionvectors2}, \, d \;\mapsto\; \frac{(dc)\cap (abn)}{\lr{a\,b\,c\,n}}. 
\end{equation*}
\end{definition}
The vector $(ij)\cap (rsq):=v_i \lr{j \, r \, s\, q}-v_j \lr{i \, r \, s\, q}= - v_r \lr{i \, j \, s\, q}+v_s \lr{i \, j \, r\, q}-v_q \lr{i \, j \, r\, s} $ is in the intersection of the $2$-plane and the $3$-plane spanned by $v_i,v_j$ and $v_r,v_s,v_q$, respectively.

 We show\footnote{We will sometime omit the dependence on the indices $D=\{a,b,c,d,n\}$ in $\Psi$ (and $\rPsi$) for brevity.} that $\Psi$ is in fact a \emph{quasi-homomorphism} (see \cite{Fraser}) from the cluster algebra\footnote{$\C[\widehat{\Gr}_{4,N_L}^{\circ}] \times \C[\widehat{\Gr}_{4,N_R}^{\circ}]$ is a cluster algebra where each seed 
is the disjoint union of a seed of each factor.} $\C[\widehat{\Gr}_{4,N_L}^{\circ}] \times \C[\widehat{\Gr}_{4,N_R}^{\circ}]$ to a sub-cluster algebra of $\C[\widehat{\Gr}_{4,n}^{\circ}]$. See \cite[Definition 3.1, Proposition 3.2]{Fraser} for the precise definition of a quasi-homomorphism.

\begin{theorem}\label{thm:promotion2}
Product promotion 
$\Psi$ is a quasi-homomorphism of cluster
algebras. In particular, $\Psi$ maps a cluster variable (respectively, cluster)
of  $\C[\widehat{\Gr}_{4,N_L}^{\circ}] \times \C[\widehat{\Gr}_{4,N_R}^{\circ}]$, to a cluster variable (respectively, sub-cluster) of 
$\C[\widehat{\Gr}_{4,n}^{\circ}]$, up to multiplication by Laurent monomials in $\mathcal{T'}:=\{ 
\lr{a\,b\,c\,n},
\lr{a\,b\,c\,d},
\lr{b\,c\,d\,n}, 
\lr{a\,c\,d\,n}\}$. 
\end{theorem}

\begin{remark}
\cref{def:product_promotion} and \cref{thm:promotion2} extend also to the degenerate cases, e.g. for $a=1$ (\emph{upper promotion}), where $\Psi:\C(\widehat{\Gr}_{4,N_R}) \to \C(\widehat{\Gr}_{4,n})$, see \cite[Section 4.3]{Even-Zohar:2023del}. 
\end{remark}

\begin{definition}\label{def:rPsi} Let $x$ be a cluster variable of $\C[\widehat{\Gr}_{4,N_L}^{\circ}]$ or $\C[\widehat{\Gr}_{4,N_R}^{\circ}]$. 
 We define the \emph{rescaled product promotion} $\rPsi(x)$ of $x$ to be the cluster variable of $\Gr_{4,n}$ obtained from $\Psi(x)$ by removing\footnote{If $x= \lr{bcdn}$, then $\rPsi(x)=\Psi(x)= x$.} the Laurent monomial in $\mathcal{T'}$ (c.f. \Cref{thm:promotion2}).
\end{definition}

The fact that product promotion is a cluster quasi-homomorphism may be of independent interest
in the study of the cluster structure on $\Gr_{4,n}$.
Much of the work thus far on the cluster structure of the Grassmannian
has focused on cluster variables which are polynomials in Pl\"ucker coordinates
with low degree; by constrast, the cluster variables we obtain can have arbitrarily high degree
in Pl\"ucker coordinates (e.g. see the \emph{chain polynomials} in \cite[Theorem 8.3]{Even-Zohar:2023del}). We introduce the following notation:
\begin{align}
	\lr{a\,b\,c \br d\,e \br f\,g\,h}  &:= 
\lr{a,b,c, (d \, e) \cap (f \, g\, h)}=\lr{a\,b\,c\,d}\,\lr{e\,f\,g\,h} - \lr{a\,b\,c\,e}\,\lr{d\,f\,g\,h} \label{eq1:quadratic}.
\end{align}

\begin{example}\label{ex:psi}
For $N_L$ and $N_R$ as in \cref{ex:bcw_tile}, the only Pl\"ucker which changes is: $
  \Psi(\lr{1 \, 2 \, 4 \, 7})= \lr{1 \, 2 \, 7 | 3 \, 4| 5\, 6\, 7}/ \lr{3 \, 4\, 6\, 7}$, and $\rPsi(\lr{1 \, 2 \, 4 \, 7})=\lr{1 \, 2 \, 7 | 3 \, 4| 5\, 6\, 7}$ which is a quadratic cluster variable in $\mbox{Gr}_{4,7}$, e.g. obtained by mutating $\lr{2367}$ in $\Sigma_{4,7}$ of \cref{fig:gr_quiver}.   
\end{example}






Using rescaled product promotion and the operations in \cref{prop:pre_cluster}, 
we associate to each BCFW tile $\gt{S}$ a collection of compatible cluster variables $\Irr(S)$ for $\Gr_{4,n}$. 

\begin{definition}[Cluster variables for BCFW tiles]\label{def:coord_cluster_vars}
Let $S \subset \Grk$ be a BCFW cell. We define the set of \emph{coordinate cluster variables} $\Irr(S)$ for $S$ recursively as follows:
\begin{itemize}[noitemsep]
\vspace{-0.3em}
\item If $S =S_{L} \bowtie S_{R}$ with indices $D_k=(a_k, b_k, c_k, d_k, n_k)$, then \vspace{-0.5em}
\begin{equation}
    \Irr(S)=\rPsi_{D_k}(\Irr(S_L) \cup \Irr(S_R)) \cup \lbrace \lr{I}, I \in {D_k \choose 4} \rbrace,
\end{equation} \vspace{-1em}
\item If $S=\begin{cases}
\pre_i(S')& \\
\cyc(S')& \\
\refl(S')
\end{cases}$ then $\Irr(S)=\begin{cases}
\Irr(S')& \\
\cyc^{-1}(\Irr(S'))& \\
\refl(\Irr(S'))
\end{cases},$
\end{itemize}
and for the base case $k=0$, we set $\Irr(S)=\emptyset$. Here, $\cyc^{-1}=\cyc^{n-1}$.  

 
For a BCFW cell $S$, $\Irr(S)$ depends on the sequence of operations in \cref{def:BCFW_cell} used to build $S$, but we will drop this dependence for brevity. 

Note that $\Irr(S)$ is a collection of compatible cluster variables for $\mbox{Gr}_{4,n}$ \cite[Lemma 7.6]{Even-Zohar:2023del}. 
\end{definition} 




\begin{example}\label{ex:coord_clust_vars} From \cref{ex:bcw_tile}, $S_{ex}=S_L \bowtie S_R$ and
 $\Irr(S_L)=\{\lr{I}, I \in {D_L \choose 4} \}, \Irr(S_R)=\emptyset$, where $D_L=\{1,2,3,4,7\}$. Then by \cref{ex:psi} the coordinate cluster variables $\Irr(S_{ex})$ are: $\rPsi(\Irr(S_L))=\Irr(S_L) \setminus \{\lr{1247}\} \cup \{\lr{127|34|567}$\} together with $\{\lr{I}, I \in {D \choose 4} \}$.
\end{example}

Given a cluster variable $x$ in $\Gr_{4,n}$, we will denote as $x(Y)$ the functionary on $\Gr_{k,k+4}$ (cf. \cref{def:functionary}) obtained by identifying Pl\"ucker coordinates $\lr{I}$ in $\Gr_{4,n}$ with twistor coordinates $\llrr{I}$ in $\Gr_{k,k+4}$ (cf. \cref{def:tw_coords}). 
Then interpreting each cluster variable as a functionary,
we describe each BCFW tile as the semialgebraic subset of
$\Gr_{k,k+4}$ where the coordinate cluster variables take on particular signs.

\begin{theorem}[Sign description of BCFW tiles]\label{th:sign_description}
	Let $\gt{S}$ be a BCFW tile. For each element $x$ of $\Irr(S)$, the functionary $x(Y)$ has a definite sign $s_x$ on $\gto{S}$ and 
	\[\gto{S}= \{Y \in \Gr_{k,k+4}: s_x \, x(Y) >0 \text{ for all } x \in \Irr(S) \}.\]
\end{theorem}

\begin{example}
The open tile $\gto{ex}:=\tilde{Z}(S_{ex})$, with $S_{ex}$ from \cref{ex:bcw_tile}, is the semialgebraic set in $\mbox{Gr}_{2,6}$ where the functionaries $x(Y)$, with $x \in \Irr(S_{ex})$ of \cref{ex:coord_clust_vars} are positive, except when $x \in \{\lr{3567},\lr{3457},\lr{2347},\lr{3567}\}$, for which $x(Y)$ are negative.
\end{example}

 One can  study facets of tiles (see \cref{def:facet2}) by describing associated functionaries which vanish on them. Given a functionary $F(\llrr{I})$ on $\Gr_{k,k+4}$, we can obtain a polynomial $F(\lr{I})$ in the Pl\"ucker coordinates in $\Gr_{4,n}$. Then the coordinate cluster variables in \cref{def:coord_cluster_vars} are a key tool in the proof of cluster adjacency conjecture for BCFW tiles:


\begin{theorem}[Cluster adjacency for BCFW tiles]
	\label{thm:clusteradjacency}
	Let $\gt{S}$ be a BCFW tile of 
	$\mathcal{A}_{n,k,4}^Z$. Each facet $\gt{S'}$ of $\gt{S}$ lies on a hypersurface cut out by a functionary $F_{S'}(\llrr{I})$ such that $F_{S'}(\lr{I})$ is in $\Irr(S)$. Thus $\{F_{S'}(\lr{I}): \gt{S'} \text{ a facet of }\gt{S}\}$ is a collection of compatible cluster variables of $\Gr_{4,n}$.
\end{theorem}

Finally, we show how to use BCFW tiles to tile $\mathcal{A}^Z_{n,k,4}$ (\cref{def:tiling}). \Cref{thm:bcfw_tiles,th:sign_description} are important ingredients to prove our last main result \cref{thm:BCFWtiling}. We use \cref{not:LR_cluster}, fix $n\geq k+4$, and define $b_{min}:=2$ if $k_L=0$ and otherwise $b_{min}:=k_L+3$. 

\begin{definition}[BCFW collections]\label{def:BCFWoutput}

We say that a collection $\mathcal{T}$ of $4k$-dimensional BCFW cells in 
$\Gr_{k,n}^{\geq 0}$  
	is a \emph{BCFW collection of cells} for 
$\mathcal{A}_{n,k,4}$ if it has the following recursive form: \vspace{-0.5em}
\begin{itemize}[noitemsep]
\item If $k=0$ or $k=n-4$, $\mathcal{T}$ is the single BCFW cell 
		$\Gr_{0,n}^{>0}$ or $\Gr_{n-4,n}^{>0}$, respectively.
\item If $\mathcal{T}=\{S\}$ is a BCFW collection of cells, 
	so is $\{\refl S\}_{S \in \mathcal{T}}$ and $\{\cyc S\}_{S \in \mathcal{T}}$. \vspace{0.2em}
\item Otherwise $\mathcal{T} = 
	\mathcal{T}_{pre} \bigsqcup_{b,k_L,k_R} \mathcal{T}_{k_L,k_R,b,n}$,
	where
		\begin{itemize}
\item $b$ ranges from 
	$b_{min}$ to $n-3-k_R$, and $k_L,k_R$ as in \cref{not:LR_cluster};
\item 
$\mathcal{T}_{pre}=\{\pre_{d}(S)\}_{S \in \mathcal{C}},$ where 
			$\mathcal{C}$ is a BCFW collection of cells for $\mathcal{A}_{[n]\setminus\{d\},k, 4}$;
\item  $\mathcal{T}_{k_L,k_R,b,n}=\{S_L \bcfw S_R\}_{(S_L,S_R) \in \mathcal{C}_L \times \mathcal{C}_R}$
				where 
				$\mathcal{C}_L$ and 
				$\mathcal{C}_R$ are BCFW collections of cells for	$\mathcal{A}_{N_L,k_L,4}$ 
				and $\mathcal{A}_{N_R,k_R,4}$.
		\end{itemize}
		\end{itemize}
	\end{definition}

\begin{figure}
	\centering
	\includegraphics[width=0.8\textwidth]{bcfw_tiling.pdf}
	\caption{\label{fig:bcfw_tiling} 
	BCFW tiling for $\mathcal{A}_{n,k,4}$. On the right: the first term is obtained by tiling $\mathcal{A}_{[n] \setminus \{d\},k,4}$ (from $\mathcal{T}_{pre}$); the second term is the union over $b,k_L,k_R$ as in \cref{def:BCFWoutput} of the collections of tiles obtained by tiling $\mathcal{A}_{N_L,k_L,4}$ and $\mathcal{A}_{N_R,k_R,4}$ (from $\mathcal{T}_{k_L,k_R,b,n}$).}
\end{figure}

\begin{theorem}[BCFW tilings]\label{thm:BCFWtiling}
Every BCFW collection of cells $\mathcal{T}=\{S \}$ as in \cref{def:BCFWoutput} gives rise to a tiling 
	$\{Z_{S} \}_{S \in \mathcal{T}}$ 
	of the amplituhedron
$\mathcal{A}_{n,k,4}^Z$, which we refer to as a \emph{BCFW tiling}. 
\end{theorem}
 
See \cref{fig:bcfw_tiling} for an illustration. This generalizes the main result
of \cite{even2021amplituhedron}, which proved the same result for the \emph{standard} BCFW cells, and proves
the main conjecture of \cite{arkani-hamed_trnka}.

Non-BCFW tiles are also expected to satisfy cluster adjacency, have a sign description in terms of cluster variables, and appear in tilings of $\mathcal{A}^Z_{n,k,4}$, see \cite[Section 12.2]{Even-Zohar:2023del}. 


\vskip .2cm



%\noindent{\bf Acknowledgements:~} 
\acknowledgements{TL is supported by SNSF grant Dynamical Systems, grant no.~188535.
MP is supported by Harvard CMSA and at IAS by the grant number DE-SC0009988. 
MSB is supported by the National Science Foundation under Award No.~DMS-2103282.
RT (incumbent of the Lillian and George Lyttle Career Development Chair) was supported by the ISF grant No.~335/19 and 1729/23.
LW is supported by the National Science Foundation under Award No. 
DMS-1854316 and DMS-2152991. Any opinions, findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the National Science
Foundation.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}
