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%%% We can change these whenever we want, e.g., would \rowm_{\mathcal{I}} be better ror
%%% \rowJ ?? --TR
\def\rowm{\rho}
\def\rowA{\rowm_{\mathcal{A}}}
\def\rowJ{\rowm_{\mathcal{J}}}
\renewcommand{\topfraction}{0.89}
\renewcommand{\bottomfraction}{0.89}
\renewcommand{\textfraction}{0.11}
\renewcommand{\floatpagefraction}{0.8}

\usepackage{lipsum}

%% define your title in the usual way
\title{Birational antichain toggling and rowmotion}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Michael Joseph\thanks{\href{mailto:mjosephmath@gmail.com}{mjosephmath@gmail.com}}\addressmark{1}
\and Tom
Roby\thanks{\href{mailto:tomrobyUConn@gmail.com}{tomrobyUConn@gmail.com}}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Technology and Mathematics, Dalton State College, Dalton, GA 30720,  USA\\
 \addressmark{2}Department of Mathematics, University of Connecticut, Storrs, 06269-1009, USA}

%% 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{We define a new combinatorial dynamical system on node-labelings of a poset $P$ by
rational functions called \textit{birational antichain rowmotion} (\textit{BAR-motion} for
short). It is analogous to the previously-studied \textit{birational
rowmotion} action (here called \textit{birational order rowmotion}, or \textit{BOR-motion}).
The latter action is detropicalized from an extension of combinatorial rowmotion on
order ideals of $P$ to Stanley's \textit{order polytope} $\calo\calp(P)$.  Analogously
BAR-motion detropicalizes the extension of rowmotion on \textit{antichains} of $P$ to the 
\textit{chain polytope} $\calc (P)$.
 
 We study BAR-motion by defining a birational antichain \textit{toggle group} generated by involutions called \textit{toggles}.
 This lifts Striker's toggle group on the set of antichains of a poset to the birational
 realm.  We construct an explicit isomorphism between this group and Einstein and Propp's
 group of birational order toggles, 
 lifting an analogous one between the toggle groups of
 order ideals and antichains at the combinatorial level.
 
 
 For certain nice families of posets, the order of BOR-motion is known to be
 finite, and can often be easily computed.  We lift Stanley's transfer
 map between $\calc (P)$ and $\calo\calp(P)$ to a birational transfer 
 map, which allows us to easily deduce certain properties of one kind of birational rowmotion
 from the other.  We take advantage of this to derive the periodicity and order of BAR-motion on certain
 root and minuscule posets.  We also lift a refined homomesy result of Propp and the second
 author from the combinatorial setting to the birational one, using an analogue of the
 ``Stanley--Thomas'' word, which cyclically rotates equivariantly with BAR-motion. 
% Going the other way, we generalize the periodicity and order
%  results of Grinberg-Roby for so-called ``skeletal posets'' to the larger class of 
%  \textit{series-parallel} posets.
 }

%% 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{
 birational rowmotion, 
 dynamical algebraic combinatorics,
 homomesy, 
 periodicity,
 poset,
 toggling.
 }

%% 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

%% 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{bibliography.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}

\textit{Combinatorial rowmotion} is a particular permutation of the set of order ideals $\calj(P)$ of a finite poset $P$ or of the set of antichains $\cala(P)$
of $P$.  It was first studied as a map on $\cala(P)$
by Brouwer and Schrijver~\cite{brouwer1974period}, and goes by several names; in recent literature, the name 
``rowmotion,'' due to Striker and Williams~\cite{strikerwilliams} (who summarize the history), has stuck. 

Rowmotion has proven to be of great interest in dynamical algebraic combinatorics.  On
several ``nice'' posets (e.g., positive root poset or minuscule posets, such as a product of
two chains), rowmotion exhibits various phenomena including \emph{periodicity} (of a relatively
small order), \emph{cyclic sieving} (as defined by Reiner, Stanton, and White~\cite{csp}),
and \emph{homomesy} (where a natural statistic, e.g.~cardinality, has the same average over every
orbit)~\cite{ast,shahrzad,indepsetspaper,propproby,rush-wang-homomesy-minuscule,vorland-3-chains}. 
Rowmotion is related to Auslander--Reiten translation on certain quivers~\cite{Yil17}.   
Quite surprisingly, some of these features extend to the piecewise-linear (order polytope)
level and can be lifted further to the birational level.~\cite{einpropp}.  One sometimes
gets periodicity of the same order as the combinatorial map, and homomesy extends as well.  
Birational rowmotion is related to $Y$-systems of type $A_m \times A_n$ described in
Zamolodchikov periodicity~\cite[\S4.4]{robydac} and to the \emph{$R$-systems} of
Galashin--Pylyavskyy~\cite{r-systems}. 

The lifting of order ideal rowmotion (herein denoted $\rowJ$) to BOR-motion (birational order rowmotion) proceeds by first writing $\rowJ$ as a product of
involutions called \textit{toggles}, which act on $\calj (P)$, the set of order ideals of a
poset. These toggles are then extended to Stanley's \textit{order polytope} $\calo\calp(P)$, which can then be lifted (via detropicalization) to toggles at
the birational level~\cite{einpropp}.  Striker defined \emph{antichain toggles} that act
on $\cala (P)$, the set of antichains of a poset~\cite{strikergentog}, as part of a broader
study of toggling in general.  The first author gave an explicit isomorphism between these
two different toggle groups for the same poset $P$, and extended these results to the
piecewise-linear level, where $\cala (P)$ extends to Stanley's \emph{chain polytope} $\calc
(P)$~\cite{antichain-toggling}.  In this work we lift this as before to obtain a new
operation called \emph{Birational Antichain Rowmotion} or \emph{BAR-motion} for short.

We show that \emph{BAR-motion} is periodic on $P=[a]\times[b]$ (product of two chains), with the same order as BOR-motion (or indeed
order ideal rowmotion), and lift a homomesy result of ~\cite{propproby} for fibers of the
poset $P=[a]\times [b]$ (product of two chains) to BAR-motion, which we conjecture holds for
all $a$, but currently only have a proof for $a\leq 2$.  Our main tools for doing this
include a lifting of Stanley's transfer map between $\calo (P)$ and $\calc (P)$, and a
generalization of the Stanley--Thomas word, which provides an equivariant projection of
BAR-motion to cyclic rotation.  (See \cref{sec:ST2b}.)

%\twr{Anything else to say here? }

The authors are grateful for useful conversations with
David Einstein,
Darij Grinberg, 
James Propp, 
Richard Stanley, 
Jessica Striker,
and Corey Vorland.

\subsection{Rowmotion in the combinatorial realm}
We assume familiarity with basic notions from the theory of posets (chains, antichains, order
ideals, order filters, linear extensions, etc.)  as given in \cite[Ch. 3]{ec1ed2}.
Throughout this paper $P$ will denote a finite poset. 

% \begin{definition}\hspace{-8 in}.
% \begin{itemize}
% \item An \textbf{order ideal} (resp.\ \textbf{order filter}) of $P$ is a subset $I\subseteq P$ such that if $x\in I$ and $y<x$ (resp.\ $y>x$) in $P$, then $y\in I$.  We denote the sets of order ideals and order filters of $P$ as $\calj(P)$ and $\calf(P)$ respectively.
% \item An \textbf{antichain} (resp.\ \textbf{chain}) of $P$ is a subset $S\subseteq P$ in which any two elements are incomparable (resp.\ comparable).  The set of antichains of $P$ is denoted $\cala(P)$.
% %\item For a subset $S\subseteq P$, an element $x\in S$ is a \textbf{maximal} (resp.\ \textbf{minimal}) element of $S$ if $S$ does not contain any $y>x$ (resp.\ $y<x$).
% \end{itemize}
% \end{definition}

%Let $P$ denote a finite poset throughout the paper.
To define rowmotion, following the notation of Einstein-Propp~\cite{einpropp}, we define the following natural bijections between the sets $\calj(P)$ of all
\emph{order ideals} of $P$, $\calf(P)$ of all \emph{order filters} of $P$, and $\cala(P)$ of
all \emph{antichains} of $P$. 
\begin{itemize}
\item The map $\Theta: 2^P \ra 2^P$ where $\Theta(S) = P\sm S$ is the complement of $S$ (so $\Theta$ sends order ideals to order filters and vice versa).
\item The \textbf{up-transfer} $\up:\calj(P) \ra \cala(P)$, where $\up(I)$ is the set of maximal elements of $I$.  For an antichain $A\in\cala(P)$,
$\up^{-1}(A)=\{x\in P: x\leq y \text{ for some } y\in A\}$.
\item The \textbf{down-transfer} $\down:\calf(P) \ra \cala(P)$, where $\down(F)$ is the set of minimal elements of $F$.  For an antichain $A\in\cala(P)$,
$\down^{-1}(A)=\{x\in P: x\geq y \text{ for some } y\in A\}$.
\end{itemize}

\begin{definition}\label{def:OIrow}
\textbf{Order ideal rowmotion} is the map $\rowJ: \calj(P) \ra \calj(P)$ given by the composition
$\rowJ = \up^{-1}\circ\down\circ\Theta$.
\textbf{Antichain rowmotion} is the map $\rowA: \cala(P) \ra \cala(P)$ given by the composition
$\rowA = \down\circ\Theta\circ\up^{-1}$.
\end{definition}

\begin{example}\label{ex:a3-row}
Below we show examples of $\rowJ$  and $\rowA$ on the positive root poset
$\Phi^+(A_3)$. In each step, the elements of the subset of the poset are given by the
filled-in circles.
%\twr{Mike: Could you add $\rowJ$ and $\rowA$ at the start of the appropriate lines?}
\begin{center}
\begin{tikzpicture}[xscale=0.5,yscale=0.5]
\node at (-5,-3) {$\rowJ:$};
\begin{scope}[shift={(0,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw (-1,1) circle [radius=0.2];
\draw[fill] (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[fill] (2,0) circle [radius=0.2];
\end{scope}
\node at (3.5,-3) {$\stackrel{\Theta}{\longmapsto}$};
\begin{scope}[shift={(7,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw[fill] (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw (-2,0) circle [radius=0.2];
\draw (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\node at (10.5,-3) {$\stackrel{\down}{\longmapsto}$};
\begin{scope}[shift={(14,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw (-2,0) circle [radius=0.2];
\draw (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\node at (17.5,-3) {$\stackrel{\up^{-1}}{\longmapsto}$};
\begin{scope}[shift={(21,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\end{tikzpicture}
\end{center}

\begin{center}
\begin{tikzpicture}[xscale=0.5,yscale=0.5]
\node at (-5.5,2) {$\rowA:$};
\begin{scope}
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw (-1,1) circle [radius=0.2];
\draw[fill] (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\node at (3.5,1) {$\stackrel{\up^{-1}}{\longmapsto}$};
\begin{scope}[shift={(7,0)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw (-1,1) circle [radius=0.2];
\draw[fill] (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[fill] (2,0) circle [radius=0.2];
\end{scope}
\node at (10.5,1) {$\stackrel{\Theta}{\longmapsto}$};
\begin{scope}[shift={(14,0)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw[fill] (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw (-2,0) circle [radius=0.2];
\draw (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\node at (17.5,1) {$\stackrel{\down}{\longmapsto}$};
\begin{scope}[shift={(21,0)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw (-2,0) circle [radius=0.2];
\draw (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\end{scope}
\end{tikzpicture}
\end{center}
\end{example}


\subsection{The order ideal toggle group}
%%An equivalent way to express order ideal rowmotion on a finite poset $P$ is via permutations on $\calj(P)$ called \emph{toggles}.  
The map $\rowJ$ can also be written a composition of involutions on $\calj (P)$ called
\emph{toggles}, as first shown by Cameron and Fon-Der-Flaass~\cite{cameronfonder}. 
%%% WE CAN COMMENT THIS OUT IF NEED BE: 
They also
proved that for any finite connected poset $P$, the toggle group $\tog_\calj(P)$ generated by
$\{t_e:e\in P\}$
is either the symmetric or alternating group on the set $\calj(P)$.  
   %%%introduced a \textbf{toggle group} on the set $\calj(P)$ 

\begin{definition}[\cite{cameronfonder}]\label{def:comb-row}
For $e\in P$, the \textbf{order ideal toggle} corresponding to $e$ is the map
$t_e: \calj(P)\ra \calj(P)$
defined by
$t_e(I)=\left\{\begin{array}{ll}
I\cup\{e\} &\text{if $e\not\in I$ and $I\cup\{e\}\in \calj(P)$,}\\
I\sm\{e\} &\text{if $e\in I$ and $I\sm\{e\}\in \calj(P)$,}\\
I &\text{otherwise.}
\end{array}\right.$\\
Let $\tog_\calj(P)$ denote the \textbf{toggle group} of $\calj(P)$ generated by the toggles $\{t_e : e\in P\}$.
\end{definition}

The toggle $t_e$ either adds or removes $e$ from the order ideal if the resulting set is still an order ideal, and otherwise does nothing.
 The following proposition presents basic properties of order ideal toggles.
%\iffalse
\begin{prop}[\cite{cameronfonder}]\label{prop:J-toggle-inv-commute}
Each toggle $t_x$ is an involution (i.e., $t_x^2$ is the identity).  Two toggles $t_x,t_y$ commute if and only if neither $x$ nor $y$ covers the other.
\end{prop}
%\fi

%Cameron and Fon-Der-Flaass proved that on any finite connected poset, the toggle group $\tog_\calj(P)$ generated by
%$\{t_e:e\in P\}$
%is either the symmetric or alternating group on $\calj(P)$~\cite{cameronfonder}.
\iffalse
\begin{defn}[{\cite[\S3.5]{ec1ed2}}]
A sequence $(x_1,x_2,\dots,x_n)$ containing all of the elements of a finite poset $P$ exactly once is called a \textbf{linear extension} of $P$ if it is order-preserving, that is, whenever $x_i<x_j$ in $P$ then $i<j$.
\end{defn}
\fi
\begin{prop}[\cite{cameronfonder}]\label{prop:row-toggles}
For any linear extension $(x_1,x_2,\dots,x_n)$ of $P$, order ideal rowmotion is given
by $\rowJ = t_{x_1} t_{x_2} \cdots t_{x_n}$.
\end{prop}

\iffalse
\begin{example}\label{ex:row-toggles}
For the poset of \cref{ex:a3-row}, as labeled below, $(a,b,c,d,e,f)$ gives a linear extension.
We show the effect of applying $t_at_bt_ct_dt_et_f$ to the order ideal considered in \cref{ex:a3-row}.
In each step, we indicate the element whose toggle we apply next in {\color{red}red}.
Notice that the outcome is the same order ideal we obtained by the three step process, demonstrating \cref{prop:row-toggles}.

\begin{center}
\begin{tikzpicture}[scale=0.34]
\begin{scope}[shift={(0,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw[red] (0,2) circle [radius=0.2];
\draw (-1,1) circle [radius=0.2];
\draw[fill] (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[fill] (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\node at (3.5,-3) {$\stackrel{t_f}{\longmapsto}$};
\begin{scope}[shift={(7,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw (-1,1) circle [radius=0.2];
\draw[red,fill] (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[fill] (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\node at (10.5,-3) {$\stackrel{t_e}{\longmapsto}$};
\begin{scope}[shift={(14,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[red] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[fill] (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\node at (17.5,-3) {$\stackrel{t_d}{\longmapsto}$};
\begin{scope}[shift={(21,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw[red,fill] (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\node at (24.5,-3) {$\stackrel{t_c}{\longmapsto}$};
\begin{scope}[shift={(28,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[red,fill] (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\node at (31.5,-3) {$\stackrel{t_b}{\longmapsto}$};
\begin{scope}[shift={(35,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[red,fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {3};
\end{scope}
\node at (38.5,-3) {$\stackrel{t_a}{\longmapsto}$};
\begin{scope}[shift={(42,-4)}]
\draw[thick] (-0.1, 1.9) -- (-0.9, 1.1);
\draw[thick] (0.1, 1.9) -- (0.9, 1.1);
\draw[thick] (-1.1, 0.9) -- (-1.9, 0.1);
\draw[thick] (-0.9, 0.9) -- (-0.1, 0.1);
\draw[thick] (0.9, 0.9) -- (0.1, 0.1);
\draw[thick] (1.1, 0.9) -- (1.9, 0.1);
\draw (0,2) circle [radius=0.2];
\draw[fill] (-1,1) circle [radius=0.2];
\draw (1,1) circle [radius=0.2];
\draw[fill] (-2,0) circle [radius=0.2];
\draw[fill] (0,0) circle [radius=0.2];
\draw (2,0) circle [radius=0.2];
\node[above] at (0,2) {$f$};
\node[left] at (-1,1) {$d$};
\node[right] at (1,1) {$e$};
\node[below] at (-2,0) {$a$};
\node[below] at (0,0) {$b$};
\node[below] at (2,0) {$c$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{example}
\fi

\subsection{The antichain toggle group}
Toggling makes sense in a broader context, as formalized by Striker~\cite{strikergentog}. We can define \emph{antichain toggles} on $\cala(P)$, by replacing $\calj(P)$ with $\cala(P)$ in the definition.  Removing any element from an antichain always yields an antichain, giving a simpler second case.  
\begin{defn}[\cite{strikergentog}]\label{def:comb-tau}
Let $e\in P$.  Then the \textbf{antichain toggle} corresponding to $e$ is the map $\tau_e: \cala(P)\ra \cala(P)$
defined by
$$\tau_e(A)=\left\{\begin{array}{ll}
A\cup\{e\} &\text{if $e\not\in A$ and $A\cup\{e\}\in\cala(P)$,}\\
A\sm\{e\} &\text{if $e\in A$,}\\
A &\text{otherwise.}
\end{array}\right.$$
Let $\tog_\cala(P)$ denote the \textbf{toggle group} of $\cala(P)$ generated by the toggles $\{\tau_e : e\in P\}$.
\end{defn}
%Striker proved that for a finite connected poset $P$, the group $\tog_\cala(P)$ is either the symmetric or alternating group on $\cala(P)$~\cite[Cor.~3.13]{strikergentog}.
%\iffalse
%The analogue of Proposition~\ref{prop:J-toggle-inv-commute} shows that antichain toggles
%commute less easily.
\begin{prop}[{\cite{strikergentog}}]\label{prop:tau-commute}
Each antichain toggle $\tau_x$ is an involution.
Two toggles $\tau_x,\tau_y$ commute if and only if $x=y$ or $x$ and $y$ are incomparable.
\end{prop}
%\fi
The first author constructed an explicit isomorphism between $\tog_\calj(P)$ and \linebreak
$\tog_\cala(P)$~\cite{antichain-toggling}. 

\begin{defn}
For $e\in P$, we define the following.
\begin{itemize}
\item Set $t_e^*\in\tog_\cala(P)$ by $t_e^* := \tau_{e_1} \tau_{e_2}\cdots \tau_{e_k} \tau_e \tau_{e_1} \tau_{e_2}\cdots \tau_{e_k}$ where $e_1,\dots,e_k$ are the elements of $P$ covered by $e$.  (If $e$ is a minimal element of $P$, then $k=0$ and $t_e^*=\tau_e$.)
\item Set $\eta_e\in\tog_\calj(P)$ by $\eta_e:=t_{x_1}t_{x_2}\cdots t_{x_k}$
where $(x_1,x_2,\dots,x_k)$ is a linear extension of the subposet $\{x\in P : x<e\}$ of
$P$.\footnote{Any two linear extensions of a poset differ by a sequence of transpositions
between adjacent incomparable elements~\cite{etienne-84}.  So $\eta_e$ is well-defined.} 
\item Let $\tau_e^* := \eta_e t_e \eta_e^{-1}\in\tog_\calj(P)$.
\end{itemize}
\end{defn}

The following theorem gives an explicit isomorphism from $\tog_\cala(P)$ to $\tog_\calj(P)$ via $\tau_e\mapsto \tau_e^*$, with inverse given by $t_e\mapsto t_e^*$.

\begin{thm}[{\cite[Theorems 2.15 and 2.19]{antichain-toggling}}]\label{thm:atar}
The following diagrams commute.
\begin{center}
\begin{tikzpicture}[yscale=2/3]
\node at (0,1.8) {$\calj(P)$};
\node at (0,0) {$\cala(P)$};
\node at (3.25,1.8) {$\calj(P)$};
\node at (3.25,0) {$\cala(P)$};
\draw[semithick, ->] (0,1.3) -- (0,0.5);
\node[left] at (0,0.9) {$\up$};
\draw[semithick, ->] (0.7,0) -- (2.5,0);
\node[below] at (1.5,0) {$t_e^*$};
\draw[semithick, ->] (0.7,1.8) -- (2.5,1.8);
\node[above] at (1.5,1.8) {$t_e$};
\draw[semithick, ->] (3.25,1.3) -- (3.25,0.5);
\node[right] at (3.25,0.9) {$\up$};
\begin{scope}[shift={(7,0)}]
\node at (0,1.8) {$\calj(P)$};
\node at (0,0) {$\cala(P)$};
\node at (3.25,1.8) {$\calj(P)$};
\node at (3.25,0) {$\cala(P)$};
\draw[semithick, ->] (0,1.3) -- (0,0.5);
\node[left] at (0,0.9) {$\up$};
\draw[semithick, ->] (0.7,0) -- (2.5,0);
\node[below] at (1.5,0) {$\tau_e$};
\draw[semithick, ->] (0.7,1.8) -- (2.5,1.8);
\node[above] at (1.5,1.8) {$\tau_e^*$};
\draw[semithick, ->] (3.25,1.3) -- (3.25,0.5);
\node[right] at (3.25,0.9) {$\up$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{thm}

Note that in \cref{thm:atar}, the left commutative diagram together with the basic properties of toggles
(\cref{prop:J-toggle-inv-commute,prop:tau-commute}) are enough to prove the right commutative diagram
using group theory alone.  A consequence of this isomorphism is that antichain rowmotion is also a product of antichain toggles in an order specified by a linear extension, but in the \emph{opposite} order from order ideal
rowmotion.

\begin{prop}[{\cite[Prop.~2.24]{antichain-toggling}}]\label{prop:row-toggles-anti}
For any linear extension $(x_1,x_2,\dots,x_n)$ of $P$,  antichain rowmotion is given by
$\rowA = \tau_{x_n} \cdots \tau_{x_2} \tau_{x_1}$.
\end{prop}

\subsection{Birational dynamics}

The toggling perspective allows us to extend these maps from the combinatorial realm
(on finite sets) to the piecewise-linear realm (polytopes whose vertices correspond to these
sets), and then lift to the birational realm by detropicalizing the operations~\cite{einpropp}.
%A different direction to generalize Cameron and Fon-Der-Flaass's toggling notion is by
%extending from the combinatorial realm (uses finite sets) to the piecewise-linear realm
%(polytopes whose vertices describe the given set) and furthermore the birational realm by
%detropicalization of operations~\cite{einpropp}. 
Piecewise-linear dynamics begin with two polytopes introduced by Stanley~\cite{Sta86}, the \textit{order polytope} and the \textit{chain polytope} of $P$.
The vertices of these polytopes are the sets $\calf(P)$ of order filters and $\cala(P)$ of antichains (associating a subset of $P$ with its indicator function
labeling).  Einstein and Propp defined piecewise-linear toggle operations on the order
polytope that match the order ideal toggle $t_e$ when restricted to the vertices (though here
we use order \emph{filters} instead of order ideals)~\cite[\S3,4]{einpropp}.

\iffalse
\begin{definition}[\cite{Sta86}]
The \textbf{order polytope} of $P$ is the set $OP(P)$ of labelings $f:P\ra [0,1]$ that are order-preserving (i.e., if $a\leq b$ in $P$, then $f(a)\leq f(b)$).
The \textbf{chain polytope} of $P$ is the set $\calc(P)$ of labelings $f:P\ra [0,1]$ such that the sum of the labels across every chain is at most 1.
\end{definition}

By associating a subset of $P$ with its indicator functions, the sets $\calf(P)$ and $\cala(P)$ describe the vertices of $OP(P)$ and $\calc(P)$ respectively.  Similarly, we can define an \textbf{order-reversing polytope} $OR(P)$ to consist of all labelings $f:P\ra [0,1]$ that are order-reversing (i.e., if $a\leq b$ in $P$, then $f(a)\geq f(b)$).  The vertices of $OR(P)$ are the order ideals of $P$.
To define toggles, there is no important difference in defining them over $OR(P)$ as opposed to $OP(P)$.  The piecewise-linear order toggles are typically defined on $OP(P)$.

\begin{defn}[\cite{einpropp}]
Let $e\in P$.
For $f\in OP(P)$, we extend $f$ to a labeling on the poset $\hat{P}$ which
consists of adjoining a minimal element $\hat{0}$ and a maximal element $\hat{1}$
to $P$ and defining $f(\hat{0})=0$ and $f(\hat{1})=1$.
The \textbf{piecewise-linear order toggle} $T_e:OP(P)\ra OP(P)$ is
$$(T_e(f))(x) =
\left\{\begin{array}{ll}
f(x) &\text{if }x\not=e\\
\max\limits_{y\lessdot e} f(y) + 
\min\limits_{y\gtrdot e} f(y) - f(e)
&\text{if }x=e\end{array}\right..
$$
We use the notation $x\lessdot y$ to mean ``$y$ covers $x$'' and $x\gtrdot y$ to mean ``$x$ covers $y$''.  In the above, we use cover relations in $\hat{P}$ so that every element of $P$ covers some element of $\hat{P}$ and is covered by an element of $\hat{P}$.
\end{defn}
\fi

This paper focuses on the generalization to the birational realm.
Let $\kk$ be a field.  To define birational toggles, we ``detropicalize'' the operations from the piecewise-linear toggles, by replacing addition with multiplication and 
the $\max$ operation with addition.  
The additive identity 0 lifts to 1, and 1 lifts to a generic fixed constant $C\in\kk$.
% Therefore, the additive identity 0 is replaced with
% the multiplicative identity 1.
% However, 1 is no special element under the operations of $\max$ and addition, so it is replaced with a generic fixed constant $C\in\kk$.
This gives the following definition of the birational order toggle.  (The definition
in~\cite{einpropp} is slightly more general than we need here; setting $\alpha=1$ and
$\omega=C$ in their version gives ours.)

\begin{defn}[{\cite[Definition~5.1]{einpropp}}]
Let $\kk^P$ be the set of $\kk$-labelings of the elements of $P$.  We extend $P$ to the poset $\hat{P}$ by adding a minimal element $\hat{0}$ and maximal element $\hat{1}$.
For $e\in P$, the \textbf{birational order toggle} at $e$ is the
birational map
$T_e: \kk^P \dra \kk^P$ given by
$$(T_e(f))(x) =
\left\{\begin{array}{ll}
f(x) &\text{if }x\not=e\\
\frac{\sum\limits_{y\in\hat{P},y\lessdot x} f(y)}
{f(e) \sum\limits_{y\in\hat{P},y\gtrdot x} \frac{1}{f(y)}}
&\text{if }x=e\end{array}\right.
$$ where we set $f(\hat{0})=1$ and $f(\hat{1})=C$.
The notation $y\lessdot x$ means ``$x$ covers $y$'.
%and $y\gtrdot x$
%means ``$y$ covers $x$''.
\end{defn}
The birational order toggles $\{T_e: e\in P\}$
generate a group, which we will call $\btog_O(P)$.
These are involutions whose commutation is like with order ideal toggles.
%It is not hard to show the same basic properties hold for the birational order toggles as for the order ideal toggles.

\begin{prop}\label{prop:BTogO-toggle-inv-commute}
Each toggle $T_x$ is an involution (i.e., $T_x^2$ is the identity).  Two toggles $T_x,T_y$ commute if and only if neither $x$ nor $y$ covers the other.
\end{prop}

\begin{defn}[{\cite[Definition~5.2]{einpropp}}]
Let $(x_1,x_2,\dots,x_n)$ be any linear extension of $P$.
The birational analogue of order ideal rowmotion, which we will call
\textbf{birational order rowmotion} (or \textbf{BOR-motion}),
is $\BOR= T_{x_1} T_{x_2} \cdots T_{x_n}$.  (Compare with
\cref{prop:row-toggles}.)
\end{defn}

Any property of periodicity or homomesy satisfied by birational toggling and rowmotion
also holds in the piecewise-linear realm (by tropicalization) and furthermore in the combinatorial realm (by restriction).
What is more surprising, however, is that for certain families of posets
(e.g. type A and B root posets and products of two chains),
many properties (particularly periodicity) that hold in the combinatorial realm extend to the birational realm; see~\cite{einpropp, grinberg-roby,musiker-roby}.
Furthermore, studying birational rowmotion can shed light on some patterns
in the combinatorial realm, such as why the posets where rowmotion has nice properties are
usually graded posets; see~\cite[\S3--6]{grinberg-roby}.

\subsection{Birational transfer maps}
Stanley defined a bijection between the order and chain polytopes of $P$ called
the \emph{transfer map}.  By detropicalizing the operations, we get the birational analogue of the
up-transfer map.  Einstein and Propp prove~\cite[\S6]{einpropp} that $\BOR=\Theta\circ\up^{-1}\circ\down$ where the birational transfer maps are defined below.
The order of composition differs from order ideal toggling since birational rowmotion $\rowJ$ is
defined through the order filter perspective. 
%
\begin{defn}[{\cite[\S6]{einpropp}}]\label{def:bir-trans}
For a labeling $f\in \kk^P$ and element $x\in P$,
\begin{itemize}
\item $(\Theta f)(x)=\frac{C}{f(x)}$,
\item $(\down f)(x)=\frac{f(x)}{\sum\limits_{y\lessdot x} f(y)}$ (with $f(\hat{0})=1$),
\item $(\up^{-1} f)(x) = \sum\left\{ f(y_1)f(y_2)\cdots f(y_k):x=y_1
\lessdot y_2\lessdot \cdots \lessdot y_k\lessdot \hat{1}\right\}$.
\end{itemize}
\end{defn}

The maps $\Theta$, $\up$, and $\down$ are called \textbf{complementation}, \textbf{up-transfer},
and \textbf{down-transfer}.  They generalize those above 
\cref{def:OIrow}.  We use the same symbols in both the combinatorial and birational
realms, allowing context to clarify which is meant. 

\section{Birational antichain toggling and rowmotion}

\subsection{Birational antichain rowmotion}
Now we will combine the different generalizations of toggling and study a birational analogue of antichain toggling and rowmotion.
%%Again, we let $\kk$ be a field throughout the remainder of this abstract.

\begin{defn}
\textbf{Birational antichain rowmotion} (or \textbf{BAR-motion}) is the
birational map $\BAR= \down\circ\Theta\circ\up^{-1}$ where $\down,\Theta,\up^{-1}$ are defined in
\cref{def:bir-trans}.
\end{defn}

\begin{ex}\label{ex:2x3-3step}
\cref{fig:2x3} shows the poset $P=[2]\times[3]$ and a generic labeling $g\in\kk^P$.  
\cref{fig:2x3-BAR} show one iteration of BAR-motion on this labeling.

\begin{figure}
\begin{center}
\begin{tikzpicture}[xscale=4/3,yscale=10/13]
%\begin{small}
\node at (0,0) {$(1,1)$};
\node at (-1,1) {$(2,1)$};
\node at (1,1) {$(1,2)$};
\node at (0,2) {$(2,2)$};
\node at (2,2) {$(1,3)$};
\node at (1,3) {$(2,3)$};
\draw[thick] (-0.2,0.3) -- (-0.8,0.7);
\draw[thick] (0.2,0.3) -- (0.8,0.7);
\draw[thick] (1.2,1.3) -- (1.8,1.7);
\draw[thick] (0.2,2.3) -- (0.8,2.7);
\draw[thick] (-0.2,1.7) -- (-0.8,1.3);
\draw[thick] (0.2,1.7) -- (0.8,1.3);
\draw[thick] (1.2,2.7) -- (1.8,2.3);
\begin{scope}[shift={(6,0)}]
\node at (0,0) {$u$};
\node at (-1,1) {$v$};
\node at (1,1) {$w$};
\node at (0,2) {$x$};
\node at (2,2) {$y$};
\node at (1,3) {$z$};
\draw[thick] (-0.2,0.2) -- (-0.8,0.8);
\draw[thick] (0.2,0.2) -- (0.8,0.8);
\draw[thick] (1.2,1.2) -- (1.8,1.8);
\draw[thick] (0.2,2.2) -- (0.8,2.8);
\draw[thick] (-0.2,1.8) -- (-0.8,1.2);
\draw[thick] (0.2,1.8) -- (0.8,1.2);
\draw[thick] (1.2,2.8) -- (1.8,2.2);
\end{scope}
%\end{small}
\end{tikzpicture}
\end{center}
\caption{Left: The poset $[2]\times[3]$.
Right: A generic labeling on $[2]\times[3]$, used in
examples throughout this abstract.}
\label{fig:2x3}
\end{figure}

\begin{figure}
\begin{center}
\begin{tikzpicture}[xscale=1.87,yscale=1.25]
%\begin{small}
\begin{scope}[yscale=5/7]
\node at (0,0) {$u$};
\node at (-1,1) {$v$};
\node at (1,1) {$w$};
\node at (0,2) {$x$};
\node at (2,2) {$y$};
\node at (1,3) {$z$};
\draw[thick] (-0.2,0.2) -- (-0.8,0.8);
\draw[thick] (0.2,0.2) -- (0.8,0.8);
\draw[thick] (1.2,1.2) -- (1.8,1.8);
\draw[thick] (0.2,2.2) -- (0.8,2.8);
\draw[thick] (-0.2,1.8) -- (-0.8,1.2);
\draw[thick] (0.2,1.8) -- (0.8,1.2);
\draw[thick] (1.2,2.8) -- (1.8,2.2);
\end{scope}
\draw[ultra thick, ->] (0,-0.5) -- (0,-1.1);
\node[left] at (0,-0.8) {$\BAR$};
\draw[ultra thick, ->] (2.5,0.5) -- (3.5,0.5);
\node[above] at (3,0.5) {$\up^{-1}$};
\node[above] at (3,-2.7) {$\down$};
\draw[ultra thick, ->] (3.5,-2.7) -- (2.5,-2.7);
\draw[ultra thick, ->] (5,-0.5) -- (5,-1.1);
\node[left] at (5,-0.8) {$\Theta$};
\begin{scope}[yscale=3/7,shift={(5,0)},xscale=5/6]
\node at (0,0) {$u(vx+wx+wy)z$};
\node at (-1,1) {$vxz$};
\node at (1,1) {$w(x+y)z$};
\node at (0,2) {$xz$};
\node at (2,2) {$yz$};
\node at (1,3) {$z$};
\draw[thick] (-0.3,0.3) -- (-0.7,0.7);
\draw[thick] (0.3,0.3) -- (0.7,0.7);
\draw[thick] (1.3,1.3) -- (1.7,1.7);
\draw[thick] (0.3,2.3) -- (0.7,2.7);
\draw[thick] (-0.3,1.7) -- (-0.7,1.3);
\draw[thick] (0.3,1.7) -- (0.7,1.3);
\draw[thick] (1.3,2.7) -- (1.7,2.3);
\end{scope}
\begin{scope}[shift={(0,-4.1)}]
\node at (0,0) {$\frac{C}{u(vx+wx+wy)z}$};
\node at (-1,1) {$\frac{u(vx+wx+wy)}{vx}$};
\node at (1,1) {$\frac{u(vx+wx+wy)}{w(x+y)}$};
\node at (0,2) {$\frac{vw(x+y)}{vx+wx+wy}$};
\node at (2,2) {$\frac{w(x+y)}{y}$};
\node at (1,3) {$\frac{xy}{x+y}$};
\draw[thick] (-0.3,0.3) -- (-0.7,0.7);
\draw[thick] (0.3,0.3) -- (0.7,0.7);
\draw[thick] (1.3,1.3) -- (1.7,1.7);
\draw[thick] (0.3,2.3) -- (0.7,2.7);
\draw[thick] (-0.3,1.7) -- (-0.7,1.3);
\draw[thick] (0.3,1.7) -- (0.7,1.3);
\draw[thick] (1.3,2.7) -- (1.7,2.3);
\end{scope}
\begin{scope}[shift={(5,-4.1)},xscale=5/6]
\node at (0,0) {$\frac{C}{u(vx+wx+wy)z}$};
\node at (-1,1) {$\frac{C}{vxz}$};
\node at (1,1) {$\frac{C}{w(x+y)z}$};
\node at (0,2) {$\frac{C}{xz}$};
\node at (2,2) {$\frac{C}{yz}$};
\node at (1,3) {$\frac{C}{z}$};
\draw[thick] (-0.3,0.3) -- (-0.7,0.7);
\draw[thick] (0.3,0.3) -- (0.7,0.7);
\draw[thick] (1.3,1.3) -- (1.7,1.7);
\draw[thick] (0.3,2.3) -- (0.7,2.7);
\draw[thick] (-0.3,1.7) -- (-0.7,1.3);
\draw[thick] (0.3,1.7) -- (0.7,1.3);
\draw[thick] (1.3,2.7) -- (1.7,2.3);
\end{scope}
%\end{small}
\end{tikzpicture}
\end{center}
\caption{One iteration of $\BAR$-motion on $[2]\times[3]$.}
\label{fig:2x3-BAR}
\end{figure}
\end{ex}

\subsection{Birational antichain toggle group}
As in the combinatorial realm, we can obtain an equivalent expression of BAR-motion through
toggle operations, which we now introduce.

\begin{defn}
Let $e\in P$ and $g\in \kk^P$.  Then we define
$\mc_e(P)$ as the set of all maximal chains of $P$ through $e$ and $\Upsilon_e g=
\sum\limits_{(y_1,\dots,y_k)\in \mc_e(P)} g(y_1) \cdots g(y_k)$.
\end{defn}

\begin{defn}
Let $e\in P$.  The \textbf{birational antichain toggle} is the rational map $\tau_e:\kk^P \dra \kk^P$ defined as follows:

$$\big(\tau_e(g)\big)(x) = \left\{\begin{array}{ll}
\dfrac{C}{\sum\limits_{(y_1,\dots,y_k)\in \mc_e(P)} g(y_1) \cdots g(y_k)} &\text{if $x=e$}\\
g(x) &\text{if $x\not=e$}
\end{array}\right.
=
\left\{\begin{array}{ll}
\dfrac{C}{\Upsilon_e g} &\text{if $x=e$}\\\\
g(x) &\text{if $x\not=e$.}
\end{array}
\right.$$

Let $\btog_A(P)$ denote the group generated by the toggles $\{\tau_e: e\in P\}$.  We call $\btog_A(P)$ the \textbf{birational antichain toggle group}.
\end{defn}

The definition of the birational antichain toggle comes directly from detropicalizing
the operations in the toggles on the chain polytope (as defined
in~\cite[\S3]{antichain-toggling}), which correspond to the antichain toggles when
restricting to the vertices of the chain polytope. 
Note that we use the same notation for (combinatorial) antichain toggles $\tau_e$ and birational antichain toggles.  In the remainder of this abstract, we are working in the birational realm
(except briefly in \cref{sec:ST2b} when we discuss the combinatorial results we are lifting to the birational realm).
\iffalse
As the definition of $\tau_e$ only refers to labels of elements for chains through $e$, changing the label at an element incomparable with $e$ does not change the label of $(\tau_e f)(e)$.
\fi
It is not hard to verify the following basic properties of antichain toggles hold
for the birational liftings too.

\begin{prop}\label{prop:bir-tau-commute}
Each birational antichain toggle $\tau_x$ is an involution.
Two toggles $\tau_x,\tau_y$ commute if and only if $x=y$ or $x$ and $y$ are incomparable.
\end{prop}

We construct an isomorphism between the birational toggle groups
$\btog_O(P)$ and $\btog_A(P)$ analogous to one for the combinatorial toggle groups
$\tog_\calj(P)$ and $\tog_\cala(P)$. 

\begin{defn}
For $e\in P$, we define the following.
\begin{itemize}
\item Set $T_e^*\in\btog_A(P)$ by $T_e^* := \tau_{e_1} \tau_{e_2}\cdots \tau_{e_k} \tau_e \tau_{e_1} \tau_{e_2}\cdots \tau_{e_k}$ where $e_1,\dots,e_k$ are the elements of $P$ covered by $e$.  (If $e$ is a minimal element of $P$, then $k=0$ and $t_e^*=\tau_e$.)
\item Set $\eta_e\in\btog_O(P)$ by $\eta_e:=T_{x_1}T_{x_2}\cdots T_{x_k}$
where $(x_1,x_2,\dots,x_k)$ is a linear extension of the subposet $\{x\in P : x<e\}$ of $P$.
\item Let $\tau_e^* := \eta_e T_e \eta_e^{-1}\in\btog_O(P)$.
\end{itemize}
\end{defn}

\begin{thm}\label{thm:T-star}
Let $e\in P$.  Then the following diagrams commute on the domains in which the maps are defined.
So there is an isomorphism from $\btog_A(P)$ to $\btog_O(P)$ given by $\tau_e \mapsto \tau_e^*$
with inverse given by $T_e\mapsto T_e^*$.
\begin{center}
\begin{tikzpicture}[yscale=2/3]
\node at (0,1.8) {$\kk^P$};
\node at (0,0) {$\kk^P$};
\node at (0,-1.8) {$\kk^P$};
\node at (3.25,1.8) {$\kk^P$};
\node at (3.25,0) {$\kk^P$};
\node at (3.25,-1.8) {$\kk^P$};
\draw[semithick, dashed, ->] (0,1.3) -- (0,0.5);
\node[left] at (0,0.9) {$\up^{-1}$};
\draw[semithick, dashed, ->] (0.7,-1.8) -- (2.5,-1.8);
\node[below] at (1.5,-1.8) {$T_e$};
\draw[semithick, dashed, ->] (0.7,1.8) -- (2.5,1.8);
\node[above] at (1.5,1.8) {$T_e^*$};
\draw[semithick, dashed, ->] (3.25,1.3) -- (3.25,0.5);
\node[right] at (3.25,0.9) {$\up^{-1}$};
\draw[semithick, dashed, ->] (0,-0.5) -- (0,-1.3);
\node[left] at (0,-0.9) {$\Theta$};
\draw[semithick, dashed, ->] (3.25,-0.5) -- (3.25,-1.3);
\node[right] at (3.25,-0.9) {$\Theta$};
\begin{scope}[shift={(6,0)}]
\node at (0,1.8) {$\kk^P$};
\node at (0,0) {$\kk^P$};
\node at (0,-1.8) {$\kk^P$};
\node at (3.25,1.8) {$\kk^P$};
\node at (3.25,0) {$\kk^P$};
\node at (3.25,-1.8) {$\kk^P$};
\draw[semithick, dashed, ->] (0,1.3) -- (0,0.5);
\node[left] at (0,0.9) {$\up^{-1}$};
\draw[semithick, dashed, ->] (0.7,-1.8) -- (2.5,-1.8);
\node[below] at (1.5,-1.8) {$\tau_e^*$};
\draw[semithick, dashed, ->] (0.7,1.8) -- (2.5,1.8);
\node[above] at (1.5,1.8) {$\tau_e$};
\draw[semithick, dashed, ->] (3.25,1.3) -- (3.25,0.5);
\node[right] at (3.25,0.9) {$\up^{-1}$};
\draw[semithick, dashed, ->] (0,-0.5) -- (0,-1.3);
\node[left] at (0,-0.9) {$\Theta$};
\draw[semithick, dashed, ->] (3.25,-0.5) -- (3.25,-1.3);
\node[right] at (3.25,-0.9) {$\Theta$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{thm}

The proof of \cref{thm:T-star} is very similar to the piecewise-linear realm proof
in~\cite[Thm.~3.19]{antichain-toggling} but with detropicalized operations.
%The need for the complementation operation $\Theta$ in the diagrams stems from the technicality that birational order toggles are viewed from the order filter perspective, not order ideal perspective.

Using the isomorphism between the two toggle groups, we can prove that for a linear extension $(x_1,x_2,\dots,x_n)$ of $P$,
$\BOR= T_{x_1} T_{x_2} \cdots T_{x_n} = \tau_{x_n}^* \cdots \tau_{x_2}^* \tau_{x_1}^*$ using
a purely group-theoretic proof analogous to the one found in~\cite[Thm.~3.21]{antichain-toggling}.
This gives the following birational generalization of \cref{prop:row-toggles-anti}.

\begin{prop}\label{prop:BAR-tog}
Let $(x_1,x_2,\dots,x_n)$ be any linear extension of a finite poset $P$.  Then $\BAR=\tau_{x_n}\cdots \tau_{x_2} \tau_{x_1}$.
\end{prop}

\begin{ex}\label{ex:2x3}
Reconsider the poset $P=[2]\times[3]$ and labeling of \cref{fig:2x3}.  We perform
$\BAR$ along the linear extension $\Big((1,1),(2,1),(1,2),(2,2),(1,3),(2,3)\Big)$.  There
are three maximal chains through the bottom element $(1,1)$ of $P$: 

\begin{itemize}
\item $(1,1)\lessdot(2,1)\lessdot(2,2)\lessdot(2,3)$ with product of labels $uvxz$
\item $(1,1)\lessdot(1,2)\lessdot(2,2)\lessdot(2,3)$ with product of labels $uwxz$
\item $(1,1)\lessdot(1,2)\lessdot(1,3)\lessdot(2,3)$ with product of labels $uwyz$
\end{itemize}

For $\Upsilon_{(1,1)}g$, we add up the products of the labels on these three maximal chains, and get $\Upsilon_{(1,1)}g=uvxz+uwxz+uwyz=u(vx+wx+wy)z$.
Then to apply the toggle $\tau_{(1,1)}$,
we change the label of $(1,1)$ from $u$ to
$\frac{C}{u(vx+wx+wy)z}$.  This is shown in \cref{fig:2x3-toggle}.
Now we apply $\tau_{(2,1)}$ to $\tau_{(1,1)}g$.  There is only one maximal chain through $(2,1)$.  The product of labels on that maximal chain is $\Upsilon_{(2,1)}(\tau_{(1,1)}g)=\frac{C}{u(vx+wx+wy)z}vxz$.  Thus we change the label of $(2,1)$
from $v$ to $\frac{C}{\frac{C}{u(vx+wx+wy)z}vxz}=\frac{u(vx+wx+wy)}{vx}$ as in \cref{fig:2x3-toggle}.

\begin{figure}
\begin{tikzpicture}[xscale=4/3,yscale=7/9]
%\begin{small}
\node at (0,0) {$u$};
\node at (-1,1) {$v$};
\node at (1,1) {$w$};
\node at (0,2) {$x$};
\node at (2,2) {$y$};
\node at (1,3) {$z$};
\draw[thick] (-0.2,0.2) -- (-0.8,0.8);
\draw[thick] (0.2,0.2) -- (0.8,0.8);
\draw[thick] (1.2,1.2) -- (1.8,1.8);
\draw[thick] (0.2,2.2) -- (0.8,2.8);
\draw[thick] (-0.2,1.8) -- (-0.8,1.2);
\draw[thick] (0.2,1.8) -- (0.8,1.2);
\draw[thick] (1.2,2.8) -- (1.8,2.2);
\node at (3,2.6) {$\stackrel{\tau_{(1,1)}}{\longmapsto}$};
\begin{scope}[shift={(4,0)}]
\node at (0,0) {$\frac{C}{u(vx+wx+wy)z}$};
\node at (-1,1) {$v$};
\node at (1,1) {$w$};
\node at (0,2) {$x$};
\node at (2,2) {$y$};
\node at (1,3) {$z$};
\draw[thick] (-0.2,0.2) -- (-0.8,0.8);
\draw[thick] (0.2,0.2) -- (0.8,0.8);
\draw[thick] (1.2,1.2) -- (1.8,1.8);
\draw[thick] (0.2,2.2) -- (0.8,2.8);
\draw[thick] (-0.2,1.8) -- (-0.8,1.2);
\draw[thick] (0.2,1.8) -- (0.8,1.2);
\draw[thick] (1.2,2.8) -- (1.8,2.2);
\end{scope}
\node at (7,2.6) {$\stackrel{\tau_{(2,1)}}{\longmapsto}$};
\begin{scope}[shift={(9,0)}]
\node at (0,0) {$\frac{C}{u(vx+wx+wy)z}$};
\node at (-1,1) {$\frac{u(vx+wx+wy)}{vx}$};
\node at (1,1) {$w$};
\node at (0,2) {$x$};
\node at (2,2) {$y$};
\node at (1,3) {$z$};
\draw[thick] (-0.2,0.2) -- (-0.8,0.8);
\draw[thick] (0.2,0.2) -- (0.8,0.8);
\draw[thick] (1.2,1.2) -- (1.8,1.8);
\draw[thick] (0.2,2.2) -- (0.8,2.8);
\draw[thick] (-0.2,1.8) -- (-0.7,1.3);
\draw[thick] (0.2,1.8) -- (0.8,1.2);
\draw[thick] (1.2,2.8) -- (1.8,2.2);
\end{scope}
%\end{small}
\end{tikzpicture}
\caption{The effect of applying the toggles $\tau_{(1,1)}$ and then $\tau_{(2,1)}$, as in \cref{ex:2x3}.}
\label{fig:2x3-toggle}
\end{figure}

To complete one iteration of $\BAR$ we must then apply the toggles $\tau_{(1,2)}$,
$\tau_{(2,2)}$, $\tau_{(1,3)}$, and $\tau_{(2,3)}$ in that order, giving the same result 
as in \cref{fig:2x3-BAR}. 
\end{ex}

Iterating $\BAR$ five times to the generic labeling on $[2]\times[3]$ returns the
original labeling again.  We can explain this on a general
product of two chains poset $[a]\times[b]$ with element set
$\{(i,j)\in\zz^2:1\leq i\leq a, 1\leq j\leq b\}$
with $(i_1,j_1)\leq (i_2,j_2)$ if and only if $i_1\leq i_2$ and $j_1\leq j_2$.
For the poset $[a]\times[b]$, Grinberg and the second author
proved that BOR-motion has order $a+b$~\cite{grinberg-roby}.
From the down-transfer, we can conclude the periodicity of
$\BAR$ from that of $\BOR$ because $\BOR=\down^{-1} \circ \BAR \circ \down$.

\begin{thm}\label{thm:a+b}
On $P=[a]\times[b]$, the order of BAR-motion is $a+b$.
\end{thm}

\section{Homomesy of fiber products for BAR-motion on $[2]\times [b]$}\label{sec:ST2b}

Let $\cals$ be a collection of combinatorial objects, and $f:\cals \rightarrow \kk$ a
``statistic'' on $\cals$.  We call $f$ \emph{homomesic} with respect to an invertible action
$w:\cals \rightarrow \cals$ if the (arithmetic) average of $f$ over every orbit is the
same~\cite{propproby}.  In the birational setting, to avoid dealing with taking $n$th roots,
this manifests itself as certain products across an orbit equaling a fixed constant,
independent of the initial labels.  See~\cite{propproby} for a more careful treatment of
combinatorial homomesy and~\cite{einpropp,musiker-roby} for birational homomesy. 

Propp and the second author proved a homomesy result for antichain rowmotion on
$[a]\times[b]$ in terms of \textbf{fibers}~\cite{propproby}, which we lift to the birational
realm. 

\begin{defn}[{\cite[\S3.3.2]{propproby}}]
Fix $a,b\in \zz^+$.  For $1\leq k\leq a$, the subset
$\{(k,\ell):1\leq \ell \leq b\}$ of $[a]\times[b]$ is called the \textbf{\textit{k}th positive fiber}.
For $1\leq \ell\leq b$, the subset
$\{(k,\ell):1\leq k \leq a\}$ of $[a]\times[b]$ is called the \textbf{\textit{$\ell$}th negative fiber}.
\end{defn}

On $P=[a]\times[b]$, Stanley and Thomas defined an $(a+b)$-tuple $w(A)$ corresponding to an antichain $A\in\cala(P)$ described as
$$w_i=\left\{\begin{array}{ll}
1 &\text{if $1\leq i \leq a$ and $A$ has an element in the $a$th positive fiber},\\
1 &\text{if $a+1\leq i \leq a+b$ and $A$ has NO element in the $(i-a)$th negative fiber},\\
-1 &\text{otherwise}.\\
\end{array}\right.$$
They proved that $A\mapsto w(A)$ is a bijection between antichains of $P$ and the possible tuples that can occur.  Furthermore, applying antichain rowmotion to $A$ cyclically
shifts the \textbf{Stanley--Thomas word} $w(A)$.
This proves rowmotion (in the combinatorial realm) has order $a+b$ and also that the statistics $p_i:\cala(P)\ra\zz$ and $n_i:\cala(P)\ra\zz$ where $p_i(A)$ (resp.~$n_i(A)$) is 1 if $A$ has an element
in the $i$th positive fiber (resp.~negative fiber) and 0 otherwise are homomesic with average
$b(a+b)$ for $p_i$  and $a(a+b)$ for $n_i$ on any orbit.
As cardinality of an antichain can be expressed as
$p_1+p_2+\cdots+p_a$, we see that cardinality on $\cala(P)$ is homomesic with average $ab/(a+b)$~\cite[\S3.3.2]{propproby}.

Now we define a birational analogue of the Stanley--Thomas word.  One key difference, however,
is that a $\kk$-labeling of $P$ is no longer uniquely determined by its Stanley--Thomas word.
Therefore, it can no longer be used to prove periodicity (which we already proved in \cref{thm:a+b}).
However, we can use it to prove an analogue of fiber homomesy in any situation where $\BAR$
cyclically rotates the birational Stanley--Thomas word (which so far we have only
proven for the case $a=2$).

\begin{defn}
Let $a,b\in\zz^+$, $P=[a]\times[b]$, and $g\in\kk^P$.  The \textbf{Stanley--Thomas word} $\st_g$ is the $(a+b)$-tuple
given by
$$\st_g(i)=\left\{\begin{array}{ll}
g(i,1)g(i,2)\cdots g(i,b) &\text{if }1\leq i\leq a,\\
C/\big(g(1,i-a)g(2,i-a)\cdots g(a,i-a)\big) &\text{if }a+1\leq i \leq a+b.\\
\end{array}\right.$$
\end{defn}

\begin{example}
Let $g$ be the generic labeling of $[2]\times[3]$ displayed in the right side of \cref{fig:2x3}
and the top left corner of \cref{fig:2x3-BAR}.  Then
$$\st_g = \big(\st_g(1), \st_g(2), \st_g(3), \st_g(4), \st_g(5)\big)=
\big(uwy, vxz, C/(uv),C/(wx),C/(yz)\big).$$
After applying BAR-motion to $g$
(bottom left corner of \cref{fig:2x3-BAR}),
the Stanley--Thomas word of $\BAR(g)$ is
$$\st_{\BAR(g)}=\big(C/(yz),uwy,vxz,C/(uv),C/(wx) \big)$$ which is simply a rightward cyclic shift
of $\st_g$.
\end{example}

\begin{thm}\label{thm:st2}
Let $P=[2]\times[b]$.  For a labeling $g\in\kk^P$,
$\st_{\BAR(g)}(i)=\st_g(i-1)$ for $2\leq i\leq 2+b$ and $\st_{\BAR(g)}(1)=\st_g(2+b)$.
Thus,
$$\prod\limits_{m=0}^{1+b} (\BAR^m g)(k,1) (\BAR^m g)(k,2) \cdots (\BAR^m g)(k,b)=C^b$$
for $1\leq k\leq 2$
and
$\prod_{m=0}^{1+b} (\BAR^m g)(1,\ell) (\BAR^m g)(2,\ell)=C^2$
for $1\leq \ell\leq b$.
\end{thm}

Proving this result follows from an explicit description of $\BAR(g)$ in terms of the labels
in $g$.  The details are omitted, but this can be determined on $[a]\times[b]$
by writing $\BAR=\tau_{1,b}\cdots \tau_{1,2}\tau_{1,1}\tau_{0,b}\cdots \tau_{0,2}\tau_{0,1}$ as in
\cref{prop:BAR-tog}.
We do not believe $a=2$ is necessary in \cref{thm:st2}, and we conjecture the following,
which has been verified for $a,b\leq 3$.

\begin{conj}\label{conj:st}
Let $P=[a]\times[b]$.  For a labeling $g\in\kk^P$,
$\st_{\BAR(g)}(i)=\st_g(i-1)$ for $2\leq i\leq a+b$ and $\st_{\BAR(g)}(1)=\st_g(a+b)$.
Thus,
$$\prod\limits_{m=0}^{a+b-1} (\BAR^m g)(k,1) (\BAR^m g)(k,2) \cdots (\BAR^m g)(k,b)=C^b \text{ for $1\leq k\leq a$,
and}$$
$$\prod\limits_{m=0}^{a+b-1} (\BAR^m g)(1,\ell) (\BAR^m g)(2,\ell) \cdots (\BAR^m g)(a,\ell)=C^a \text{ for $1\leq \ell\leq b$.}$$
\end{conj}


%\acknowledgements{\lipsum[20]}

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