Combinatorial, piecewise-linear, and birational \\ homomesy for products of two chains
David Einstein
Department of Mathematics
[-0.8ex] Lowell
MA
U.S.A.
James Propp\thanks{Supported by NSF Grant \#1001905.}
\url{http://jamespropp.org}
The purpose of this article is to illustrate the dynamical concept of {\em homomesy} in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and to show the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams~\cite{SW12}, in the case where the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley~\cite{Sta86}, and then lifted to birational operations on the positive orthant in $\R^{|P|}$ and indeed to a dense subset of $\C^{|P|}$. We prove that these lifted operations, like their combinatorial counterparts, have order $a+b$. In the birational setting, we prove a multiplicative homomesy theorem %% Change "a multiplicative homomesy theorem" to "multiplicative %% homomesy theorems" if I end up proving antipodal homomesy. that by tropicalization yields an additive homomesy result in the piecewise-linear setting, which in turn specializes to an additive homomesy result in the combinatorial setting already proved by the second author and Roby. %% Change "the second author" if I take on more co-authors Here the term ``additive homomesy'' refers to the phenomenon wherein the arithmetic means of some quantity over the various orbits of a dynamical system are all equal to one another, and multiplicative homomesy is an analogous multiplicative notion. In the final section, we describe results of experiments indicating directions for future work. % Or not! If I state an open problem, the referee might say % "The author should prove this result too before publishing". % keywords are optional \bigskip\noindent Keywords: antichains, dynamics, homomesy, order ideals, order polytope, piecewise-linear, promotion, rowmotion, toggle group, tropicalization.
1310.5294.tex
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\title{Combinatorial, piecewise-linear, and birational \\
homomesy for products of two chains}


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\author{
David Einstein\\
\small Department of Mathematics\\[-0.8ex]
\small University of Massachusetts Lowell\\[-0.8ex] 
\small Lowell, MA, U.S.A.\\
\small\tt deinst@gmail.com\\
\and
James Propp\thanks{Supported by NSF Grant \#1001905.}\\
\small Department of Mathematics\\[-0.8ex]
\small University of Massachusetts Lowell\\[-0.8ex] 
\small Lowell, MA, U.S.A.\\
\small\tt \url{http://jamespropp.org}\\
}

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\begin{document}

\maketitle

\begin{abstract}
The purpose of this article is to illustrate the
dynamical concept of {\em homomesy} in three kinds of dynamical systems -- 
combinatorial, piecewise-linear, and birational -- 
and to show the relationship between these three settings.
In particular, we show how the rowmotion and promotion operations
of Striker and Williams~\cite{SW12}, 
in the case where the poset $P$ is a product
of a chain of length $a$ and a chain of length $b$,
can be lifted to (continuous) piecewise-linear operations
on the order polytope of Stanley~\cite{Sta86}, 
and then lifted to birational operations
on the positive orthant in $\R^{|P|}$
and indeed to a dense subset of $\C^{|P|}$.
We prove that these lifted operations,
like their combinatorial counterparts, have order $a+b$.
In the birational setting, we prove a multiplicative homomesy theorem
%% Change "a multiplicative homomesy theorem" to "multiplicative 
%% homomesy theorems" if I end up proving antipodal homomesy.
that by tropicalization yields 
an additive homomesy result in the piecewise-linear setting,
which in turn specializes to an additive homomesy result in the
combinatorial setting already proved by the second author and Roby. 
%% Change "the second author" if I take on more co-authors
Here the term ``additive homomesy'' refers to the phenomenon wherein 
the arithmetic means of some quantity over the various orbits of a dynamical
system are all equal to one another, and multiplicative homomesy
is an analogous multiplicative notion.
In the final section, we describe results of experiments indicating
directions for future work.
% Or not!  If I state an open problem, the referee might say
% "The author should prove this result too before publishing".

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} 
antichains,
dynamics,
homomesy,
order ideals,
order polytope,
piecewise-linear,
promotion,
rowmotion,
toggle group,
tropicalization.
\end{abstract}

\noindent
{\bf Disclaimer:} This version of the article is not intended for
the general mathematical public; we are putting it on the arXiv 
because we want to establish uniformity of notation and terminology 
with specific individuals (they know who they are!)\ who are 
currently working on related projects.  All mathematical claims 
contained in this draft have, to the best of our knowledge, been 
proved, but assertions of the form ``In section X, we prove Y'' 
should be distrusted; in some cases, the proofs will be found not 
in the present article but in the article-in-progress ``Iterative 
properties of birational rowmotion'' by Darij Grinberg and Tom Roby.

\begin{section}{Introduction}
\label{sec:intro}

Numerous authors (specifically Brouwer and Schrijver~\cite{BS74},
Cameron and fon der Flaass~\cite{Fon93}\cite{CF95}, Panyushev~\cite{Pan08},
and Striker and Williams~\cite{SW12}) have studied
an operation $\row$ on the set of order ideals of a poset $P$
that, following Striker and Williams, we call {\em rowmotion}.
In exploring the properties of rowmotion,
Striker and Williams also introduced and studied 
a closely related operation $\pro$ they call {\em promotion} 
on account of its ties with promotion of Young tableaux,
which depends on the choice of an {\em rc embedding}
(a particular kind of embedding of $P$ into the poset $\N \times \N$).
Here we focus on a very particular case,
where $P$ is of the form $[a] \times [b]$ 
and the rc embedding sends $(i,j) \in P$ to $(j-i,i+j-2) \in \Z^2$,
and we explore how the cardinality of an order ideal $I \in J(P)$ 
behaves as one iterates rowmotion and promotion.

It has been shown~\cite{PR13}
that the order of $\row$ acting on $J([a] \times [b])$ is $n=a+b$.
Propp and Roby furthermore showed that the average of $|I|$
as $I$ varies over a $\row$-orbit in $J(P)$ is $ab/2$,
regardless of which orbit one takes.
That is:

\begin{theorem}
\label{thm:card}
% kludge
{\rm [PR13]}
For every ideal $I$ in $J([a] \times [b])$,
$$\frac{1}{n} \sum_{k=0}^{n-1} |\row^k (I)| = ab/2.$$
\end{theorem}

Indeed, Propp and Roby remark
(see the paragraph following the proof of Theorem 13) 
% This reference may need to be updated if we end up
% revising the article further before publishing it
% in El.J.C.!
that their arguments give a refined version of this theorem
in which $|\row^k (I)|$ is replaced by $|(\row^k I) \cap S|$
where $S$ is any {\em file} in $[a] \times [b]$,
i.e., any set consisting of all $(i,j) \in [a] \times [b]$
with $j-i$ taking on some particular value;
see Theorem~\ref{card:refined} below.
% Be sure to state and prove it!

The notion of looking at the average of a quantity over an orbit
was an outgrowth of the second author's work 
% fix as needed
on chip-firing and rotor-routing~\cite{H+08},\cite{HP10};
see in particular Proposition 3 of~\cite{PR13}.
Further inspiration came from conjectures of Panyushev~\cite{Pan08}
(later proved by Armstrong, Stump, and Thomas~\cite{AST11}).

This article presents a new proof of Theorem~\ref{thm:card} which,
although less direct than the Propp-Roby proof, 
indicates that the ``constant-averages-over-orbits phenomenon''
(also called ``homomesy'')
applies not just for actions on order ideals and antichains
but also for dynamical systems of a different character. 
Specifically, we define (continuous) piecewise-linear maps
from the order polytope of $P$ to itself 
({\em piecewise-linear rowmotion} and {\em promotion}) 
that satisfy homomesy,
and birational maps from a dense open subset of $\C^{ab}$ to itself
({\em birational rowmotion} and {\em promotion}) 
that satisfy a multiplicative version of homomesy.
We give other examples of functions that exhibit homomesy,
and in the case of combinatorial rowmotion and promotion,
we are able to show that, in a certain sense,
there are no other examples:
within a natural vector space of functions from $J([a] \times [b])$ to $\R$,
the homomesic functions are precisely those that are linear combinations
of functions of the form $I \mapsto |I \cap S|$
(where $S$ is either a file in $[a] \times [b]$
or a set of the form $\{(i,j),(a+1-i,b+1-j)\}$).

The plan of the article is as follows.
In section~\ref{sec:background},
after introducing needed preliminaries and notation,
including the general definition of (additive) homomesy 
and its specialization to the case of functions with periodic behavior,
we review some of the background 
on the rowmotion and promotion operations 
$\row,\pro: J(P) \rightarrow J(P)$.
%% We should move the definitions of the order and chain polytopes
%% into sec:background. 
% and on the order polytope $\cO(P) \subset \R^{|P|}$.
We then define (in section~\ref{sec:PL})
piecewise-linear maps $\rowP,\proP: \R^{|P|} \rightarrow \R^{|P|}$
and show that $\rowP$ and $\proP$ specialize to $\row$ and $\pro$ 
if one restricts attention to the vertices of $\cO(P)$.
Changing variables, we obtain slightly different piecewise-linear maps
$\rowH,\proH: \R^{|P|} \rightarrow \R^{|P|}$ that are 
homogeneous versions of $\rowP$, $\proP$.
Then we show (in section~\ref{sec:birational})
how $\rowH$, $\proH$ can in turn each be viewed as
a tropicalization of a birational map
from a dense open subset $U$ of $\C^{ab}$ to itself;
we call the elements of $U$ {\em $P$-arrays}. 
In section~\ref{sec:transfer},
we digress to give an alternative characterization of birational rowmotion
in terms of Stanley's {\em transfer map}
between the order polytope and chain polytope of a poset~\cite{Sta86}.
In section~\ref{sec:actions},
% New!
we construct two actions of $S_n$ on $U$.
The first (due to Darij Grinberg~\cite{G})
gives invariants for birational rowmotion;
the second gives invariants for birational promotion,
and also yields a proof of Theorem~\ref{thm:prod}:
for $v = (v_1,\dots,v_{ab}) \in U$,
the product of the coordinates of $v$ 
associated with elements of the file $S$ (denoted here by $|v|_S$)
has the property that 
$|v|_S |\pro(v)|_S |\pro^2(v)|_S \cdots |\pro^{n-1}(v)|_S = 1$.
In section~\ref{sec:pairing}, 
% Wishful!
we prove a fundamental {\em pairing property} of birational rowmotion,
and use it to show that birational rowmotion is of order $a+b$.
In section~\ref{sec:recombine}, building upon Theorem 5.4 of~\cite{SW12},
we demonstrate a concrete operation on $P$-arrays,
which we call {\em recombination},
that gives an equivariant bijection 
between birational rowmotion and promotion;
the resulting bijection between rowmotion orbits and promotion orbits
lets us deduce that birational promotion, like birational rowmotion,
is of order $a+b$, and furthermore lets us deduce
that Theorem~\ref{thm:prod} holds
when promotion is replaced by rowmotion.
In section~\ref{sec:tropic}, we use tropicalization
to deduce from Theorem~\ref{thm:prod}
a piecewise-linaer analogue (Theorem~\ref{thm:sumh})
that by a linear change of variables yields the homomesy result 
for the action of promotion on $\cO(P)$ (Theorem~\ref{thm:sum}).
This last result then yields homomesy for 
the action of promotion on $J(P)$ (Theorem~\ref{thm:card}).
Figure~\ref{fig:imply} shows the structure 
of this chain of deductions schematically
(omitting the use of recombination for passing
back and forth between rowmotion and promotion).
\begin{figure}
\label{fig:imply}
$$
\begin{array}{ccc}
\mbox{(birational)} & \mbox{Theorem~\ref{thm:prod}} & \ \\
& \Downarrow & \!\!\!\!\mbox{(tropicalization)} \\
\mbox{(piecewise-linear)} & \mbox{Theorem~\ref{thm:sumh}} & \ \\
& \Downarrow & \!\!\!\!\mbox{(change of variables)} \\
\mbox{(piecewise-linear)} & \mbox{Theorem~\ref{thm:sum}} & \ \\
& \Downarrow & \!\!\!\!\mbox{(specialization)} \\
\mbox{(combinatorial)} & \mbox{Theorem~\ref{thm:card}} & \ 
\end{array}
$$
\caption{Implications between combinatorial, piecewise-linear, 
and birational results about homomesy.}
\end{figure}
In section~\ref{sec:opposites}
we use the pairing property of section~\ref{sec:pairing}
to prove that the function that sends $f$ to $f(x) f(x')$ 
(with $x = (i,j)$ and $x' = (a+1-i,b+1-j)$)
is multiplicatively homomesic.
In section~\ref{sec:bound},
we show that all of the homomesies obtained thus far give all homomesies 
for the action of rowmotion and promotion on $J(P)$,
if one restricts to functions 
spanned by the indicator functions $1_x(I)$.
In section~\ref{sec:future},
we point out directions for future work.

% In the final section of the article,
% we return to the original conjecture of Panyushev
% (proved by Armstrong, Stump, and Thomas),
% which (in our language) asserts homomesy
% for the cardinality of antichains under the action of rowmotion,
% and give empirical support for homomesy
% of piecewise-linear and birational analogues,
% at least in the case where the root poset $P$ is of type $A$.
% Here the chain polytope, rather than the order polytope,
% is the natural setting for the piecewise-linear action.

The philosophy of lifting combinatorial actions
to piecewise-linear actions and thence to birational actions
(called ``geometric actions'' by some authors,
as in the phrase ``geometric Robinson-Schensted-Knuth'')
is not original, 
and in particular Kirillov and Berenstein's work 
on operations on Gelfand-Tsetlin patterns~\cite{KB95}
has some parallels with our constructions.
% but we are not aware of earlier work along these lines
% in the context of rowmotion and promotion.
%% But aren't K&B doing promotion in the minuscule poset of type B?
Computer experiments suggest that the homomesy phenomenon
and the implicational structure given in Figure~\ref{fig:imply}
are not unique to the poset $[a] \times [b]$
but apply to many other posets, notably root posets and minuscule posets.
For more background of homomesy,
including several examples different in nature
from the ones considered here but philosophically similar, see~\cite{PR13}.

The authors are grateful to Arkady Berenstein, Darij Grinberg, Tom Roby, 
% Richard Stanley?
and Jessica Striker for helpful conversations.

\end{section}

\begin{section}{Background}
\label{sec:background}

\begin{subsection}{Homomesy}
\label{subsec:homomesy}

Given a set $X$, an operation $T : X \rightarrow X$, 
and a function $F$ from $X$ to a field $\cK$ of characteristic 0, 
we say that $F$ is {\em homomesic} 
relative to (or under the action of) $T$, 
or that the triple $(X,T,F)$ exhibits {\em homomesy}, 
if, for all $x \in X$, the long-term average
$$\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=0}^{N-1} F(T^k(x))$$
equals some constant $c$ (independent of $x$). 
We also say in this situation that the function $F$ 
(which in this context we will sometimes call a {\em functional} on $X$)
is $c$-{\em mesic} relative to the map $T$.
In the case where the sequence $F(T^k (x))$ ($k = 0,1,2,\dots$) is periodic, 
it is easy to show that the limit is equal 
to the average of $F(T^k(x))$ over one period, 
and this will be the situation throughout this article, 
except in one paragraph in the final section.
% Make sure that that remains true.
The article~\cite{PR13} gives examples of combinatorial situations 
in which homomesy holds, 
and we have found that such situations are easy to find experimentally
once one starts to look for them,
though proving theorems that support the experimental results 
is another matter.

\end{subsection}

\begin{subsection}{Posets and toggling}
\label{subsec:posets}

We assume readers are familiar with 
the definition of a finite poset $(P, \leq)$,
as for instance given in Ch.\ 3 of~\cite{Sta11}.
For the most part, we are studying the case $P = [a] \times [b] = 
\{(i,j) \in \N \times \N: \ 1 \leq i \leq a, \ 1 \leq j \leq b\}$
with ordering defined by $(i,j) \leq (i',j')$ 
iff $i \leq i'$ and $j \leq j'$.  We put $n=a+b$. 

We write $x \coveredby y$ (``$x$ is covered by $y$'')
or equivalently $y \covers x$ (``$y$ covers $x$'') 
when $x < y$ and no $z \in P$ satisfies $x < z < y$.
We call $S \subseteq P$ a {\em filter} 
(or {\em upset} or {\em dual order ideal}) of $P$
when $x \in S$ and $y \geq x$ imply $y \in S$.
We call $S \subseteq P$ an {\em order ideal} (or {\em downset}) of $P$
when $x \in S$ and $y \leq x$ imply $y \in S$.
We call $S \subseteq P$ an {\em antichain}
when $x, y \in S$ and $x \neq y$
imply that $x$ and $y$ are {\em incomparable}
(i.e., neither $x \leq y$ nor $y \leq x$). 
The set of filters, order ideals, and antichains of $P$ 
are denoted by $\cF(P)$, $J(P)$, and $\cA(P)$, respectively.

There are natural bijections 
$\alpha_1: J(P) \rightarrow \cF(P)$, 
$\alpha_2: \cF(P) \rightarrow \cA(P)$,
and $\alpha_3: \cA(P) \rightarrow J(P)$
given by the following recipes:
\begin{compactenum}
\item[(1)] for $I \in J(P)$, let $\alpha_1(I)$ be the complement $P \setminus I$;
\item[(2)] for $F \in \cF(P)$, let $\alpha_2(F)$ be the set of minimal elements of $F$
(i.e., the set of $x \in F$ such that $y < x$ implies $y \not\in F$); and
\item[(3)] for $A \in \cA(P)$, let $\alpha_3(A)$ be the downward saturation of $A$
(i.e., the set of $y \in P$ such that $y \leq x$ for some $x \in A$).
\end{compactenum}
The composition 
$\row := \alpha_3 \circ \alpha_2 \circ \alpha_1: J(P) \rightarrow J(P)$
is not the identity map;
e.g., it sends the ``full'' order ideal ($P$ itself)
to the empty order ideal ($\eset$).

Cameron and fon der Flaass~\cite{CF95}
gave an alternative characterization of $\row$.
Given $x \in P$ and $I \in J(P)$, let $\tau_x(I)$ 
(``$I$ toggled at $x$'' in Striker and Williams' terminology)
denote the set $I \symmdiff \{x\}$ if this set is in $J(P)$ and $I$ otherwise.
Equivalently, $\tau_x(I)$ is $I$
unless $y \in I$ for all $y \coveredby x$
and $y \not\in I$ for all $y \covers x$,
in which case $\tau_x(I)$ is $I \symmdiff \{x\}$.
(We will sometimes say ``Toggling $x$ turns $I$ into $\tau_x(I)$''.)
Clearly $\tau_x$ is an involution.
It is also easy to show that $\tau_x$ and $\tau_y$ commute
unless $x \coveredby y$ or $x \covers y$.
If $x_1,x_2,\dots,x_{|P|}$ is any {\em linear extension} of $P$
(that is, a listing of the elements of $P$ such that
$x_i < x_j$ implies $i<j$), then the composition 
$\tau_{x_1} \circ \tau_{x_2} \circ \cdots \circ \tau_{x_{|P|}}$
coincides with $\row$.
In the case where the poset $P$ is {\em ranked}
(that is, where the elements can be partitioned into ranks $0, 1, 2, \dots$
such that all minimal elements of $P$ belong to rank 0
and such that $x \coveredby y$
implies that the rank of $x$ is 1 less than the rank of $y$),
one natural way to linearly extend $P$ is to list the elements by rank,
starting with rank 0 and working upward.
Given the right-to-left order of composition of
$\tau_{x_1} \circ \tau_{x_2} \circ \cdots \circ \tau_{x_{|P|}}$,
this corresponds to toggling the top rank first,
then the next-to-top rank, and so on, lastly toggling the bottom rank.
Note that when $x,y$ belong to the same rank of $P$,
the toggle operations $\tau_x$ and $\tau_y$ commute,
so even without availing ourselves of the theorem
of Cameron and fon der Flaass,
we can see that this composite operation on $J(P)$ is well-defined.
Striker and Williams, in their theory of rc posets, 
use the term ``row'' as a synonym for ``rank'',
and they refer to $\row$ as {\em rowmotion}.

For example, let $P = [2] \times [2]$,
and write $(1,1),(2,1),(1,2),(2,2)$ as $w,x,y,z$ for short,
with $w < x < z$ and $w < y < z$,
and with the rc embedding shown.
$$
\begin{array}{ccc}
\ & z &   \\[2mm]
x &   & y \\[2mm]
\ & w &   
\end{array}
$$
Under the action of $\alpha_1$, $\alpha_2$, and $\alpha_3$,
the order ideal $\{w,x\}$ gets successively mapped to
$\{y,z\}$, $\{y\}$, and $\{w,y\}$.
Under the action
of $\tau_z$, $\tau_y$, $\tau_x$, and $\tau_w$,
the order ideal $\{w,x\}$ gets successively mapped to
$\{w,x\}$, $\{w,x,y\}$, $\{w,y\}$, and $\{w,y\}$.
In both cases we obtain $\row(\{w,x\}) = \{w,y\}$.
\begin{figure}
\label{fig:rc}
\begin{center}
\begin{pspicture}(-4,-1)(4,7)
\psline[linestyle=dotted](0,0)(4,0)
\psline[linestyle=dotted](-1,1)(4,1)
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\psline[linestyle=dotted](-1,3)(4,3)
\psline[linestyle=dotted](0,4)(4,4)
\psline[linestyle=dotted](1,5)(4,5)
\psline[linestyle=dotted](-2,2)(-2,6)
\psline[linestyle=dotted](-1,1)(-1,6)
\psline[linestyle=dotted](0,0)(0,6)
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\end{center}
\caption{The standard rc embedding of the poset $P = [3] \times [4]$,
and the associated partitions of $P$ into ranks and files.  Diagonal
edges are associated with the covering relation in $P$.}
\end{figure}

We will not go into the general theory of rc posets,
as all of our work 
% (except in the last section)
%% Make a specific section-reference
concerns the special case of rc posets of the form $[a] \times [b]$.
%% Update as needed!
As an rc poset, $[a] \times [b]$ admits an embeding in the plane
that maps $(i,j)$ to the point $(j-i,i+j-2)$.
This sends the minimal element $(1,1)$ to the origin
and sends all the poset-elements of rank $m$ ($=i+j-2$)
to the horizontal line at height $m$ above the origin,
We refer to elements of $P$ that lie on a common vertical line
as belonging to the same {\em file}.
In particular, we say $(i,j)$ belongs to the 
$(j-i+a)$th\footnote{Note that 
% as $i$ ranges from $1$ to $a$ and $j$ ranges from $1$ to $b$, 
$j-i+a$ ranges from $1$ to $a+b-1=n-1$; 
this is slightly different from 
the indexing in~\cite{PR13}.} file of $P$.
% See Figure~\ref{fig:rc}.
%% kludge
See Figure 2.
Note that if $x$ and $y$ belong to the same file,
the toggle operations $\tau_x$ and $\tau_y$ commute,
since neither of $x,y$ can cover the other.
Thus the composite operation of toggling the elements of $P$
from left to right is well-defined;
Striker and Williams call this operation {\em promotion},
and we denote it by $\pro$.

\end{subsection}

\end{section}

\begin{section}{Piecewise-linear fiber-toggling}
\label{sec:PL}

Given a poset $P = \{x_1,\dots,x_n\}$, 
let $\R^P$ denote the set of functions $f : P \rightarrow \R$;
we can represent such an $f$
as a {\em $P$-array} (or {\em array} for short)
in which the values of $f(x)$ (for all $x \in P$) are arranged
according to the rc embedding of $P$ in the plane.
We will sometimes identify $\R^P$ with $\R^{|P|}$,
associating $f \in \R^P$ with $v = (f(x_1),\dots,f(x_n))$.
Let $\hat{P}$ denote the augmented poset obtained from $P$ 
by adding two extra elements $\hat{0}$ and $\hat{1}$ 
(which we sometimes denote by $x_0$ and $x_{n+1}$)
satisfying $\hat{0} < x < \hat{1}$ for all $x \in P$. 
The {\em order polytope} $\cO(P) \subset \R^P$ 
(see~\cite{Sta86})
is the set of vectors $(\hat{f}(x_1),\dots,\hat{f}(x_n))$ 
arising from functions $\hat{f} : \hat{P} \rightarrow \R$ 
that satisfy $\hat{f}(0) = 0$ and $\hat{f}(1) = 1$ 
and are {\em order-preserving}
($x \leq y$ in $P$ implies $\hat{f}(x) \leq \hat{f}(y)$ in $\R$). 
In some cases it is better to work with the augmented vector 
$(\hat{f}(x_0),\hat{f}(x_1),\dots,\hat{f}(x_n),\hat{f}(x_{n+1}))$.
In either case we have a convex compact polytope.

For example, if $P = [2] \times [2] = \{w,x,y,z\}$ as above,
then $\cO(P)$ is the set of vectors $v=(v_1,v_2,v_3,v_4) \in \R^4$ 
such that $0 \leq v_1$, $v_1 \leq v_2 \leq v_4$, 
$v_1 \leq v_3 \leq v_4$, and $v_4 \leq 1$.
% It can also be written as the convex hull of the vectors
% $(0,0,0,0)$, $(0,0,0,1)$, $(0,0,1,1)$, 
% $(0,1,0,1)$, $(0,1,1,1)$, and $(1,1,1,1)$,
% which are precisely the vectors associated with the filters of $P$.
% It is shown in~\cite{Sta86} that for any poset $P$,
% the vertices of $\cO(P)$ correspond to the indicator functions
% of the filters of $P$.

Given a convex compact polytope $\polytope$ in $\R^n$
(we are only concerned with the case $\polytope=\cO(P)$ here
but the definition makes sense more generally),
we define the {\em piecewise-linear toggle operation} 
$\phi_i$ ($1 \leq i \leq n$)
as the unique map from $\polytope$ to itself
whose action on the 1-dimensional fibers of $\polytope$ 
in the $i$th coordinate direction
is the linear map that switches the two endpoints of the fiber.
That is, given $v = (v_1,\dots,v_n) \in \polytope$,
we define 
\begin{equation}
\label{eq:phiv}
\phi_i(v) = (v_1,\dots,v_{i-1},L+R-v_i,v_{i+1},\dots,v_n),
\end{equation}
where the real numbers $L$ and $R$
are respectively the left and right endpoints of the set 
$\{t \in \R: \ (v_1,\dots,v_{i-1},t,v_{i+1},\dots,v_n)
\in \polytope\}$,
which is a bounded interval because $K$ is 
convex and compact.\footnote{Note that $L$ and $R$ depend on
$v_1,\dots,v_{i-1},v_{i+1},\dots,v_n$,
though our notation supresses this dependence.}
Since $L+R-(L+R-v_i)=v_i$, each toggle operation is an involution.

Similar involutions were studied by Kirillov and Berenstein~\cite{KB95}
in the context of Gelfand-Tsetlin triangles.
Indeed, one can view their action as an instance of our framework,
where instead of looking at the rectangle posets $[a] \times [b]$
one looks at the triangle posets
with elements $\{(i,j): \ 1 \leq i \leq j \leq n\}$
and covering-relations $(i,j-1) \coveredby (i,j)$ ($1 \leq i \leq j \leq n$)
and $(i+1,j+1) \coveredby (i,j)$ ($1 \leq i \leq j \leq n-1$);
Kirillov and Berenstein use the term ``elementary transformations''
in their Definition 0.1, whereas we use the term ``fiber-toggles''.

In the case where $\polytope$ is the order polytope of $P$,
and a particular element $x \in P$ has been indexed as $x_i$,
we write $\phi_i$ as $\phi_x$. 
The $L$ and $R$ that appear in (\ref{eq:phiv})
are given by
\begin{equation}
\label{eq:L}
L = \max \{v_j: \ 0 \leq j \leq n+1, \ x_j \coveredby x_i\}
\end{equation}
and
\begin{equation}
\label{eq:R}
R = \min \{v_j: \ 0 \leq j \leq n+1, \ x_j \covers x_i\} .
\end{equation}
Using these definitions of $L$ and $R$,\footnote{If 
$v$ is in $\cO(P)$ one also has 
$L = \max \{v_j : x_j \leq x_i\}$ 
and 
$R = \min \{v_j : x_j \geq x_i\}$, 
but the formulas (\ref{eq:L}) and (\ref{eq:R}) 
turn out to be the ``right'' ones to use 
in the complement of the order polytope in $\R^P$,
as well as the right ones vis-a-vis lifting
the action to the birational setting.}
we see that equation (\ref{eq:phiv}) defines 
an involution on all of $\R^P$, not just $\cO(P)$.
It is easy to show that $\phi_x$ and $\phi_y$ commute
unless $x \coveredby y$ or $x \covers y$.
These piecewise-linear toggle operations $\phi_x$ are analogous to
the combinatorial toggle operations $\tau_x$ 
(and indeed $\phi_x$ generalizes $\tau_x$ in a sense to be made precise below),
so it is natural to define 
piecewise-linear rowmotion $\rowP: \cO(P) \rightarrow \cO(P)$
as the composite operation accomplished by toggling from top to bottom
(much as ordinary rowmotion $\row: J(P) \rightarrow J(P)$
can be defined as the composite operation obtained by
toggling from top to bottom).  Likewise we can define
piecewise-linear promotion $\proP: \cO(P) \rightarrow \cO(P)$
as the composite operation accomplished by toggling from left to right.

Continuing the example $P = [2] \times [2] = \{w,x,y,z\}$
from subsection~\ref{subsec:posets},
let $v = (.1,.2,.3,.4) \in \cO(P)$,
corresponding to the order-preserving function $f$
that maps $w,x,y,z$ to $.1,.2,.3,.4$, respectively.
Under the action of $\phi_z$, $\phi_y$, $\phi_x$, and $\phi_w$,
the vector $v = (.1,.2,.3,.4)$ gets successively mapped to
$(.1,.2,.3,.9)$, $(.1,.2,.7,.9)$, $(.1,.8,.7,.9)$, 
and $(.6,.8,.7,.9) = \rowP(v)$.

We will see
% Provide pointer!
that $\rowP$ and $\proP$ are of order $n$.
Specifically, we will show as a corollary of Theorem~\ref{thm:pairing}
that $\rowP$ is of order $n$,
and by then applying recombination (see~\ref{thm:recombine})
we will conclude that $\proP$ is of order $n$ as well.
% If the method of proof of Theorem~\ref{thm:pairing}
% turns out to be the sort that could be applied to
% promotion as well, a remark to that effect might be
% included, here or elsewhere in the article.

% It is believed (although at the time of this writing has not yet been proved)
% that for $P = [a] \times [b]$, $\proP$ is of order $n$.
% It is perhaps curious that we are able to prove homomesy of the action
% without even showing that the action is a finite cyclic action!
% %% Change this if the situation changes
% We can, however, show that $\proP$ and $\rowP$ have the same order,
% and that functions that are homomesic for one are homomesic for the other;
% see subsection~\ref{subsec:rowmotion}.

The vertices of $\cO(P)$ are precisely the 0,1-valued functions $f$ on $P$
with the property that $x \leq y$ in $P$ implies $f(x) \leq f(y)$ in $\{0,1\}$.
That is, they are precisely the indicator functions of filters.
Filters are in bijection with order ideals
by way of the complementation map,
so the vertices of $\cO(P)$ are in bijection 
with the elements of lattice $J(P)$.
Each toggle operation acts as a permutation on the vertices of $\cO(P)$. 
Indeed, if we think of each vertex $\cO(P)$
as determining a cut of the poset $P$
into an upset (filter) $\Sup$ and a downset (order ideal) $\Sdown$
(the pre-image of 1 and 0, respectively,
under the order-preserving map from $P$ to $\{0,1\}$),
then the effect of the toggle operation $\phi_x$ ($x \in P$)
is just to move $x$ from $\Sup$ to $\Sdown$ (if $x$ is in $\Sup$)
or from $\Sdown$ to $\Sup$ (if $x$ is in $\Sdown$),
unless this would violate the property
that $\Sup$ must remain an upset and $\Sdown$ must remain a downset.
In particular, we can see that when our point $v \in \cO(P)$ is a vertex
associated with the cut $(\Sup,\Sdown)$,
the effect of $\phi_x$ on $\Sdown$ is just
toggling the order ideal $\Sdown$ at the element $x \in P$.

It is not hard to show that each toggle operation preserves the quantity
$\min \{f(y) - f(x): \: x,y \in \hat{P}, x \coveredby y\}$.
Therefore $\rowP$ and $\proP$ preserve this quantity as well.

Define $|v|$ as the sum of the entries of $v$
(since all the entries are non-negative,
this is the $L^1$-norm of $v$).  
Then we will show:

\begin{theorem}
\label{thm:sum}
For every $v$ in $\R^P$,
$$ \frac{1}{n} \sum_{k=0}^{n-1} |\proP^k (v)| = ab/2.$$
\end{theorem}

Theorem~\ref{thm:card} is just the special case of this theorem
where $v$ is a vertex of $\cO(P)$.

One can generalize the above construction
by allowing $f(\hat{0})$ and $f(\hat{1})$
to have fixed values in $\R$ other than 0 and 1 respectively.
The nicest choice is to take $f(\hat{0}) = f(\hat{1}) = 0$.
In this case, we write the promotion operation as $\proH$
(and likewise we write the homogeneous piecewise-linear
rowmotion operation as $\rowH$).
The maps $\proP$ and $\proH$ are closely related.
Suppose that $P$ has $r+1$ ranks, numbered 0 (bottom) through $r$ (top).
Given an arbitrary $f$ in $\R^P$
(considered as a map from $\hat{P}$ to $\R$
that sends $\hat{0}$ to 0 and $\hat{1}$ to 1),
define $\overline{f}(x) = f(x) - m/r$
where $x$ belongs to rank $m$. 
Then $\overline{f}$ sends $\hat{0}$ and $\hat{1}$ to 0,
and each function from $\hat{P}$ to $\R$
that sends $\hat{0}$ and $\hat{1}$ to 0
arises as $\overline{f}$ for a unique $f$ in $\cO(P)$.
Furthermore, the map $f \mapsto \overline{f}$
commutes with promotion.\footnote{More generally, 
this way of relating $\proP$ and $\proH$
works whenever the poset $P$ is ranked.}

In this way it can be seen that Theorem~\ref{thm:sum} is equivalent to
the following claim:

\begin{theorem}
\label{thm:sumh}
For every $v$ in $\R^P$,
$$\sum_{k=0}^{n-1} |\proH^k (v)| = 0.$$
\end{theorem}
%% Give details of proof, here or later!

Note that if $a$ or $b$ is 1,
so that $[a] \times [b]$ is just a chain with $n=ab$ elements,
then our piecewise-linear maps are all linear,
and the effect of the map $\phi_i$ ($1 \leq i \leq n$) 
is just to swap the $i$th and $i+1$st elements of
the difference-vector $(v_1-v_0,v_2-v_1,\dots,v_{n+1}-v_{n})$,
a vector of length $n+1$ whose entries sum to 0.
Consequently $\proH$ is just a cyclic shift of a vector
whose entries sum to 0,
and the claim of Theorem~\ref{thm:sumh} follows.

It might be possible to prove Theorem~\ref{thm:sum}
by figuring out how the map $\proP$ dissects the order polytope into pieces
and re-arranges them via affine maps.
Likewise, it might be possible to prove Theorem~\ref{thm:sumh} 
by giving a precise analysis 
of the piecewise-linear structure of the map $\proH$.
However, we will take a different approach,
proving the result in the piecewise-linear setting
by proving it in the birational setting and then tropicalizing.

\end{section}

\begin{section}{Birational toggling}
\label{sec:birational}

The definition of the toggling operation
involves only addition, subtraction, min, and max.
As a result of this
one can define birational transformations on $(\R^+)^P$
that have some formal resemblance to the toggle operations on $\cO(P)$.
This transfer makes use of a dictionary in which 
0, addition, subtraction, max, and min
are respectively replaced by 
1, multiplication, division, addition, and parallel addition (defined below),
% Check: Grinberg-Roby uses 
%       ordinary addition for the down-neighbors of v
% and 
%       parallel addition for the up-neighbors of v; 
% PL-toggling uses
%       max for the down-neighbors of v 
% and 
%       min of the up-neighbors of v.
% So: + corresponds to max and || corresponds to min.
resulting in a subtraction-free rational expression.\footnote{The 
authors are indebted to Arkady Berenstein for pointing out
the details of this transfer of structure
from the piecewise-linear setting to the birational setting.}
Parallel addition can be expressed in terms of the other operations,
but taking a symmetrical view of the two forms of addition 
turns out to be fruitful.
Indeed, in setting up the correspondence we have a choice to make:
by ``series-parallel duality'',
one could equally well use a dictionary 
that switches the roles of addition and parallel addition.
We hope the choice that we have made here
will prove to be at least as convenient as
the other option would have been.

For $x,y$ satisfying $x+y \neq 0$, we define 
the parallel sum of $x$ and $y$ as $x \psum y = xy/(x+y)$.
In the case where $x$, $y$ and $x+y$ are all nonzero,
$xy/(x+y)$ is equal to $1/(\frac1x+\frac1y)$,
which clarifies the choice of notation and terminology:
if two electrical resistors of resistance $x$ and $y$ are connected in parallel,
the resulting circuit element has an effective resistance of $x \psum y$.
Note that if $x$ and $y$ are in $\R^+$, 
then $x+y$ and $x \psum y$ are in $\R^+$ as well.
Also note that $\psum$ is commutative and associative,
so that a compound parallel sum $x \psum y \psum z \psum \cdots$
is well-defined; it is equal to product $x y z \cdots$
divided by the sum of all products that omit exactly one of the variables,
and in the case where $x,y,z,\dots$ are all positive,
it can also be written as $1/(\frac1x+\frac1y+\frac1z+\cdots)$.
We thus have the reciprocity relation
\begin{equation}
\label{eq:reciprocity}
(x \psum y \psum z \psum \cdots) 
(\frac{_1}{^x}+\frac{_1}{^y}+\frac{_1}{^z}+\cdots) = 1.
\end{equation}
The identity 
\begin{equation}
\label{eq:duality}
(x \psum y) (x+y) = xy
\end{equation}
also plays an important role.

Recall formulas (\ref{eq:phiv}), (\ref{eq:L}) and (\ref{eq:R}) above.
Instead of taking the maximum of the $x_j$'s satisfying $x_j \coveredby x_i$,
we can take their ordinary (or ``series'') sum,
and instead of taking the minimum of the $x_j$'s 
satisfying $x_j \covers x_i$,
we can take their parallel sum.
Proceeding formally, given a non-empty set $S = \{s_1,s_2,\dots\}$,
let $\bigsig^+ S$ denote $s_1 + s_2 + \cdots$ 
and $\bigsig^{\psum} S$ denote $s_1 \psum s_2 \psum \cdots$.
Then for $v = (v_0,v_1,\dots,v_n,v_{n+1}) \in (\R^+)^{\hat{P}}$ 
with $v_0 = v_{n+1} = 1$
% (recall that $x_0 = \hat{0}$ and $x_{n+1} = \hat{1}$)
and for $1 \leq i \leq n$ we can take
\begin{equation}
\label{eq:phib}
\phi_i(v) = (v_0,v_1,\dots,v_{i-1},LR/v_i,v_{i+1},\dots,v_n,v_{n+1}),
\end{equation}
with
\begin{equation}
\label{eq:Lb}
L = \bigsig^{+} \{v_j: \ 0 \leq j \leq n+1, \ x_j \coveredby x_i\}
\end{equation}
and
\begin{equation}
\label{eq:Rb}
R = \bigsig^{\psum} \{v_j: \ 0 \leq j \leq n+1, \ x_j \covers x_i\}.
\end{equation}
We call the maps $\phi_i$ given by (\ref{eq:phib}) 
{\em birational toggle operations},
as opposed to the piecewise-linear toggle operations
treated in the previous section.\footnote{We use 
the same symbol $\phi_i$ for both,
but this should cause no confusion.}
As the $0$th and $n+1$st coordinates of $v$ 
aren't affected by any of the toggle operations,
we can just omit those coordinates,
reducing our toggle operations to actions on $(\R^+)^P$.
Since $LR/(LR/v_i) = v_i$,
each birational toggle operation is an involution
on the orthant $(\R^+)^{P}$.
The birational toggle operations are analogous to
the piecewise-linear toggle operations
(in a sense to be made precise below),
so it is natural to define 
{\em birational rowmotion} $\rowB: (\R^+)^P \rightarrow (\R^+)^P$
as the composite operation accomplished by toggling from top to bottom,
and to define
{\em birational promotion} $\proB: (\R^+)^P \rightarrow (\R^+)^P$
as the composite operation accomplished by toggling from left to right.

It is not hard to show that 
each birational toggle operation preserves the quantity 
$\bigsig^{+} \{f(x) / f(y): \: x,y \in \hat{P}, x \coveredby y\}$
(or if one prefers the reciprocal quantity
$\bigsig^{\psum} \{f(y) / f(x): \: x,y \in \hat{P}, x \coveredby y\}$).
Therefore $\rowB$ and $\proB$ also preserve this quantity.

Continuing our running example $P = [2] \times [2] = \{w,x,y,z\}$,
let $v = (1,2,3,4) \in \R^P$,
corresponding to the positive function $f$
that maps $w,x,y,z$ to $1,2,3,4$, respectively.
Under the action of $\phi_z$, $\phi_y$, $\phi_x$, and $\phi_w$,
the vector $v = (1,2,3,4)$ gets successively mapped to
$(1,2,3,\frac{5}{4})$,
$(1,2,\frac{5}{12},\frac{5}{4})$,
$(1,\frac{5}{8},\frac{5}{12},\frac{5}{4})$,
and $(\frac{1}{4},\frac{5}{8},\frac{5}{12},\frac{5}{4}) = \rowB(v)$.
We can check that the quantity 
$\bigsig^{+} \{f(x) / f(y): \: x,y \in \hat{P}, x \coveredby y\}$
retains the value $\frac{85}{12}$ throughout the process. 
%% Maybe better to use \proB instead of \rowB here and elsewhere?

For most of the purposes of this article,  
it would suffice to take $\proB$ to be a map from $(\R^+)^P$ to itself. 
However, $\proB$ can be extended to a map
from a dense open subset of $\R^P$ to itself,
and indeed, from a dense open subset $U$ of $\C^P$ to itself.
All expressions we consider are well-defined on the open orthant $(\R^+)^P$,
and all the theorems we prove amount to identities
that are valid when all variables lie in this orthant;
this implies that the identities hold
outside of some singular variety in $\C^P$.
Identifying the singular subvariety on which $\proB$ 
(or one of its powers) is undefined seems like an interesting question, 
but it is one that we leave to others.
Alternatively, Tom Roby has pointed out that one can replace $\R^+$ by
a ring of rational functions in formal indeterminates
indexed by the elements of $P$,
thereby avoiding the singularity issue
(once one checks that the rational functions in question
can be expressed as ratios of polynomials with positive coefficients).
%% If Tom becomes a coauthor, change the preceding sentence.
Experiments in Sage conducted by Darij Grinberg
have taken this formal approach, 
and there is strong evidence there for the conjecture that
for $P = [a] \times [b]$, the operation $\proB$ is of order $n$.

We focus on the ``multiplicatively homogeneous'' case
$f(\hat{0}) = f(\hat{1}) = 1$, since
(just as in the piecewise-linear setting) no generality is lost.
at least if we restrict to vectors $v$ in the positive orthant,
so that $r$th roots are globally well-defined,
where $r+1$ is the number of ranks of $P$.
Given an arbitrary $f: \hat{P} \rightarrow \R^+$,
let $\alpha = f(\hat{0})^{1/r}$
and $\omega = f(\hat{1})^{1/r}$,
and define $\overline{f}$ by 
$\overline{f}(x) = f(x) / \alpha^{r-m} \omega^{m}$
for $x$ belonging to rank $m$.
Then $\overline{f}$ sends $\hat{0}$ and $\hat{1}$ to 1,
and for any choice of $\alpha,\omega \in \R^+$,
each function from $\hat{P}$ to $\R^+$
that sends $\hat{0}$ and $\hat{1}$ to 1
arises as $\overline{f}$ for a unique $f : \hat{P} \rightarrow \R^+$
sending $\hat{0}$ to $\alpha$ and $\hat{1}$ to $\omega$.
Furthermore, the map $f \mapsto \overline{f}$ commutes with promotion.

Define $|v|$ to be the product of the entries of $v$.
As in the piecewise-linear case,
our proof-method will enable us to demonstrate
(multiplicative) homomesy of the action
even allowing for the possibility that the orbits are infinite.
That is, even though we cannot yet show that
the sequence $v, \proB(v), \proB^2(v), \dots$ is periodic,
we will show that the sequence $|v|, |\proB(v)|, |\proB^2(v)|, \dots$,
is periodic with period $n$.  Indeed, we will prove more:

\begin{theorem}
\label{thm:prod}
For every $v$ in $(\R^+)^P$
corresponding to an $f: \hat{P} \rightarrow \R^+$
with $f(\hat{0})=f(\hat{1})=1$,
$$\prod_{k=0}^{n-1} |\proB^k (v)| = 1.$$
\end{theorem}

In other words, the geometric mean of the values of $|\proB(v)|$
as $v$ traces out an orbit in $(\R^+)^P$ is equal to 1 for every orbit.
Theorem~\ref{thm:prod} also applies to a dense open subset of $\R^P$,
and indeed to a dense open subset of $\C^P$,
but the paraphrase in terms of geometric means does not hold in general
since $z \mapsto z^{1/N}$ is not single-valued on $\C$.

\end{section}

\begin{section}{Birational promotion and Stanley's transfer map}

\label{sec:transfer}

Although most of our work with rowmotion
treats it as a composition of $|P|$ toggles
(from the top to the bottom of $P$),
we noted in subsection~\ref{subsec:posets}
that $\row$ can also be defined
as a composition of three operations
$\alpha_1$, $\alpha_2$, $\alpha_3$.\footnote{Indeed 
this was the way in which Brouwer and Schrijver
originally defined this operation,
in the context of the Boolean lattices 
$[2] \times [2] \times \cdots \times [2]$.}
Here we point out that this alternative definition
can be lifted to the piecewise-linear setting
and indeed to the birational setting.

For the piecewise-linear setting, we first recall Richard Stanley's definition
of the chain polytope $\cC(P)$ of a poset $P$~\cite{Sta86}.
A {\em chain} in a poset $P$ is a totally ordered subset of $P$,
and a {\em maximal chain} in a poset $P$
is a chain that is not a proper subset of any other chain.
If the poset $P$ is ranked, then the maximal chains in $P$ 
are precisely those chains that contain an element of every rank.
The {\em chain polytope} of a poset $P$
is the set of maps from $P$ to $[0,1]$
such that for every chain $C$ in $P$,
\begin{equation}
\label{eq:chain}
\sum_{x \in C} f(x) \leq 1.
\end{equation}
Just as the vertices of the order polytope of $P$
correspond to the indicator functions of filters of $P$,
the vertices of the chain polytope of $P$ 
correspond\footnote{One direction of this claim is easy:
since every antichain intersects every chain of $P$
in at most one element of $P$,
the indicator function of an antichain must correspond to a point in $\cC(P)$.
For the other direction,
see Theorem 2.2 of~\cite{Sta86}.}
to the indicator functions 
of antichains in $P$.

Stanley defines a {\em transfer map} $\Phi: \cO(P) \rightarrow \R^P$
via the formula
\begin{equation}
\label{eq:transfer}
(\Phi f)(x) = \min \{ f(x)-f(y): y \in \hat{P}, \ x \covers y \}
\end{equation}
for all $x \in P$ (recall that we have $f(\hat{0}) = 0$).
Stanley proves that $\Phi$ is a bijection between $\cO(P)$ and $\cC(P)$
that carries the vertices of the former to the vertices of the latter.
The inverse of $\Phi$ is given by\footnote{This is not 
precisely the definition of $\Psi$ that Stanley gives,
but the two definitions are easily seen to be equivalent.}
\begin{equation}
\label{eq:inverse}
(\Psi g)(x) = \max \{ g(y_1)+g(y_2)+\cdots+g(y_k): \hat{0} \coveredby y_1
\coveredby y_2 \coveredby \cdots \coveredby y_k = x \} .
\end{equation}

Let $\tilde{\cO}(P)$ be the set of order-reversing maps from $P$ to $[0,1]$.
We now define bijections 
$\alpha_1: \tilde{\cO}(P) \rightarrow \cO(P)$, 
$\alpha_2: \cO(P) \rightarrow \cC(P)$,
and $\alpha_3: \cC(P) \rightarrow \tilde{\cO}(P)$
given by the following recipes:
\begin{compactenum}
\item[(1)] for $f \in \tilde{\cO}(P)$, let $\alpha_1(f)$ be defined by 
$(\alpha_1(f))(x) = 1-f(x)$;
\item[(2)] for $f \in \cO(P)$, let $\alpha_2(f) = \Phi(f)$; and
\item[(3)] for $f \in \cC(P)$, let $\alpha_3(f)$ be defined by
\begin{equation}
\label{eq:longsum}
(\alpha_3 f)(x) = \max \{ f(y_1)+f(y_2)+\cdots+f(y_k): \ x = y_1
\coveredby y_2 \coveredby \cdots \coveredby y_k \coveredby \hat{1} \} .
\end{equation}
\end{compactenum}
Note that the third of these operations is just the map $\Psi$ 
``turned upside down''.
It is not hard to check that (\ref{eq:longsum})
can be replaced by the recursive definition
% \begin{equation}
% \label{eq:shortinverse}
% (\alpha_2 f)(x) = f(x) + 
% \max \{ (\alpha_2 f)(y): \ y \in \hat{P}, \ y \coveredby x \}
% \end{equation}
% and that likewise (\ref{eq:longsum})
% can be replaced by
\begin{equation}
\label{eq:shortsum}
(\alpha_3 f)(x) = f(x) + 
\max \{ (\alpha_3 f)(y): \ y \in \hat{P}, \ y \covers x \}
\end{equation}
which turns out to be the form most suitable for lifting
to the birational setting.

It readily follows from Theorem~\ref{thm:alphas} below (by tropicalization) 
that $\rowP = \alpha_1 \circ \alpha_3 \circ \alpha_2$.
Moreover, if we make $\alpha_1$ homogeneous
by redefining $\alpha_1(f)$ so that
$(\alpha_1(f))(x)$ is $-f(x)$ instead of $1-f(x)$,
then the composition of the three maps gives $\rowH$.

%% Give numerical example of thm:alphas with [2] x [2].

\begin{theorem}
\label{thm:alphas}
For any poset $P$, the birational rowmotion operation $\rowB$ 
coincides with the composition $\alpha_1 \circ \alpha_3 \circ \alpha_2$,
where $\alpha_1, \alpha_2, \alpha_3: \R^P \rightarrow \R^P$ are defined by
\begin{eqnarray}
(\alpha_1 f)(x) & = & 1/f(x), \label{eq:alph1}\\
(\alpha_2 f)(x) & = & 
\bigsig^{\psum} \{ f(x)/f(y): \ y \in x^- \}, \ {\rm and} \label{eq:alph2}\\
(\alpha_3 f)(x) & = & 
f(x) \ \bigsig^{+} \{ (\alpha_3 f)(y): \ y \in x^+ \}\label{eq:alph3},
\end{eqnarray}
where $x^+$ denotes $\{ y \in \hat{P}: \ y \covers x \}$
and $x^-$ denotes $\{ y \in \hat{P}: \ x \covers y \}$.
(Note that definition (\ref{eq:alph3}) is recursive.)
\end{theorem}

\begin{proof}
We wish to prove that $(\alpha_1 \alpha_3 \alpha_2 f)(x) = (\rowB f)(x)$
for all $x$ in $P$.
We prove the claim by backward induction in the poset $P$,
i.e., from top to bottom.
The base case $x = \hat{1}$ is trivial
(both sides of the equation equal 1),
so we need only prove the induction step.
Assume that every $y \in x^+$ satisfies the induction hypothesis.

We have from equation \eqref{eq:alph2} 
$$(\alpha_2 f)(x) = \bigsig^{\psum} \{ f(x)/f(y): y \in x^- \}
= f(x) \bigsig^{\psum} \{ 1/f(y): y \in x^- \}.$$
Next, applying $\alpha_3$ to $\alpha_2 f$ using
\eqref{eq:alph3} we have
\begin{eqnarray*}
(\alpha_3 \alpha_2 f)(x) & = & 
(\alpha_2 f)(x) \bigsig^{+} \{(\alpha_3 \alpha_2 f)(y): y \in x^+ \} \\
& = & f(x) \bigsig^{\psum} \{ 1/f(y): y \in x^- \} 
\bigsig^{+} \{(\alpha_3 \alpha_2 f)(y): y \in x^+ \} \\
& = & f(x) \frac{\bigsig^{\psum} \{ 1/f(y): y \in x^- \}} 
{\bigsig^{\psum} \{(\alpha_1 \alpha_3 \alpha_2 f)(y): y \in x^+ \}}
\end{eqnarray*}
where the last line follows from 
(\ref{eq:reciprocity}) and (\ref{eq:alph1}).
Now applying $\alpha_1$ to both sides we get
\begin{align*}
(\alpha_1 \alpha_3 \alpha_2 f)(x) &= 
\frac{\bigsig^{\psum} \{(\alpha_1 \alpha_3 \alpha_2 f)(y): y \in x^+ \}}
{f(x) \bigsig^{\psum} \{ 1/f(y): y \in x^- \}}\\
&=\frac{1}{f(x)}\bigsig^{\psum} \{(\alpha_1 \alpha_3 \alpha_2 f)(y): y \in
x^+ \}\bigsig^{+}\{f(y): y \in x^-\}.
\end{align*}
But by the induction hypothesis, $(\alpha_1 \alpha_3 \alpha_2 f)(y)$
for $y \in x^+$ is just $(\rowB f)(y)$, so we get
\[
(\alpha_1 \alpha_3 \alpha_2 f)(x) = 
\frac{1}{f(x)}\bigsig^{\psum} \{(\rowB f)(y): y \in
x^+ \}\bigsig^{+}\{f(y): y \in x^-\}=(\rowB f)(x)
\]
\end{proof}

\end{section}

\begin{section}{Symmetric group actions}

{\sf{Move this remark someplace else:}}
It should be noted
while the leftmost file and top rank each consist 
of just a single element of $P$,
the two associated toggle-operations on $J(P)$,
viewed as permutations of $J(P)$, do not have the same cycle-structure:
the former acts nontrivially on many elements of $J(P)$
while the latter acts nontrivially on only two of them.

\label{sec:actions}

\begin{subsection}{Rank-toggling and rowmotion}

\label{subsec:rankaction}

{\sf{This subsection needs to be written.}}

% I wonder if we can get a homomesy for rowmotion from Darij's construction.

\end{subsection}

\begin{subsection}{File-toggling and promotion}

Recall that $P = [a] \times [b]$ can be partitioned into files
numbered 1 through $n-1$ from left to right.
Given $f: \hat{P} \rightarrow \R^+$
with $f(\hat{0}) = f(\hat{1}) = 1$,
let $p_i$ ($1 \leq i \leq n-1$) be the product of the numbers $f(x)$
with $x$ belonging to the $i$th file of $P$,
let $p_0 = p_n = 1$,
and for $1 \leq i \leq n$ let $q_i = p_i/p_{i-1}$.
Call $q_1,\dots,q_n$ the {\em quotient sequence} associated with $f$,
and denote it by $Q(f)$.
This is analogous to the difference sequence introduced in~\cite{PR13}.
Note that the product $q_1 \cdots q_n$ telescopes to $p_n/p_0=1$.
For $i$ between 1 and $n-1$,
let $\phi_i^*$ be the product of the commuting involutions $\phi_x$
for all $x$ belonging to the $i$th file.
Lastly, given a sequence of $n$ numbers $w = (w_1,\dots,w_n)$,
and given $1 \leq i \leq n-1$, define 
$\sigma_i (w) = (w_1,\dots,w_{i-1},w_{i+1},w_{i},w_{i+2},\dots,w_n)$;
that is, $\sigma_i$ switches the $i$th and $i+1$st entries of $w$.

\begin{lemma}
\label{lem:swap}
For all $1 \leq i \leq n-1$,
$$Q(\phi_i^* f) = \sigma_i Q(f).$$
That is, toggling the $i$th file of $f$
swaps the $i$th and $i+1$st entries
of the quotient sequence of $f$.
\end{lemma}

\begin{proof}
Let $f'=\phi_i^* f$,
let $p'_i$ ($1 \leq i \leq n-1$) be the product of the numbers $f'(x)$
with $x$ belonging to the $i$th file of $P$,
let $p'_0 = p'_n = 1$,
and for $1 \leq i \leq n$ let $q'_i = p'_i/p'_{i-1}$.
We have $p'_j = p_j$ for all $j \neq i$
(since only the values associated with elements of $P$
of the $i$th file are affected by $\phi_i$),
so we have $q'_j = p'_i/p'_{i-1} = p_i/p_{i-1} = q_j$ 
for all $j$ other than $i$ and $i+1$.
The product $q'_1 \cdots q'_n$ telescopes to 1 as before.
The lemma asserts that $q'_i = q_{i+1}$ and $q'_{i+1} = q_i$.
It suffices to prove just one of the two assertions,
since we know ahead of time that $q'_i q'_{i+1} = q_i q_{i+1}$.
% Is this sufficiently clear?
Expressed in terms of the $p_j$'s, the assertion $q'_i = q_{i+1}$ 
amounts to the claim $p'_{i}/p'_{i-1} = p_{i+1}/p_{i}$,
or equivalently $p_{i} p'_{i} = p'_{i-1} p_{i+1}$,
which is just the claim $p_{i} p'_{i} = p_{i-1} p_{i+1}$.

At this point we write $p_{i} p'_{i}$ as the product of
$f(x) f'(x)$ as $x$ varies over the $i$th file of $P$.
For each $x$ in the $i$th file of $P$, $f(x) f'(x) = L_x R_x$
where $L_x = \bigsig^{+} \{f(w): \ w \coveredby x\}$
and $R_x = \bigsig^{\psum} \{f(y): \ y \covers x\}$ (with $w,y \in \hat{P}$).
Now we note a key property of the structure of $P = [a] \times [b]$:
if $x^+$ and $x^-$ are vertically adjacent elements of a given file, 
with $x^+$ above $x^-$,
the $w$'s that contribute to $L_{x^+}$ are precisely
the $y$'s that contribute to $R_{x^-}$.
So, when we take the product of $f(x) f'(x) = L_x R_x$
over all $x$ in the $i$th file, the factors $L_{x^+}$ and $R_{x^-}$ combine
to give $\prod_w f(w)$
where $w$ varies over the elements satisfying $x^- \coveredby w \coveredby x^+$
(here we are using the identity (\ref{eq:duality})).
The only factors that do not combine in this way are
$L_x$ where $x$ is the bottom element of the $i$th file
and $R_x$ where $x$ is the top element of the $i$th file.
Both of these factors can be written in the form $f(z)$
where $z$ is a single element of $\hat{P}$
belonging to either the $i-1$st or $i+1$st file.
By examining cases, it is easy to check
that every element of the $i-1$st file or $i+1$st file
makes a single multiplicative contribution,
so that $p_{i} p'_{i}$ is the product of $f(z)$
as $z$ varies over the union of the $i-1$st and $i+1$st files of $P$.
But this product is precisely $p_{i-1} p_{i+1}$.
So we have proved that $p_{i} p'_{i} = p_{i-1} p_{i+1}$, 
which concludes the proof.
\end{proof}

{\sf{Perhaps the rest of this subsection should be moved elsewhere.}}

Recalling that $\proB$ is the composition
$\phi_{n-1}^* \circ \cdots \circ \phi_{1}^*$,
we have:

\begin{corollary}
\label{cor:shift}
$Q(\proB f)$ is the leftward cyclic shift of $Q(f)$.
\end{corollary}

\begin{proof}[Proof of Theorem~\ref{thm:prod}]
For each $i$ between 1 and $n-1$, view $q_i$ as a function of $f$.
Corollary~\ref{cor:shift} tells us that the values
$q_i(\proB^0 f), q_i(\proB^1 f), q_i(\proB^2 f), \dots q_i(\proB^{n-1} f)$
are respectively equal to the numbers
$q_{i}(f),q_{i+1}(f),\dots,q_{i-1}(f)$, which multiply to 1.
Therefore $q_i$ (viewed as a function of $f$)
is multiplicatively homomesic under the action of $\proB$
(with average value 1 on all orbits),
for all $1 \leq i \leq n-1$.
Therefore the same is true of $p_1 = q_1$,
$p_2 = q_1 q_2$, $p_3 = q_1 q_2 q_3$, etc.
Hence $p_1 p_2 \cdots p_{n-1} = |f|$ is also
multiplicatively 1-homomesic, as claimed.
\end{proof}

The preceding argument does more than confirm Theorem~\ref{thm:prod}. 
If we define $|v|_i$ to be the quantity $p_i(f)$
(where as earlier $f$ and $v$ are two different symbols 
for the same mathematical object),
the argument tells us that the $n-1$ functions $v \mapsto |v|_i$
have the property that
$$\prod_{k=0}^{n-1} |\proB^k (v)|_i = 1.$$
(The original theorem asserts this just 
for the product $|v|_1 |v|_2 \cdots |v|_{n-1} = |v|$.)
%% Fold the preceding sentence into the article differently

Also note that, although the theorem asserts homomesy
only for $v$'s in $(\R^+)^P$,
the same holds for every $v$ in a dense open subset of $\R^P$,
whose complement consists of points
for which the orbit $v,\proB(v),\dots,\proB^{n-1}(v)$
is not well-defined because of some denominator vanishing.
The same is true for $\C^P$.

As a final remark, it should be mentioned that
although the operations $\phi_i^*$ generate an $S_n$-action 
on the $Q$-vectors associated with arrays $f$,
they do not generate an $S_n$-action on the arrays $f$ themselves;
for instance, in the case $P = [2] \times [2]$,
% This example needs to be fixed, now that I've redefined the operations.
$(\phi_1^* \phi_{2}^*)^3$ does not send $(.3,.4,.5,.7)$ to 
itself.\footnote{We are grateful to Darij Grinberg 
for pointing this out to us.}
%% though its 6th power does appear to always be the identity.
%% Can we prove this?

\end{subsection}

\end{section}

\begin{section}{Pairing}

{\sf{This section needs to be written.}}

\label{sec:pairing}

\end{section}

\begin{section}{Recombination}

\label{sec:recombine}

%% This section needs a real theorem, with real notation

We have explained rowmotion and promotion
as toggling by ranks (from top to bottom) 
and by files (from left to right).
A different way to understand these operations
is by toggling {\em fibers}\footnote{We prefer
to avoid words like ``row'', ``column'' and ``diagonal''
since each of these words has two different meanings
according to whether one imbeds $[a] \times [b]$ as an rc poset
or as a subposet of the first quadrant.}
of $[a] \times [b]$,
that is, sets of the form 
$[a] \times \{j\}$ or $\{i\} \times [b]$.
We refer to fibers as being positive or negative
according to whether their slope in the rc embedding
is $+1$ or $-1$.

We illustrate these ideas in the context of $P = [3] \times [3]$.
%% Maybe illustrations should be decorated with arrows,
%% as per Tom's illustration.
Here we see two orders in which one can 
toggle all the elements of $P$ to obtain $\row$.
The first is just the standard order for rowmotion 
(from top to bottom rank by rank, from left to right within each rank).
The second toggles 
the elements in the topmost positive fiber from top to bottom,
then the elements in the middle positive fiber from tom to bottom,
and then the elements in the bottommost positive fiber from top to bottom.
Since the element of $P$ marked $3$ in the left frame
neither covers nor is covered by
the element of $P$ marked $4$,
the associated toggles commute,
and the same goes for the two toggles
associated with the elements of $P$ marked $6$ and $7$.
Hence the composite operation on the left (``rowmotion by ranks'')
coincides with the composite operation on the right (``rowmotion by fibers'').
$$
\begin{array}{ccccccccccc}
  &   & 1 &   &   &       &   &   & 1 &   &   \\
  & 2 &   & 3 &   &       &   & 2 &   & 4 &   \\ 
4 &   & 5 &   & 6 & \ \ \ & 3 &   & 5 &   & 7 \\
  & 7 &   & 8 &   &       &   & 6 &   & 8 &   \\
  &   & 9 &   &   &       &   &   & 9 &   &  
\end{array}
$$

There is a similar picture for promotion.
Here we see two other orders in which one can toggle all the elements of $P$.
The first is just the standard order for promotion 
(from left to right file by file, from top to bottom within each file).
The second toggles 
the elements in the topmost positive fiber from left to right, 
then the elements in the middle positive fiber from left to right, 
and then the elements in the bottommost positive fiber from left to right.
As before, the 3 and the 4 can be swapped,
as can the 6 and the 7.
Hence the composite operation on the left (``promotion by files'')
coincides with the composite operation on the right (``promotion by fibers'').
$$
\begin{array}{ccccccccccc}
  &   & 4 &   &   &       &   &   & 3 &   &   \\
  & 2 &   & 7 &   &       &   & 2 &   & 6 &   \\ 
1 &   & 5 &   & 9 & \ \ \ & 1 &   & 5 &   & 9 \\
  & 3 &   & 8 &   &       &   & 4 &   & 8 &   \\
  &   & 6 &   &   &       &   &   & 7 &   &  
\end{array}
$$
Note that in both cases we divide the poset into positive fibers,
and toggle them from top to bottom;
the only difference is whether we toggle the elements within each fiber
from left to right (promotion) or right to left (rowmotion).

%% Discuss more systematically the eight things you can do along these lines
%% which variously give (ordinary or inverse) promotion or rowmotion.
%% Maybe make a table?

Promotion and rowmotion are even more intimately linked
than the preceding discussion might suggest; 
this link manifests itself if we look at the negative fibers.

Here is a partial orbit of $\rowB$ for $P=[3] \times [3]$:
%% Make this a numbered figure
$$
\begin{array}{ccccccccccccccccccc}
  &   &   &a_7&   &   &   &   &   &b_7&   &   &   &   &   &c_7&   &   &     \\[3mm]
  &   &a_4&   &a_8&   &   &   &b_4&   &b_8&   &   &   &c_4&   &c_8&   &     \\[3mm]
f=&a_1&   &a_5&   &a_9&\mR&b_1&   &b_5&   &b_9&\mR&c_1&   &c_5&   &c_9&\dots\\[3mm]
  &   &a_2&   &a_6&   &   &   &b_2&   &b_6&   &   &   &c_2&   &c_6&   &     \\[3mm]
  &   &   &a_3&   &   &   &   &   &b_3&   &   &   &   &   &c_3&   &   &
\end{array}
$$

Now consider the array $g=\Delta f$ formed by recombining
the bottom negative fiber of $f$ (consisting of $a_1,a_2,a_3$),
the middle negative fiber of $\rowB f$ (consisting of $b_4,b_5,b_6$),
and the top negative fiber of $\rowB^2 f$ (consisting of $c_7,c_8,c_9$),
as shown in the left frame of the figure below:
%% Make this a numbered figure
$$
\begin{array}{cccccccccccc}
  &  &   &c_7&   &   &   &   &   &d_7&   &   \\[3mm]
  &  &b_4&   &c_8&   &   &   &c_4&   &d_8&   \\[3mm]
g=&a_1&  &b_5&   &c_9&\mP&b_1&   &c_5&   &d_9\\[3mm]
  &  &a_2&   &b_6&   &   &   &b_2&   &c_6&   \\[3mm]
  &  &   &a_3&   &   &   &   &   &b_3&   &   
\end{array}
$$
We claim that $\proB g$ coincides with the array formed by recombining
the bottom negative fiber of $\rowB f$ (consisting of $b_1,b_2,b_3$),
the middle negative fiber of $\rowB^2 f$ (consisting of $c_4,c_5,c_6$),
and the top negative fiber of $\rowB^3 f$ (consisting of $d_7,d_8,d_9$),
as shown in the right frame of the figure above.
For example, in both cases, 
when one computes $c_5$ in terms of previously-computed entries,
the governing relation is $b_5 c_5 = (b_2 + b_6)(c_4 \psum c_8)$.
% Check that this is right.
%% Give more details of this example, showing the arrays at the point in time 
%% that this particularly toggle is taking place.  

More generally, given $P = [a] \times [b]$
and given an array $f: P \rightarrow (\R^+)^n$,
let $\Delta f: P \rightarrow (\R^+)^n$ be the array
whose $b$ negative fibers, read from bottom to top,
have the same values as the corresponding negative fibers
in $f$, $\rowB f$, $\rowB^2 f$, \dots, $\rowB^{b-1} f$;
that is, for $x=(i,j) \in P$,
put $(\Delta f)(x) = (\rowB^{j-1}f)(x)$.

\begin{theorem}
\label{thm:delta}
$$\Delta \rowB f = \proB \Delta f.$$
That is, $\Delta$ is an equivariant bijection between
the action of rowmotion on $(\R^+)^P$
and the action of promotion on $(\R^+)^P$.
%% Check that this is right.
\end{theorem}

\begin{proof}
We will show that
\begin{equation}
\label{eq:restate}
(\Delta \rowB f)(i,j) = (\proB \Delta f)(i,j)
\end{equation}
for all $(i,j) \in [a] \times [b]$ by left-to-right induction 
(starting with $(a,1)$ and ending with $(1,b)$).
The reader may find it handy to consult the figure
$$
\begin{array}{ccc}
(i+1,j) &       & (i,j+1) \\[3mm]
        & (i,j) &         \\[3mm]
(i,j-1) &       & (i-1,j)
\end{array}
$$
which shows the vicinity of a typical element $x=(i,j)$
in the poset $[a] \times [b]$ (away from the boundary).
In the boundary cases where one or more of the pairs
$(i+1,j),(i,j+1),(i,j-1),(i-1,j)$ does not belong to $[a] \times [b]$,
the pair(s) may be ignored.

Note that $(i,j-1)$ and $(i+1,j)$ are to the left of $(i,j)$.
If we now assume that (\ref{eq:restate}) holds for $(i,j-1)$ 
(which we take to be vacuously true if $(i,j-1) \not\in [a] \times [b]$)
and for $(i+1,j)$,
then 
\begin{equation}
\label{eq:iha}
(\Delta \rowB f)(i,j-1) = (\proB \Delta f)(i,j-1)
\end{equation}
and 
\begin{equation}
\label{eq:ihb}
(\Delta \rowB f)(i+1,j) = (\proB \Delta f)(i+1,j).
\end{equation}

Note that $\rowB$ can be described directly via the recurrence
\begin{equation}
\label{eq:rowrecur}
(\rowB f)(x) \ = \ \frac{1}{f(x)} \ \bigsig^{+} \{f(y): \: y \in x^-\}
\ \bigsig^{\psum} \{(\rowB f)(y): \: y \in x^+\}
\end{equation}
where in lieu of including $\hat{0}$ and $\hat{1}$
we interpret $\bigsig^{\psum}$ and $\bigsig^+$ of the empty set as 1.
We rewrite (\ref{eq:rowrecur}) as
\begin{equation}
\label{eq:rowij}
(\rowB f)(i,j) \ = \ \frac{1}{f(i,j)} 
\ \bigsig^{+} \{f(i-1,j),f(i,j-1)\}
\ \bigsig^{\psum} \{(\rowB f)(i+1,j),(\rowB f)(i,j+1)\},
\end{equation}
where terms $f(\cdot,\cdot)$ and $(\rowB f)(\cdot,\cdot)$
are to be ignored if the arguments 
do not belong to $[a] \times [b]$
(and $\bigsig^{+}$ and $\bigsig^{\psum}$ of the empty set are taken to be 1).
Likewise, promotion can be described by the recurrence
\begin{equation}
\label{eq:proij}
(\proB f)(i,j) \ = \ \frac{1}{f(i,j)} 
\ \bigsig^{+} \{f(i-1,j),(\proB f)(i,j-1)\}
\ \bigsig^{\psum} \{(\proB f)(i+1,j),f(i,j+1)\}.
\end{equation}

From the definition of $\Delta$, we have
\begin{equation}
\label{eq:deltaij}
(\Delta f)(i,j) = (\rowB^{j-1} f)(i,j).
\end{equation}
Hence
\begin{eqnarray*}
(\Delta \rowB f)(i,j) 
& = & (\rowB^{j-1} \rowB f)(i,j) 
\ \mbox{(by (\ref{eq:deltaij}))} \\
& = & (\rowB \rowB^{j-1} f)(i,j) \\
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{(\rowB^{j-1} f)(i-1,j),(\rowB^{j-1} f)(i,j-1)\} \\
& & \ \times \ \bigsig^{\psum} 
    \{(\rowB \rowB^{j-1} f)(i+1,j),(\rowB \rowB^{j-1} f)(i,j+1)\} 
\ \mbox{(by (\ref{eq:rowij}))} \\ 
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{(\rowB^{j-1} f)(i-1,j),(\rowB^{j-1} f)(i,j-1)\} \\
& & \ \times \ \bigsig^{\psum} \{(\rowB^{j} f)(i+1,j),(\rowB^{j} f)(i,j+1)\} .
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
(\proB \Delta f)(i,j) 
& = & \frac{1}{(\Delta f)(i,j)} 
\ \times \ \bigsig^{+} \{(\Delta f)(i-1,j),(\proB \Delta f)(i,j-1)\} \\
& & \ \times \bigsig^{\psum} \{(\proB \Delta f)(i+1,j),(\Delta f)(i,j+1)\} 
\ \mbox{(by (\ref{eq:proij}))} \\
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{ (\rowB^{j-1} f)(i-1,j), (\proB \Delta f)(i,j-1)\} \\
& & \ \times \bigsig^{\psum} \{(\proB \Delta f)(i+1,j), (\rowB^{j} f)(i,j+1) \} 
\ \mbox{(by (\ref{eq:deltaij}))} \\
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{ (\rowB^{j-1} f)(i-1,j), (\Delta \rowB f)(i,j-1)\} \\
& & \ \times \bigsig^{\psum} \{(\Delta \rowB f)(i+1,j), (\rowB^{j} f)(i,j+1) \} 
\ \mbox{(by (\ref{eq:iha}) and (\ref{eq:ihb}))} \\ 
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{ (\rowB^{j-1} f)(i-1,j), (\rowB^{j-2} \rowB f)(i,j-1)\} \\
& & \ \times \bigsig^{\psum} \{(\rowB^{j-1} \rowB f)(i+1,j), (\rowB^{j} f)(i,j+1) \} 
\ (\mbox{by (\ref{eq:deltaij})}) \\
& = & \frac{1}{(\rowB^{j-1} f)(i,j)} 
\ \times \ \bigsig^{+} \{ (\rowB^{j-1} f)(i-1,j), (\rowB^{j-1} f)(i,j-1)\} \\
& & \ \times \bigsig^{\psum} \{(\rowB^{j} f)(i+1,j), (\rowB^{j} f)(i,j+1) \} 
\end{eqnarray*}
Comparing right-hand sides of the equation blocks,
we conclude that $(\Delta \rowB f)(i,j) = (\proB \Delta f)(i,j)$,
which was to be proved.
\end{proof}

For the next theorem, it is helpful to have 
\begin{equation}
\label{eq:deltainv}
(\Delta^{-1} g)(i,j) = (\proB^{-(j-1)} g)(i,j)
\end{equation}
as a companion to (\ref{eq:deltaij}).\footnote{This 
is just a re-indexing of
our original description of the relation
between the functions $f$ and $g$
that preceded the statement and proof
of Theorem~\ref{thm:delta}.)}

\begin{theorem}
For any functional $F$ on $(\R^+)^P$
of the form $Ff = \prod_{x \in P} f(x)^{a_x}$
with $a_x \in \Z$ for all $x \in P$,
$$\prod_{k=0}^{n-1} F(\rowB^k f) = \prod_{k=0}^{n-1} F(\proB^k f).$$
In particular, if $F$ is homomesic with respect to promotion,
$F$ is homomesic with respect to rowmotion.
\end{theorem}

\begin{proof}
First consider $F_x f = f(x)^a$ for some particular $x \in P$, say $x=(i,j)$.
Theorem~\ref{thm:delta} implies $\rowB^k f = \Delta^{-1} \proB^k \Delta f$.
Then setting $g_j = \proB^{-(j-1)} \Delta f$ we have
\begin{eqnarray*}
F_x (\rowB^k f) 
& = & ((\rowB^k f)(x))^a \\
& = & ((\Delta^{-1} \proB^k \Delta f)(x))^a \\
& = & (\proB^{-(j-1)} \proB^k \Delta f)(x))^a 
\ \mbox{(by (\ref{eq:deltainv}))} \\
& = & (\proB^k \proB^{-(j-1)} \Delta f)(x))^a \\
& = & F_x (\proB^k g)
\end{eqnarray*}
so that
$$\prod_{k=0}^{n-1} F_x (\rowB^k f) = \prod_{k=0}^{n-1} F_x (\proB^k g).$$
%% Here's where I give a proof that's more analytic than I'd like.
%% Unfortunately, given that I don't know of a proof of the periodicity
%% of promotion/rowmotion as of late July 2013, I don't see a slick
%% work-around.
Oops; I just discovered that the proof is wrong!

The original sketch said:
``The recombination picture also tells us 
that if we look at any particular poset element $x \in P$,
the values for $f(x)$ that we see as $f$ ranges over the rowmotion orbit 
of some $f_0$ will be the same as
the values for $g(x)$ that we see as $g$ ranges over the promotion orbit
of $g_0 = \Delta f_0$.
In particular, if the geometric mean of $f(x)$ 
as $f$ ranges over the rowmotion orbit of $f_0$
is the same value $c$ for every $f_0$.
then the geometric mean of $g(x)$ 
as $g$ ranges over the rowmotion orbit of $g_0$
is the same value $c$ for every $g_0$.
This applies not just to the elementary functionals 
that send $f$ to $f(x)$ for some $x \in P$
but also to more general functionals 
obtained as products of elementary functionals
or even more generally, ratios of such products.
In this way, we can see that the file-products $q_i$
discussed in the proof of Theorem~\ref{thm:prod}
are homomesic under promotion as well as rowmotion.''
But trying to fill in the details,
I learned that I was missing a key detail.
So, ignore that end-of-proof symbol:
\end{proof}

This recombination picture works equally well 
in the piecewise-linear setting and the combinatorial setting,
and in the context of rowmotion and promotion on $J(P)$
our $\Delta$ is essentially the same equivariant bijection
as the one given in Theorem 5.4 of~\cite{SW12},
which Striker and Williams use to show
that rowmotion and promotion, viewed as permutations of $J(P)$,
have the same cycle-structure.
%% Do they explicitly do this?

% To see why recombination holds let us index the negative fibers 
% $1$ to $b$ from bottom to top, and let $d_i$ denote the action 
% of toggling the $i$th negative fiber from the top down.  
% Then left-to-right promotion can be written $d_1 d_2 \cdots d_b$ 
% and rowmotion can be written $d_b d_{b-1} \cdots d_1$.  
% Exactly paralleling theorem 5.4 of Striker and Williams 
% we get a conjugacy from rowmotion to promotion as 
% $D=\prod_{i=1}^{b-1}\prod_{j=i}^1 d_j^{-1}$.  
% This can be seen as reversing promotion on the bottom $b-1$ negative fibers, 
% then the bottom $b-2$, $\ldots$, and finally just the bottom fiber.  
% Reading consecutive fibers from consecutive iterations of rowmotion 
% achieves the same effect.

%% Tom says he felt the need for an example here.  He worked out
%% the case of B_2, and showed that it's a cone in R^4 of a cone
%% over a square.  Maybe include this once I understand it?

%% Save this remark for a later article in which we prove it!
% It is also worth pointing out that
% this recombination picture works for all rc-embedded posets,
% and not just for products of two chains.
%% Are we quite sure that it works even for improperly rc-embedded posets?

\end{section}

\begin{section}{Tropicalization}
\label{sec:tropic}

Theorem~\ref{thm:prod} is nothing more than 
a complicated identity involving the operations
of multiplication, division, addition, and parallel addition.
As such, Theorem~\ref{thm:prod} can be tropicalized to yield
an identity involving the operations
of addition, subtraction, min, and max.
The resulting identity is nothing other than Theorem~\ref{thm:sumh}.

\begin{lemma}
\label{lem:tropic}
Suppose $E_1(t_1,\dots,t_r)$ and $E_2(t_1,\dots,t_r)$ 
are subtraction-free rational functions,
expressed in terms of the number 1 
and the operations $+$, $\psum$, $\times$, and $\div$,
such that $E_1(t_1,\dots,t_r) = E_2(t_1,\dots,t_r)$ 
for all $t_1,\dots,t_r$ in $\R^+$.
Let $e_i(t_1,\dots,t_r)$ (for $1 \leq i \leq 2$)
be the result of replacing 
$1$, $+$, $\psum$, $\times$, and $\div$
by $0$, $\min$, $\max$, $+$, and $-$, respectively.
Then $e_1(t_1,\dots,t_r) = e_2(t_1,\dots,t_r)$ 
for all $t_1,\dots,t_r$ in $\R$.
\end{lemma}

\begin{proof}
Write $e_i(t_1,\dots,t_m) = -\lim_{N \rightarrow \infty}
\frac{1}{N} \log E_i(e^{-Nt_1},\dots,e^{-Nt_m})$.\footnote{We 
are indebted to Colin McQuillan and Will Sawin 
for clarifying this point; see~\cite{MO13}.}
%% Check!  I got the sign wrong; maybe I got other things wrong too.
\end{proof}

It is worth observing that the converse of the Lemma is false.
That is, there are tropical identities like
$\max(t,t)=t$ whose ``detropicalization'' is false.
Thus, while Theorem~\ref{thm:sumh} is a corollary of Theorem~\ref{thm:prod},
Theorem~\ref{thm:prod} cannot be derived as a corollary of Theorem~\ref{thm:sumh},
at least not using any methods we are aware of.

At the same time, we should mention that in a certain sense,
Theorem~\ref{thm:sumh} can be proved 
without relying on Theorem~\ref{thm:prod}.
Specifically, one can tropicalize each of the steps 
in the proof of Theorem ~\ref{thm:prod},
using (for instance) the identity
\begin{equation}
\min(x,y)+\max(x,y) = x+y
\end{equation}
in place of the identity (\ref{eq:duality}).

At the end of section~\ref{sec:proof},
we proved a refined version of Theorem~\ref{thm:prod},
and this tropicalizes, yielding a refined version of Theorem~\ref{thm:sumh}:
Using our usual conflation of $f$ and $v$,
let us redefine $|v|_i$ to be the sum of $f(x)$
over all $x$ belonging to the $i$th file of $P$.
Then the $n-1$ functions $v \mapsto |v|_i$
have the property that for every $v$ in $\R^P$, 
$$\sum_{k=0}^{n-1} |\proH^k (v)|_i = 0.$$

These $n-1$ functions are also homomesic for $\proP$,
although now each file has its own (non-zero) value $c_i$
with the property that the average of $|v|_i$ over every orbit is 
$ai/n$ if $i \leq b$ and $b(n-i)n$ if $i > b$.
%% Give proof?
Thus we obtain a refined version of Theorem~\ref{thm:sum}.

Lastly, if we specialize to the vertices of $\cO(P)$,
we obtain a refined version of Theorem~\ref{thm:card}
(also found in~\cite{PR13}):
for every ideal $I$ in $J([a] \times [b])$,
if we let $P_i$ denote the $i$th file of $P = [a] \times [b]$, then
$$\frac{1}{n} \sum_{k=0}^{n-1} |\pro^k (I) \cap P_i| = 
\left\{ \begin{array}{cl} \frac{ai}{n} & \mbox{if $i \leq b$,} \\[.3mm]
\frac{b(n-i)}{n} & \mbox{if $i > b$.} \end{array} \right.$$

\end{section}

\begin{section}{More homomesies} 
\label{sec:opposites}

{\sf{This section needs to be written.}}

Recall that $P_i$ ($1 \leq i \leq a+b-1$)
denotes the $i$th file of $P = [a] \times [b]$.
We have shown that for $P = [a] \times [b]$,
the functions $$p_i: f \mapsto \sum_{\mbox{$x \in P_i$}} f(x)$$
(and their analogues in the birational setting)
are homomesic under the action of rowmotion and promotion.
These are not, however, the only combinations of
the local evaluation operations $f \mapsto f(x)$ ($x \in P$)
that exhibit homomesy.
Specifically, we now show that for all $a,b$,
the functions $f \mapsto f(x) + f(x')$ are homomesic
under rowmotion and promotion,
where $x=(i,j)$ and $x'=(i',j')$ are opposite elements of $P$,
that is, $i+i'=a+1$ and $j+j'=b+1$.
(In the birational setting, we mean $f \mapsto f(x) f(x')$.)

\end{section}

\begin{section}{No more homomesies}
\label{sec:bound},

{\sf{This section needs to be written.}}

\end{section}

\begin{section}{Directions for future work}
\label{sec:future}

\begin{subsection}{Other invariants}

{\sf{This subsection needs to be junked, although parts of it
might belong elsewhere in the article.}}

We have mentioned that 
for piecewise-linear rowmotion and promotion, the quantity
$$\min \{f(y) - f(x): \: x,y \in \hat{P}, \ x \coveredby y\}$$
is invariant, and that for birational rowmotion and promotion,
the quantity 
$$\bigsig^+ \{f(x) / f(y): \: x,y \in \hat{P}, \ x \coveredby y\}$$
is invariant.
There is a sense in which the space of invariant functionals
and the space of homomesic functionals
are dual to one another (see section 2.2 of~\cite{PR13}),
so it is natural to complement our program 
of finding all the homomesic functionals for $\proB$ and $\rowB$
by the program of finding all the invariants.
In the article~\cite{KB95}, Kirillov and Berenstein show that
when $P$ is the triangle poset mentioned in section~\ref{sec:PL},
there is more than one invariant functional;
see for instance Lemma 3.1 of their paper
and the paragraph that follows its proof.
Berenstein has mentioned (in private communication)
that the motivation for the Kirillov-Berenstein piecewise-linear invariants
comes from the birational setting, which admits birational counterparts
of the piecewise-linear invariants;
this seems likely to be the case for rectangle posets as well.

\end{subsection}

\begin{subsection}{Other actions}

We have seen in the case of $P = [a] \times [b]$
that rowmotion (toggling all the elements from top to bottom)
and promotion (toggling all the elements from left to right)
have the same cycle structure as permutations of $J(P)$;
this is a special case of Theorem 5.4 of~\cite{SW12},
valid for all rc-embedded ranked posets.
Furthermore, the arguments of~\ref{subsec:rowmotion},
although stated and proved in the case $[a] \times [b]$,
readily extend as well to rc-embedded ranked posets $P$,
showing that the homomesies of rowmotion
are all homomesies of promotion 
and vice versa.\footnote{There is some fine print here:
the homomesies in question must be generated by
local evaluation operations $f \mapsto f(x)$ ($x \in P$),
where ``generated'' must be interpreted
additively in the piecewise-linear case and
multiplicatively in the birational case.}
%% This fine print will be especially important if we expand the article
%% to include homomesy results related to the chain polytope, i.e.,
%% homomesy results generalizing cardinality of antichains.

Since top-to-bottom toggling and left-to-right toggling
give operations that in $J(P)$ are indistinguishable vis-a-vis their cycle-structure
and in all three settings (combinatorial, piecewise-linear, and birational)
are indistinguishable vis-a-vis their homomesies,
one might wonder if this indistinguishability remains true
if one toggles the elements of $P$ in an arbitrary order
(toggling each element once).

Here the answer is ``no'', and we can see this
even in as simple a case as $P = [2] \times [2] = \{w,x,y,z\}$.
Although $\tau = \tau_y \tau_z \tau_x \tau_w: J(P) \rightarrow J(P)$ 
(being a permutation of a finite set) is of finite order, 
we will see shortly that
$\phiP = \phi_y \phi_z \phi_z \phi_x: \cO(P) \rightarrow \cO(P)$ 
is of infinite order,
which implies that the homogeneous version $\phiH$
and the birational version $\phiB$ of this map
are of infinite order as well.

To see that $\phiP$ is not of finite order, consider the rational points
of the form $(1/d,k/d,k/d,k/d)$ in $\cO$.
Then it can be checked that as long as $k$ is between 4 and $d$,
the map $\phiP$ sends $v_k=(1,k,k,k)/d$ successively to
$$
\begin{array}{ll}
(k-1,k-1,d-1,d)/d,            & (0,d-k+1,0,d-1)/d,  \\
(0,k-2,k-1,k-1)/d,            & (k-2,k-1,d-1,d)/d,  \\
(1,d-k+2,1,d-1)/d,            & (0,k-3,k-3,k-2)/d,  \\
(k-3,k-2,d,d)/d,              & (1,d-k+3,1,d)/d,    \\
(0,k-3,k-4,k-3)/d,            & (k-4,k-4,d-1,d-1)/d,\\ 
(0,d-k+3,1,d)/d,\ \mbox{and}  & (1,k-2,k-2,k-2)/d=v_{k-2},
\end{array}
$$
so that $\phiP^{12}$ sends $v_{k}$ to $v_{k-2}$.
Therefore there are rational points in $\cO(P)$ whose orbits
are arbitrarily large.

On the other hand, it appears that some form of homomesy applies
even to maps of infinite order.
Empirical evidence suggests that for the action of $\phiP$ 
on $\cO(P) \cap \Q^P$ (which consist of finite orbits of unbounded size),
the function $f: v \mapsto v_1 - v_2 - v_3 + v_4$
is homesic with average 0 on all orbits.
Indeed, we conjecture that even if $v$ has infinite $\phiP$-orbit,
$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1} f(\phiP^k(v)) = 0$.

\end{subsection}

\begin{subsection}{Other homomesies}

A different sort of homomesy, 
already studied in the combinatorial literature, relates to
cardinality of antichains rather than cardinality of order ideals.
Instead of $\alpha_3 \circ \alpha_2 \circ \alpha_1: J(P) \rightarrow J(P)$
we can consider
$\alpha_2 \circ \alpha_1 \circ \alpha_3: \cA(P) \rightarrow \cA(P)$.
Various conjectures of Panyushev~\cite{Pan08}, 
some of which were proved by Armstrong et al.~\cite{AST11},
assert that the function $A \mapsto |A|: \ \cA(P) \rightarrow \Z$
is homomesic under the action of $\alpha_2 \circ \alpha_1 \circ \alpha_3$,
for $P$ a root poset.
Experiments suggest that for $P = [a] \times [b]$,
this homomesy lifts to the piecewise-linear setting.
Putting it differently, we use the transfer map
to move the dynamics on $\cO$ to $\cC$,
where there are different local evaluation operations
(which cannot be expressed as linear combinations of
the local evaluation operations for $\cO$),
and different homomesies.
It is natural to suspect that what works for piecewise-linear setting
also works in the birational setting,
if one detropicalizes the maps in the natural way.
We have checked that this works for some small values of $a$ and $b$.

\end{subsection}

\begin{subsection}{Other posets}

Experiments also suggest that much of this theory carries over
from the poset $[a] \times [b]$ (a minuscule poset of type A)
to other minuscule posets, and root posets as well.
For instance, Shahrzad Haddadan (in work to be described elsewhere)
has shown that if $P$ is a root poset of type A,
then the functional $I \mapsto |I_+| - |I_-|$
is homomesic for the action of rowmotion and promotion on $J(P)$,
where $I_+$ and $I_-$ denote the elements of $I$
of even and odd rank, respectively.

\end{subsection}

\end{section}

%%% BIBLIOGRAPHY BELOW HERE
\bibliographystyle{abbrvnat}
% use the following instead if you encounter problems 
%\bibliographystyle{alpha}
\bibliography{sample}
\label{sec:biblio}

\begin{thebibliography}{10}

% \bibitem[Arm06]{Arm06}
% D.~Armstrong, \emph{Generalized noncrossing partitions and combinatorics of
%   {C}oxeter groups}, Mem. Amer. Math. Soc. \textbf{202} (2006), no.~949.

\bibitem[AST11]{AST11}
D.~Armstrong, C.~Stump and H.~Thomas, \emph{A uniform bijection between
nonnesting and noncrossing partitions}, preprint, available at 
\url{http://arxiv.org/abs/1101.1277} (2011).

\bibitem[BS74]{BS74}
A.~Brouwer and A.~Schrijver, \emph{On the period of an operator, defined on
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\bibitem[CF95]{CF95}
P.~Cameron and D.~G.~Fon-Der-Flaass, \emph{Orbits of Antichains
Revisited}, Europ. J. Comb. \textbf{16} (1995), 545--554.  

\bibitem[Fon93]{Fon93}
D.~G.~Fon-Der-Flaass, \emph{Orbits of Antichains
in Ranked Posets}, Europ. J. Comb. \textbf{14} (1993), 17--22.  

\bibitem[HP10]{HP10}
A.~Holroyd and J.~Propp, \emph{Rotor walks and Markov chains}, in 
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\bibitem[H+08]{H+08}
A.~Holroyd, L.~Levine, K.~M\'esz\'aros,
Y.~Peres, J.~Propp, and D.~Wilson, \emph{Chip-Firing and Rotor-Routing on Directed Graphs}, 
Progress in Probability, \textbf{60} (2008), 331--364.  

\bibitem[KB95]{KB95}
A.~N.~Kirillov and A.~D.~Berenstein,
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and combinatorics of Young tableaux}, Algebra i Analiz {\bf 7} (1995), 92-–152;
English transl., St.\ Petersburg Math.\ J.\ {\bf 7} (1996), no. 1, 77–127;
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\bibitem[MO13]{MO13}
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\bibitem[Pan08]{Pan08}
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\bibitem[PR13]{PR13}
J.~Propp and T.~Roby, \emph{Homomesy in products of two chains},
submitted; \href{jamespropp.org/propp-roby.pdf}.
%% Revise reference later

\bibitem[RSW04]{RSW04}
V.~Reiner, D.~Stanton, and D.~White, \emph{The cyclic sieving phenomenon},
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%\bibitem[Sta09]{Sta09}
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\end{thebibliography}

\end{document}

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%\bibitem[Sta09]{Sta09}
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