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\author{Darij Grinberg (MIT) \\
\textit{joint work with Tom Roby (UConn)}}
\title[Birational rowmotion]{The order of birational rowmotion}

\date{17 June 2014 \\ Arbeitsgemeinschaft Diskrete Mathematik, Vienna}

\begin{document}

\frame{\titlepage
\textbf{slides:} {\red \url{http://mit.edu/~darij/www/algebra/vienna2014.pdf}} \\
\textbf{paper:} {\red \url{http://mit.edu/~darij/www/algebra/skeletal.pdf}} or {\red \tt arXiv:1402.6178v3}
}

\begin{frame}
\frametitle{\ \ \ \ Introduction: Posets}

\begin{itemize}

\item A \textbf{poset} (= partially ordered set) is a set $P$ with a reflexive, transitive and antisymmetric relation.

\item We use the symbols $<$, $\leq$, $>$ and $\geq$ accordingly.

\item We draw posets as Hasse diagrams:
\[
\begin{array}{l|r}
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\left(2,1\right) \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] \\
& \left(1,1\right) &
}
\phantom{xx}
&
\phantom{xx}
\xymatrixrowsep{1.5pc}
\xymatrix{
\delta \ar@{-}[rd] & & \\
& \gamma \ar@{-}[ld] \ar@{-}[rd] & \\
\alpha & & \beta
}
\end{array}
\]

\item We only care about finite posets here.

\item We say that $u \in P$ \textbf{is covered by} $v \in P$ (written $u \lessdot v$) if we have $u < v$ and there is no $w \in P$ satisfying $ u < w < v$.
\item We say that $u \in P$ \textbf{covers} $v \in P$ (written $u \gtrdot v$) if we have $u > v$ and there is no $w \in P$ satisfying $ u > w > v$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Introduction: Posets}

\begin{itemize}

\item An \textbf{order ideal} of a poset $P$ is a subset $S$ of $P$ such that if $v \in S$ and $w \leq v$, then $w \in S$.

\item Examples (the elements of the order ideal are marked in red):
\[
\begin{array}{l|r}
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\phantom{xx}
&
\phantom{xx}
\xymatrixrowsep{1.5pc}
\xymatrix{
\delta \ar@{-}[rd] & & \\
& \red{\gamma} \ar@{-}[ld] \ar@{-}[rd] & \\
\red{\alpha} & & \red{\beta}
}
\end{array}
\]
\hrulefill
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& 3 \ar@{-}[ld] \ar@{-}[rd] & \red{5} \ar@{-}[rd] & 6 \ar@{-}[d] & \red{7} \ar@{-}[ld] \\
1 & & \red{2} & \red{4} & &
}
\]

\item We let $J(P)$ denote the set of all order ideals of $P$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion}

\begin{itemize}

\item \textbf{Classical rowmotion} is the rowmotion studied by Striker-Williams (\texttt{arXiv:1108.1172}). It has appeared many times before, under different guises:
\begin{itemize}
\item Brouwer-Schrijver (1974) (as a permutation of the antichains),
\item Fon-der-Flaass (1993) (as a permutation of the antichains),
\item Cameron-Fon-der-Flaass (1995) (as a permutation of the monotone Boolean functions),
\item Panyushev (2008), Armstrong-Stump-Thomas (2011) (as a permutation of the antichains or ``nonnesting partitions'', with relations to Lie theory).
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: the standard definition}

\begin{itemize}

\item Let $P$ be a finite poset. \\
\textbf{Classical rowmotion} is the map $\mathbf{r} : J(P) \to J(P)$ which sends
\only<1>{{\red{every order ideal $S$}}}
\only<2-4>{every order ideal $S$ }to the order ideal obtained as follows: \\
\only<1>{Let $M$ be the set of minimal elements of the complement $P \setminus S$.}
\only<2-3>{{\red{Let $M$ be the set of minimal elements of the complement $P \setminus S$.}}}
\only<4>{Let $M$ be the set of minimal elements of the complement $P \setminus S$.} \\
\only<1-3>{Then, $\mathbf{r}(S)$ shall be the order ideal generated by these elements (i.e., the set of all $w\in P$ such that there exists an $m\in M$ such that $w\leq m$).}
\only<4>{{\red{Then, $\mathbf{r}(S)$ shall be the order ideal generated by these elements (i.e., the set of all $w\in P$ such that there exists an $m\in M$ such that $w\leq m$).}}}

\end{itemize}

{\bf Example:}

{\only<1>{ Let $S$ be the following order ideal ($\newmoon$ = inside order ideal):
\[
\xymatrixcolsep{1.5pc}
\xymatrix{
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\newmoon \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \\
& \newmoon & & \newmoon &
}
\] }}

{\only<2>{ Mark $M$ (= minimal elements of complement) {\blue blue}.
\[
\xymatrixcolsep{1.5pc}
\xymatrix{
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\newmoon \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \\
& \newmoon & & \newmoon &
}
\] }}

{\only<3>{ Forget about the old order ideal:
\[
\xymatrixcolsep{1.5pc}
\xymatrix{
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\fullmoon \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \\
& \fullmoon & & \fullmoon &
}
\] }}

{\only<4>{ $\mathbf r(S)$ is the order ideal generated by $M$ (``everything below $M$''):
\[
\xymatrixcolsep{1.5pc}
\xymatrix{
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\fullmoon \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \ar@{-}[rd] & & \blue \newmoon \ar@{-}[ld] \\
& \blue \newmoon & & \blue \newmoon &
}
\] }}

\pause \pause \pause

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: properties}

Classical rowmotion is a permutation of $J(P)$, hence has finite order. This order can be fairly large.

\pause
However, \textbf{for some types of $P$}, the order can be explicitly computed or bounded from above.

See Striker-Williams for an exposition of known results.

\begin{itemize}

\item If $P$ is a $p \times q$-rectangle:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
\left(2,1\right) \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \\
& \left(1,1\right) & &
}
\]
(shown here for $p=2$ and $q=3$), then $\ord\left(\mathbf{r}\right) = p+q$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: properties}

{\bf Example:}

\begin{overprint}
\onslide<1>
Let $S$ be the order ideal of the $2\times 3$-rectangle given by:
\onslide<2>
$\mathbf{r}(S)$ is
\onslide<3>
$\mathbf{r}^2(S)$ is
\onslide<4>
$\mathbf{r}^3(S)$ is
\onslide<5>
$\mathbf{r}^4(S)$ is
\onslide<6>
$\mathbf{r}^5(S)$ is
\end{overprint}

\begin{overprint}
\onslide<1>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\onslide<2>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
\left(2,1\right) \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\onslide<3>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & {\red \left(1,3\right)} \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\onslide<4>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& {\red \left(2,2\right)} \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\onslide<5>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & {\red \left(1,3\right)} \ar@{-}[ld]\\
\left(2,1\right) \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\onslide<6>
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
\end{overprint}

\vspace{0.2cm}

\begin{overprint}
\onslide<6>
which is precisely the $S$ we started with.
\end{overprint}

\vspace{0.2cm}

\begin{overprint}
\onslide<6>
$\ord(\mathbf r) = p+q = 2+3 = 5$.
\end{overprint}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: properties}

Further posets for which classical rowmotion has small order:

(Still see Striker-Williams for references.)

\begin{itemize}

\item If $P$ is a $\Delta$-shaped triangle with sidelength $p-1$:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \\
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\fullmoon & & \fullmoon & & \fullmoon
}
\]
(shown here for $p=4$), then $\ord\left(\mathbf{r}\right) = 2p$ (if $p > 2$).

\item In this case, $\mathbf{r}^p$ is ``reflection in the $y$-axis'' (i.e., the central vertical axis).

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: properties}

Yet further posets for which classical rowmotion has small order:

(Still see Striker-Williams for references.)

\begin{itemize}

\item If $P$ is the poset of all positive roots of a finite Weyl group $W$, then $\mathbf{r}^{2h} = \id$,
where $h$ is the Coxeter number of $W$. (Armstrong-Stump-Thomas, {\tt arXiv:1101.1277v2}.)

\item This includes the triangles from previous slide, but also these kind of beasts:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & & & \fullmoon \ar@{-}[ld] \\
& & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
& & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \\
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\fullmoon & & \fullmoon & & \fullmoon
}
\]
(for $B_3$).
\end{itemize}

\end{frame}


\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: the toggling definition}

There is an alternative definition of classical rowmotion, which splits
it into many little steps.

\begin{itemize}

\item If $P$ is a poset and $v \in P$, then the \textbf{$v$-toggle}
is the map $\mathbf{t}_v : J(P) \to J(P)$ which takes every order
ideal $S$ to:
\begin{itemize}
\item $S \cup \left\{v\right\}$, if $v$ is not in $S$ but all elements of
$P$ covered by $v$ are in $S$ already;
\item $S \setminus \left\{v\right\}$, if $v$ is in $S$ but none of the
elements of $P$ covering $v$ is in $S$;
\item $S$ otherwise.
\end{itemize}

\item Simpler way to state this: $\mathbf{t}_v\left(S\right)$ is:
\begin{itemize}
\item $S \bigtriangleup \left\{v\right\}$ (symmetric difference)
if this is an order ideal;
\item $S$ otherwise.
\end{itemize}
(``Try to add or remove $v$ from $S$; if this breaks the
order ideal axiom, leave $S$ fixed.'')

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Classical rowmotion: the toggling definition}

\begin{itemize}

\item Let $\left(v_1,v_2,...,v_n\right)$ be a \textbf{linear extension} of $P$; this means a list of all elements of $P$ (each only once) such that $i < j$ whenever $v_i < v_j$.

\item Cameron and Fon-der-Flaass showed that
\[
\mathbf r = \mathbf t_{v_1} \circ \mathbf t_{v_2} \circ ... \circ \mathbf t_{v_n}.
\]

\end{itemize}

\textbf{Example:}

\only<1>{Start with this order ideal $S$: \phantom{$t_{(2,2)}$}
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\only<2>{First apply $\mathbf t_{(2,2)}$, which changes nothing:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\only<3>{Then apply $\mathbf t_{(1,2)}$, which adds $(1,2)$ to the order ideal:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\left(2,1\right)} \ar@{-}[rd] & & \red{\left(1,2\right)} \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\only<4>{Then apply $\mathbf t_{(2,1)}$, which removes $(2,1)$ from the order ideal:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\left(2,1\right) \ar@{-}[rd] & & \red{\left(1,2\right)} \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\only<5>{Finally apply $\mathbf t_{(1,1)}$, which changes nothing:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\left(2,1\right) \ar@{-}[rd] & & \red{\left(1,2\right)} \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\only<6>{So this is $\mathbf r(S)$: \phantom{$t_{(2,2)}$}
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\left(2,1\right) \ar@{-}[rd] & & \red{\left(1,2\right)} \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\pause \pause \pause \pause \pause 

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Goals of the talk}

\begin{itemize}

\item define \textbf{birational rowmotion} (a generalization of classical rowmotion introduced by David Einstein and James Propp, based on ideas of Anatol Kirillov and Arkady Berenstein).

\item show how some properties of classical rowmotion generalize to birational rowmotion.

\item ask some questions and state some conjectures.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: definition}

\begin{itemize}

\item Let $P$ be a finite poset. We define $\widehat{P}$ to be the poset obtained by adjoining two new elements $0$ and $1$ to $P$ and forcing
\begin{itemize}
\item $0$ to be less than every other element, and
\item $1$ to be greater than every other element.
\end{itemize}

\end{itemize}

{\bf Example:}
\[
P = \xymatrixrowsep{1.5pc}
\xymatrix{
\delta \ar@{-}[rd] & & \\
& \gamma \ar@{-}[ld] \ar@{-}[rd] & \\
\alpha & & \beta
} \ \ \ \ \ {\red\Longrightarrow} \ \ \ \ \  \widehat P = \xymatrixrowsep{1.5pc}
\xymatrix{
1 \ar@{-}[d] & & \\
\delta \ar@{-}[rd] & & \\
& \gamma \ar@{-}[ld] \ar@{-}[rd] & \\
\alpha \ar@{-}[rd] & & \beta \ar@{-}[ld] \\
& 0 &
}
\]

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: definition}

\begin{itemize}

\item Let $\mathbb K$ be a semifield (i.e., a field minus ``minus'').

\item A \textbf{$\mathbb K$-labelling of} $P$ will mean a function $\widehat P \to \mathbb K$.

\item The values of such a function will be called the \textbf{labels} of the labelling.

\item We will represent labellings by drawing the labels on the vertices of the Hasse diagram of $\widehat P$.

\end{itemize}

{\bf Example:} This is a $\mathbb Q$-labelling of the $2\times 2$-rectangle:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& 14 \ar@{-}[d] & \\
& 10 \ar@{-}[rd] \ar@{-}[ld] & \\
-2 \ar@{-}[rd] & & 7 \ar@{-}[ld] \\
& 1/3 \ar@{-}[d] & \\
& 12 &
}
\]

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: definition}

\begin{itemize}
\item For any $v \in P$, define the \textbf{birational $v$-toggle} as the rational map $T_v : \bbK^\whP \rato \bbK^\whP$ defined by
\begin{equation}
\left(  T_{v}f\right)  \left(  w\right)  =\left\{
\begin{array}{cc}
f\left(  w\right)  ,\ \ \ \ \ \ \ \ \ \ & \text{if }w\neq v;\\
% & \\
\dfrac{1}{f\left(  v\right)  }\cdot\dfrac{\sum\limits_{\substack{u\in
\widehat{P};\\u\lessdot v}}f\left(  u\right)  }{\sum\limits_{\substack{u\in
\widehat{P};\\u\gtrdot v}}\dfrac{1}{f\left(  u\right)  }}
,\ \ \ \ \ \ \ \ \ \ & \text{if }w=v
\nonumber
\end{array}
\right.
\end{equation}
for all $w \in \whP$.
\end{itemize}

\begin{overprint}
\onslide<1>
\begin{itemize}
\item That is,
\begin{itemize}
\item \textbf{invert} the label at $v$,
\item \textbf{multiply} it with the \textbf{sum} of the labels at vertices \textbf{covered by} $v$,
\item \textbf{multiply} it with the \textbf{harmonic sum} of the labels at vertices \textbf{covering} $v$.
\end{itemize}
\end{itemize}

\onslide<2>
\begin{itemize}
\item Notice that this is a {\bf local change} to the label at $v$; all other labels stay the same.
\item We have $T_v^2 = \id$ (on the range of $T_v$), and $T_v$ is a birational map.
\end{itemize}
\end{overprint}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: definition}

\begin{itemize}

\item We define \textbf{birational rowmotion} as the rational map
\[
R := T_{v_1} \circ T_{v_2} \circ ... \circ T_{v_n} : \bbK^\whP \rato \bbK^\whP ,
\]
where $\left(v_1,v_2,...,v_n\right)$ is a linear extension of $P$.

\item This is indeed independent on the linear extension, because:
\pause
\begin{itemize}
\item $T_v$ and $T_w$ commute whenever $v$ and $w$ are incomparable (or just don't cover each other);
\item we can get from any linear extension to any other by switching incomparable adjacent elements.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: example}

{\bf Example:}

Let us ``rowmote'' a (generic) $\bbK$-labelling of the $2\times 2$-rectangle:
\[
\begin{array}{c|c}
\text{poset\ \ \ } & \text{\ \ labelling} \\
\hline
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& 1 \ar@{-}[d] & \\
& (2,2) \ar@{-}[rd] \ar@{-}[ld] & \\
(2,1) \ar@{-}[rd] & & (1,2) \ar@{-}[ld] \\
& (1,1) \ar@{-}[d] & \\
& 0 &
}
\phantom{yyy}
&
\phantom{yyy}
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\end{array}
\]
\pause We have $R = T_{(1,1)} \circ T_{(1,2)} \circ T_{(2,1)} \circ T_{(2,2)}$ (using the linear extension $((1,1),(1,2),(2,1),(2,2))$).

That is, toggle in the order ``top, left, right, bottom''.

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: example}

{\bf Example:}

Let us ``rowmote'' a (generic) $\bbK$-labelling of the $2\times 2$-rectangle:
\[
\begin{array}{c|c}
\text{original labelling }f & 
\only<1>{\text{labelling }T_{(2,2)} f}
\only<2>{\text{labelling }T_{(2,1)}T_{(2,2)} f}
\only<3>{\text{labelling }T_{(1,2)}T_{(2,1)}T_{(2,2)} f}
\only<4>{\text{labelling }T_{(1,1)}T_{(1,2)}T_{(2,1)}T_{(2,2)} f = Rf} \\
\hline
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\phantom{yyy}
&
\phantom{yyy}
\only<1>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \red{\frac{b(x+y)}{z}} \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<2>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\frac{bw(x+y)}{xz}} \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<3>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \red{\frac{bw(x+y)}{yz}} \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<4>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \frac{bw(x+y)}{yz} \ar@{-}[ld] \\
& \red{\frac{ab}{z}} \ar@{-}[d] & \\
& a &
}
}
\end{array}
\]
\only<1>{We are using $R = T_{(1,1)} \circ T_{(1,2)} \circ T_{(2,1)} \circ \red{T_{(2,2)}}$.}
\only<2>{We are using $R = T_{(1,1)} \circ T_{(1,2)} \circ {\red{T_{(2,1)}}} \circ T_{(2,2)}$.}
\only<3>{We are using $R = T_{(1,1)} \circ {\red{T_{(1,2)}}} \circ T_{(2,1)} \circ T_{(2,2)}$.}
\only<4>{We are using $R = {\red{T_{(1,1)}}} \circ T_{(1,2)} \circ T_{(2,1)} \circ T_{(2,2)}$.}
\pause \pause \pause

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: motivation}

\begin{itemize}

\item Why is this called birational rowmotion?

\item Indeed, it generalizes classical rowmotion:
\begin{itemize}
\item Let $\TropZ$ be the \textbf{tropical semiring} over $\ZZ$. This is the set $\ZZ \cup \left\{-\infty\right\}$ with ``addition'' $\left(a,b\right)\mapsto\max\left\{a,b\right\}$ and ``multiplication'' $\left(a,b\right)\mapsto a+b$. This is a semifield.
\pause
\item To every order ideal $S \in J(P)$, assign a $\TropZ$-labelling $\tlab S$ defined by
\[
\left(  \operatorname*{tlab}S\right)  \left(  v\right)  =\left\{
\begin{array}{cc}
1,& \text{ if }v\notin S\cup\left\{  0\right\}  ;\\
0,& \text{if }v\in S\cup\left\{  0\right\}
\end{array}
\right.  .
\]
\item Easy to see:
\[
T_v \circ \tlab = \tlab \circ \mathbf t_v, \ \ \ \ \ \ \ \ \ \ R \circ \tlab = \tlab \circ \mathbf r.
\]
(And $\tlab$ is injective.)
\pause
\item If you don't like semifields, use $\mathbb Q$ and take the ``tropical limit''.
\end{itemize}

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: order}

\begin{itemize}

\item Let $\ord \phi$ denote the order of a map or rational map $\phi$. This is the smallest positive integer $k$ such that $\phi^k=\id$ (on the range of $\phi^k$), or $\infty$ if no such $k$ exists.

\item The above shows that $\ord(\mathbf{r}) \mid \ord(R)$ for every finite poset $P$.

\item Do we have equality? \\
\pause
\textbf{No!} Here are two posets with $\ord(R) = \infty$:
\[
\begin{array}{l|r}
\xymatrixrowsep{1.5pc}
\xymatrix{
\fullmoon \ar@{-}[d] & \fullmoon \ar@{-}[d] \ar@{-}[dl] & \fullmoon \ar@{-}[dl] \ar@{-}[dll] \\
\fullmoon  & \fullmoon &
}
\phantom{xx}
&
\phantom{xx}
\xymatrixrowsep{1.5pc}
\xymatrixcolsep{0.2pc}
\xymatrix{
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
\fullmoon & & \fullmoon & & \fullmoon & & \fullmoon
}
\end{array}
\]
\pause \item \textbf{Nevertheless}, equality holds for many special types of $P$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: example of finite order}

{\bf Example:}

Iteratively apply $R$ to a labelling of the $2\times 2$-rectangle.

\begin{overprint}
\onslide<1>
$R^0 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]
\onslide<2>
$R^1 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \frac{bw(x+y)}{yz} \ar@{-}[ld] \\
& \frac{ab}{z} \ar@{-}[d] & \\
& a &
}
\]
\onslide<3>
$R^2 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{bw(x+y)}{xy} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{ab}{y} \ar@{-}[rd] & & \frac{ab}{x} \ar@{-}[ld] \\
& \frac{az}{x+y} \ar@{-}[d] & \\
& a &
}
\]
\onslide<4>
$R^3 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{ab}{w} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{ayz}{w(x+y)} \ar@{-}[rd] & & \frac{axz}{w(x+y)} \ar@{-}[ld] \\
& \frac{xy}{aw(x+y)} \ar@{-}[d] & \\
& a &
}
\]
\onslide<5>
$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]
\onslide<6>
$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]
So we are back where we started.
\[
\ord(R) = 4.
\]
\end{overprint}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the graded forest case}

\begin{itemize}

\item \textbf{Theorem.} Assume that $n\in\NN$, and $P$ is a poset which is a forest (made into a poset using the ``descendant'' relation) having all leaves on the same level $n$ (i.e., each maximal chain of $P$ has $n$ vertices). Then,
\[
\ord(R) = \ord(\mathbf r) \mid \lcm\left(1,2,...,n+1\right).
\]

\end{itemize}

{\bf Example:}

This poset
\[
\xymatrix{
& \fullmoon \ar@{-}[d] & & & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon & & \fullmoon & \fullmoon
}
\]
has $\ord(R) = \ord(\mathbf r) \mid \lcm(1,2,3,4) = 12$.

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the graded forest case}

\begin{itemize}

\item Even the $ \ord(\mathbf r) \mid \lcm\left(1,2,...,n+1\right)$ part of this result seems to be new.

\item We will very roughly sketch a proof of $\ord(R) \mid \lcm\left(1,2,...,n+1\right)$. Details are in the ``Skeletal posets'' section of our paper, where we also generalize the result to a wider class of posets we call ``skeletal posets''. (These can be regarded as a generalization of forests where we are allowed to graft existing forests on roots on the top and on the bottom, and to use antichains instead of roots. An example is the $2\times 2$-rectangle.)

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: $n$-graded posets}

\begin{itemize}

\item Consider any \textbf{$n$-graded} finite poset $P$. This means that $P$ is partitioned into nonempty subsets $P_1$, $P_2$, ..., $P_n$ such that:
\begin{itemize}
\item If $u \in P_i$ and $u \lessdot v$, then $v \in P_{i+1}$.
\item All minimal elements of $P$ are in $P_1$.
\item All maximal elements of $P$ are in $P_n$.
\end{itemize}
\end{itemize}

{\bf Example:} The $2\times 2$-rectangle is a $3$-graded poset:

\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \longleftarrow P_3 \\
\left(2,1\right) \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \longleftarrow P_2 \\
& \left(1,1\right) & & \longleftarrow P_1 
}
\]


\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: homogeneous equivalence}

\begin{itemize}
\item Two $\mathbb K$-labellings $f$ and $g$ of $P$ are said to be \textbf{homogeneously equivalent} if there is a $\left(\lambda_1,\lambda_2,...,\lambda_n\right)\in \left(\mathbb K\setminus 0\right)^n$ such that
\[
g\left(v\right) = \lambda_i f\left(v\right)\ \ \ \ \ \text{for all }i\text{ and all }v\in P_i.
\]
\end{itemize}

{\bf Example:} These two labellings:
\[
\xymatrixrowsep{1pc}\xymatrixcolsep{0.20pc}\xymatrix{
& a_1 \ar@{-}[d] & \\
& z_1 \ar@{-}[rd] \ar@{-}[ld] & \\
x_1 \ar@{-}[rd] & & y_1 \ar@{-}[ld] \\
& w_1 \ar@{-}[d] & \\
& b_1 &
}
\text{ and }
\xymatrixrowsep{1pc}\xymatrixcolsep{0.20pc}\xymatrix{
& a_2 \ar@{-}[d] & \\
& z_2 \ar@{-}[rd] \ar@{-}[ld] & \\
x_2 \ar@{-}[rd] & & y_2 \ar@{-}[ld] \\
& w_2 \ar@{-}[d] & \\
& b_2 &
}
\]
are homogeneously equivalent if and only if $\dfrac{x_1}{y_1}=\dfrac{x_2}{y_2}$.

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: homogeneous equivalence and $R$}

\begin{itemize}
\item Let $\overline{\bbK^\whP}$ denote the set of all $\mathbb K$-labellings of $P$ (with no zero labels) modulo homogeneous equivalence. \\
Let $\pi : \bbK^\whP \rato \overline{\bbK^\whP}$ be the canonical projection.

\item There exists a rational map $\overline{R} : \overline{\bbK^\whP} \rato \overline{\bbK^\whP}$ such that the diagram
\[
\xymatrixcolsep{5pc}\xymatrix{
\mathbb K^{\widehat P} \ar@{-->}[r]^{R} \ar@{-->}[d]_-{\pi} & \mathbb
K^{\widehat P}
\ar@{-->}[d]^-{\pi} \\
\overline{\mathbb K^{\widehat P}} \ar@{-->}[r]_{\overline{R}} & \overline
{\mathbb K^{\widehat P}}
}
\]
commutes.

\item Hence $\ord\left(\overline R\right) \mid \ord(R)$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: interplay between $R$ and $\overline R$}

\begin{itemize}
\item But in fact, any $n$-graded poset $P$ satisfies
\[
\ord(R) = \lcm\left(n+1, \ord\left(\overline R\right)\right).
\]

\pause \item Furthermore, if $P$ and $Q$ are both $n$-graded, then the disjoint union $PQ$ of $P$ and $Q$ satisfies
\[
\ord\left(R_{PQ}\right) = \ord\left(\overline{R}_{PQ}\right) = \lcm\left(\ord\left(R_P\right), \ord\left(R_Q\right)\right)
\]
(where $R_S$ means the $R$ defined for a poset $S$).

\pause \item Finally, if $P$ is $n$-graded, and $B'_1 P$ denotes the $\left(n+1\right)$-graded poset obtained by adding a new element on top of $P$ (such that it is greater than all existing elements of $P$), then
\[
\ord\left(\overline{R}_{B'_1 P}\right) = \ord\left(\overline{R}_P\right).
\]

\pause \item Combining these, we can inductively compute $\ord\left(R_P\right)$ and $\ord\left(\overline{R}_P\right)$ for any $n$-graded forest $P$, and prove $\ord(R) \mid \lcm\left(1,2,...,n+1\right)$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: an example of the induction}
{\bf Example:} \\
Here is how we can get our forest poset using the $PQ$ and $B'_1 P$ constructions from $\varnothing$:

\begin{overprint}
\onslide<1>
\[
\xymatrix{
& \fullmoon \ar@{-}[d] & & & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon & & \fullmoon & \fullmoon
}
\]
\onslide<2>
\[
\left(
\xymatrix{
& \fullmoon \ar@{-}[d] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
\fullmoon & & \fullmoon \\
}
\right)
\cdot \left(
\xymatrix{
& & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<3>
\[
B'_1\left(\xymatrix{
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
\fullmoon & & \fullmoon \\
}\right)
\cdot \left(
\xymatrix{
& & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<4>
\[
B'_1\left(B'_1\left(\xymatrix{
\fullmoon & & \fullmoon \\
}\right)\right)
\cdot \left(
\xymatrix{
& & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<5>
\[
B'_1\left(B'_1\left(\left(\fullmoon\right) \cdot \left(\fullmoon\right)\right)\right)
\cdot \left(
\xymatrix{
& & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<6>
\[
B'_1\left(B'_1\left(\left(B'_1 \varnothing\right) \cdot \left(B'_1 \varnothing\right)\right)\right)
\cdot \left(
\xymatrix{
& & \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<7>
\[
B'_1\left(B'_1\left(\left(B'_1 \varnothing\right) \cdot \left(B'_1 \varnothing\right)\right)\right)
\cdot B'_1\left(
\xymatrix{
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & & \fullmoon \ar@{-}[d] \\
\fullmoon & & \fullmoon & \fullmoon
}
\right)
\]
\onslide<8>
\[
B'_1\left(B'_1\left(\left(B'_1 \varnothing\right) \cdot \left(B'_1 \varnothing\right)\right)\right)
\cdot B'_1\left(\left(
\xymatrix{
& \fullmoon \ar@{-}[dl] \ar@{-}[dr] & \\
\fullmoon & & \fullmoon
}\right) \cdot \left(
\xymatrix{
\fullmoon \ar@{-}[d] \\
\fullmoon
}\right)
\right)
\]
\onslide<9>
\[
B'_1\left(B'_1\left(\left(B'_1 \varnothing\right) \cdot \left(B'_1 \varnothing\right)\right)\right)
\cdot B'_1\left(B'_1\left(\left(B'_1 \varnothing\right) \left(B'_1 \varnothing\right) \right)
\cdot B'_1\left(B'_1\varnothing\right)\right)
\]
\end{overprint}
\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Classical rowmotion: the graded forest case}

\begin{itemize}

\item It remains to show $\ord(\mathbf{r}) \mid \lcm\left(1,2,...,n+1\right)$.

\pause \item This can be done by ``tropicalizing'' the notions of homogeneous equivalence, $\pi$ and $\overline R$. Details in the ``Interlude'' section of our paper.

\pause \item Actually, not as much tropicalizing as booleanizing: we only use the boolean semiring $\left\{0,1\right\}$ to get classical rowmotion. With the full force of the tropical semiring we get {\bf more} (see later)!

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Classical rowmotion: the graded forest case, precise result}

\begin{itemize}

\item \textbf{Theorem.} Assume that $n\in\NN$, and $P$ is a poset which is a forest (made into a poset using the ``descendant'' relation) having all leaves on the same level $n$ (i.e., each maximal chain of $P$ has $n$ vertices). Then,
\begin{align*}\nonumber
\ord(\overline{R}) &= \ord(\overline{\mathbf r}) \\
 &= \lcm\left\{ n - i \mid i \in \left\{0,1,\ldots, n-1\right\}; \ \left|\widehat{P}_i\right| < \left|\widehat{P}_{i+1}\right|\right\} ,
\end{align*}
where $\widehat{P}_k$ denotes the set of elements of $\widehat{P}$ which are a distance of $k$ away from $0$.

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Birational rowmotion: the rectangle case}

\begin{itemize}

\item \textbf{Theorem (periodicity):} If $P$ is the $p \times q$-rectangle (i.e., the poset $\left\{1,2,...,p\right\}\times\left\{1,2,...,q\right\}$ with coordinatewise order), then
\[
\ord\left(R\right) = p+q.
\]

\end{itemize}

\textbf{Example:} For the $2\times 2$-rectangle, this claims $\ord\left(R\right) = 2+2 = 4$, which we have already seen.

\pause
\begin{itemize}

\item \textbf{Theorem (reciprocity):} If $P$ is the $p \times q$-rectangle, and $(i, k) \in P$ and $f \in \bbK^\whP$, then
\[
f\left(\underbrace{\left(p+1-i, q+1-k\right)}_{\substack{=\text{antipode of }(i, k) \\ \text{ in the rectangle}}}\right) = \dfrac{f(0) f(1)}{\left(R^{i+k-1}f\right)\left(\left(i,k\right)\right)}.
\]

\item These were conjectured by James Propp and Tom Roby.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the rectangle case, example}

{\bf Example:} Here is the generic $R$-orbit on the $2\times 2$-rectangle again:

\[
\begin{array}{c|c}
\xymatrixrowsep{0.5pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & \only<1,3>{y} \only<2>{{\red{y}}} \ar@{-}[ld] \\
& \only<3>{{\blue{w}}} \only<1,2>{w} \ar@{-}[d] & \\
& a &
}
&
\xymatrixrowsep{0.4pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \frac{bw(x+y)}{yz} \ar@{-}[ld] \\
& \frac{ab}{z} \ar@{-}[d] & \\
& a &
}
\\
\hline
\xymatrixrowsep{0.4pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{bw(x+y)}{xy} \ar@{-}[rd] \ar@{-}[ld] & \\
\only<1,3>{\frac{ab}{y}} \only<2>{{\red {\frac{ab}{y}}}} \ar@{-}[rd] & & \frac{ab}{x} \ar@{-}[ld] \\
& \frac{az}{x+y} \ar@{-}[d] & \\
& a &
}
&
\xymatrixrowsep{0.4pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \only<3>{{\blue{\frac{ab}{w}}}} \only<1,2>{\frac{ab}{w}} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{ayz}{w(x+y)} \ar@{-}[rd] & & \frac{axz}{w(x+y)} \ar@{-}[ld] \\
& \frac{axy}{w(x+y)} \ar@{-}[d] & \\
& a &
}
\end{array}
\]

\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Birational rowmotion: the rectangle case, proof}

\begin{itemize}

\item Inspiration: Alexandre Yu. Volkov, \textit{On Zamolodchikov's Periodicity Conjecture}, \texttt{arXiv:hep-th/0606094}.

\item We will give only a very vague idea of the proof.

\item We WLOG assume that $\bbK$ is a field. (Everything is polynomial identities.)

\end{itemize}
\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Birational rowmotion: the rectangle case, proof}

\begin{itemize}
\item Let $A \in \bbK^{p\times (p+q)}$ be a matrix with $p$ rows and $p+q$ columns.

\item Let $A_i$ be the $i$-th column of $A$. Extend to all $i \in \ZZ$ by setting
\[
A_{p+q+i} = \left(-1\right)^{p-1} A_i\ \ \ \ \ \text{for all }i.
\]

\item Let $A \left[a:b \mid c:d\right]$ be the matrix whose columns are \newline {$A_a$, $A_{a+1}$, ..., $A_{b-1}$, $A_c$, $A_{c+1}$, ..., $A_{d-1}$} from left to right.

\pause
\item For every $j\in\ZZ$, we define a $\mathbb K$-labelling $\operatorname*{Grasp}\nolimits_{j}A\in
\mathbb{K}^\whP$ by
\begin{align*}
\nonumber
& \left(  \operatorname*{Grasp}\nolimits_{j}A\right)  \left(  \left(
i,k\right)  \right)  \\
&= \dfrac{\det\left(  A\left[  j+1:j+i\mid
j+i+k-1:j+p+k\right]  \right)  }{\det\left(  A\left[  j:j+i\mid
j+i+k:j+p+k\right]  \right)  }
\end{align*}
for every $(i,k)\in P$ (this is well-defined for a Zariski-generic $A$) and
$\left(  \operatorname*{Grasp}\nolimits_{j}A\right)  (0) =
\left(  \operatorname*{Grasp}\nolimits_{j}A\right)  (1) = 1$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the rectangle case, proof}

\begin{itemize}

\item The proof of $\ord(R)=p+q$ now rests on four claims:
\begin{itemize}
\item \textbf{Claim 1:} $ \operatorname*{Grasp}\nolimits_{j}A =  \operatorname*{Grasp}\nolimits_{p+q+j}A$ for all $j$ and $A$.
\item \textbf{Claim 2:} $ R\left(\operatorname*{Grasp}\nolimits_{j}A\right) =  \operatorname*{Grasp}\nolimits_{j-1}A$ for all $j$ and $A$.
\item \textbf{Claim 3:} For almost every $f \in \bbK^\whP$ satisfying $f(0) = f(1) = 1$, there exists a matrix $A \in \bbK^{p\times(p+q)}$ such that $ \operatorname*{Grasp}\nolimits_0 A = f$.
\item \textbf{Claim 4:} In proving $\ord(R)=p+q$ we can WLOG assume that $f(0) = f(1) = 1$.
\end{itemize}
\end{itemize}

\begin{overprint}
\onslide<1> \begin{itemize} \item
Claim 1 is immediate from the definitions.
\end{itemize}
\onslide<2>
\begin{itemize} \item
Claim 2 is a computation with determinants, which boils down to the three-term Pl\"ucker identities:
\begin{align*}
&  \det\left(  A\left[  a-1:b\mid c:d+1\right]  \right)  \cdot\det\left(
A\left[  a:b+1\mid c-1:d\right]  \right) \\
&  +\det\left(  A\left[  a:b\mid c-1:d+1\right]  \right)  \cdot\det\left(
A\left[  a-1:b+1\mid c:d\right]  \right) \\
&  =\det\left(  A\left[  a-1:b\mid c-1:d\right]  \right)  \cdot\det\left(
A\left[  a:b+1\mid c:d+1\right]  \right).
\end{align*}
for $A\in \mathbb{K}^{u\times v}$ and $a\leq b$ and $c\leq d$ and $b-a+d-c=u-2$.
\end{itemize}
\onslide<3>
\begin{itemize} \item
Claim 3 is an annoying (nonlinear) triangularity argument: With the ansatz $A = \left(I_p \mid B\right)$ for $B \in \bbK^{p\times q}$, the equation $ \operatorname*{Grasp}\nolimits_0 A = f$ translates into a system of equations in the entries of $B$ which can be solved by elimination.
\end{itemize}
\onslide<4>
\begin{itemize} \item
Claim 4 follows by recalling $\ord(R) = \lcm\left(n+1, \ord\left(\overline R\right)\right)$.
\end{itemize}
\onslide<5>
\begin{itemize} \item
The reciprocity statement can be proven in a similar vein.
\end{itemize}
\end{overprint}

\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Birational rowmotion: the $\Delta$-triangle case}

\begin{itemize}

\item \textbf{Theorem (periodicity):} If $P$ is the triangle $\Delta(p) = \left\{(i,k) \in \left\{1,2,...,p\right\}\times\left\{1,2,...,p\right\} \mid i+k > p+1\right\}$ with $p>2$, then
\[
\ord\left(R\right) = 2p.
\]

\end{itemize}

\textbf{Example:} The triangle $\Delta(4)$:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \\
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\fullmoon & & \fullmoon & & \fullmoon
}
\]

\pause
\begin{itemize}

\item \textbf{Theorem (reciprocity):} $R^p$ reflects any $\mathbb K$-labelling across the vertical axis.

\item These are precisely the same results as for classical rowmotion.

\pause
\item The proofs use a ``folding''-style argument to reduce this to the rectangle case.

\end{itemize}

\end{frame}

\begin{frame}

\frametitle{\ \ \ \ Birational rowmotion: the $\vartriangleright$-triangle case}

\begin{itemize}

\item \textbf{Theorem (periodicity):} If $P$ is the triangle $\left\{(i,k) \in \left\{1,2,...,p\right\}\times\left\{1,2,...,p\right\} \mid i \leq k\right\}$, then
\[
\ord\left(R\right) = 2p.
\]

\end{itemize}

\textbf{Example:} For $p = 4$, this $P$ has the form:
\[
\xymatrixrowsep{0.5pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & & \fullmoon \ar@{-}[rd] & & & \\
& & & & \fullmoon \ar@{-}[rd] \ar@{-}[ld] & & \\
& & & \fullmoon \ar@{-}[rd] & & \fullmoon \ar@{-}[rd] \ar@{-}[ld] & \\
& & & & \fullmoon \ar@{-}[rd] \ar@{-}[ld] & & \fullmoon \ar@{-}[ld] \\
& & & \fullmoon \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] & \\
& & & & \fullmoon \ar@{-}[ld] & & \\
& & & \fullmoon & & &
}.
\]

\pause

\begin{itemize}
\item Again this is reduced to the rectangle case.
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the right-angled triangle case}

\begin{itemize}

\item \textbf{Conjecture (periodicity):} If $P$ is the triangle $\left\{(i,k) \in \left\{1,2,...,p\right\}\times\left\{1,2,...,p\right\} \mid i \leq k;\ i+k > p+1\right\}$, then
\[
\ord\left(R\right) = p.
\]

\end{itemize}

\textbf{Example:} For $p = 4$, this $P$ has the form:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & & \fullmoon \ar@{-}[rd] & & & \\
& & & & \fullmoon \ar@{-}[rd] \ar@{-}[ld] & & \\
& & & \fullmoon & & \fullmoon &
}.
\]

\pause

\begin{itemize}
\item We proved this for $p$ odd.

\item Note that for $p$ even, this is a type-B positive root poset. Armstrong-Stump-Thomas did this for classical rowmotion.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the trapezoid case (Nathan Williams)}

\begin{itemize}

\item \textbf{Conjecture (periodicity):} If $P$ is the trapezoid $\left\{(i,k) \in \left\{1,2,...,p\right\}\times\left\{1,2,...,p\right\} \mid i \leq k;\ i+k > p+1;\ k\geq s\right\}$ for some $0 \leq s \leq p$, then
\[
\ord\left(R\right) = p.
\]

\end{itemize}

\textbf{Example:} For $p = 6$ and $s = 5$, this $P$ has the form:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
\fullmoon \ar@{-}[rd] & & & & \\
& \fullmoon \ar@{-}[rd] \ar@{-}[ld] & & & \\
\fullmoon \ar@{-}[rd] & & \fullmoon \ar@{-}[rd] \ar@{-}[ld] & \\
& \fullmoon \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
& & \fullmoon & & \fullmoon
}.
\]

\begin{itemize}

\item This was observed by Nathan Williams and verified for $p\leq 7$.

\item Motivation comes from Williams's ``Cataland'' philosophy.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: the root system connection (Nathan Williams)}

\begin{itemize}

\item For what $P$ is $\ord(R) < \infty$ ? This seems too hard to answer in general.

\pause
\item \textbf{Not true:} for all those $P$ that have nice and small $\ord(\mathbf{r})$'s.

\pause
\item \textbf{However} it seems that $\ord(R) < \infty$ holds if $P$ is
\textbf{the positive root poset of a coincidental-type root system} ($A_n$, $B_n$, $H_3$),
or a \textbf{minuscule heap} (see Rush-Shi, section 6).

\item But the positive root system of $D_4$ has $\ord(R) = \infty$.

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Application: Promotion on SSYTs}

\begin{itemize}
\item The following is an application of our result on rectangle-shaped posets.
\item It is well known (see Striker-Williams) that {\bf classical} rowmotion (= birational
rowmotion over the boolean semiring $\left\{0,1\right\}$) is related to
promotion on {\bf two-rowed} semistandard Young tableaux.
\item Similarly, {\bf birational} rowmotion over the tropical semiring $\TropZ$ relates
to {\bf arbitrary} semistandard Young tableaux.
\item As an application of the periodicity theorem, we obtain the classical result that promotion done
$n$ times on a rectangular semistandard Young tableau with ``ceiling'' $n$ does
nothing.
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: noncommutative generalization?}

\begin{itemize}
\item This is new and unproven, and inspired by Iyudu/Shkarin, {\tt arXiv:1305.1965v3} (Kontsevich's periodicity conjecture).

\item Work in a {\bf skew field}.  Write $\overline{m}$ for $m^{-1}$.

\item Define the $v$-toggle by
\begin{equation}
\left(  T_{v}f\right)  \left(  w\right)  =\left\{
\begin{array}{cc}
f\left(  w\right)  ,\ \ \ \ \ \ \ \ \ \ & \text{if }w\neq v;\\
% & \\
\left(\sum\limits_{\substack{u\in \widehat{P};\\u\lessdot v}}f\left(  u\right) \right)
\cdot \overline{f\left(v\right)}
\cdot \overline{\sum\limits_{\substack{u\in \widehat{P};\\u\gtrdot v}}\overline{f\left(  u\right)  }}
,\ \ \ \ \ \ \ \ \ \ & \text{if }w=v
\nonumber
\end{array}
\right.
\end{equation}
(there are other options as well -- so far unexplored).

\end{itemize}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Birational rowmotion: noncommutative generalization?}

\begin{itemize}
\item This is new and unproven, and inspired by Iyudu/Shkarin, {\tt arXiv:1305.1965v3} (Kontsevich's periodicity conjecture).

\item Work in a {\bf skew field}.  Write $\overline{m}$ for $m^{-1}$.
\end{itemize}

Iteratively apply $R$ to a labelling of the $2\times 2$-rectangle.

\begin{overprint}
\onslide<1>
$R^0 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]
\onslide<2>
$R^1 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& (x+y)\overline{z}b \ar@{-}[rd] \ar@{-}[ld] & \\
w\overline{x}(x+y)\overline{z}b \ar@{-}[rd] & & w\overline{y}(x+y)\overline{z}b \ar@{-}[ld] \\
& a\overline{z}b \ar@{-}[d] & \\
& a &
}
\]
\onslide<3>
$R^2 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& w\left(\overline{x}+\overline{y}\right)b \ar@{-}[rd] \ar@{-}[ld] & \\
a\cdot\overline{x+y}\cdot x\left(\overline{x}+\overline{y}\right)b \ar@{-}[rd] & & a\cdot\overline{x+y}\cdot y\left(\overline{x}+\overline{y}\right)b \ar@{-}[ld] \\
& a\overline{b}z\cdot\overline{x+y}\cdot b \ar@{-}[d] & \\
& a &
}
\]
\onslide<4>
$R^3 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& a\overline{w}b \ar@{-}[rd] \ar@{-}[ld] & \\
... \ar@{-}[rd] & & a \overline{b}z\cdot\overline{x+y}\cdot\overline{\overline{x}+\overline{y}}\cdot\overline{y}\cdot\left(x+y\right)\overline{w}b \ar@{-}[ld] \\
& a\overline{b}\cdot\overline{\overline{x}+\overline{y}}\cdot\overline{w}b \ar@{-}[d] & \\
& a &
}
\]
\onslide<5>
$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& a\overline{b}z\overline{a}b \ar@{-}[rd] \ar@{-}[ld] & \\
... \ar@{-}[rd] & & a\overline{b}\cdot\overline{\overline{x}+\overline{y}}\cdot\overline{x+y}\cdot y\left(\overline{x}+\overline{y}\right)\left(x+y\right)\overline{a}b \ar@{-}[ld] \\
& a\overline{b}w\overline{a}b \ar@{-}[d] & \\
& a &
}
\]
\onslide<6>
$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& a\overline{b}z\overline{a}b \ar@{-}[rd] \ar@{-}[ld] & \\
a\overline{b}x\overline{a}b \ar@{-}[rd] & & a\overline{b}y\overline{a}b \ar@{-}[ld] \\
& a\overline{b}w\overline{a}b \ar@{-}[d] & \\
& a &
}
\]
(after nontrivial simplifications).
\onslide<7>
$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& a\overline{b}z\overline{a}b \ar@{-}[rd] \ar@{-}[ld] & \\
a\overline{b}x\overline{a}b \ar@{-}[rd] & & a\overline{b}y\overline{a}b \ar@{-}[ld] \\
& a\overline{b}w\overline{a}b \ar@{-}[d] & \\
& a &
}
\]
That is, all of our labels got conjugated by $a\overline b$. {\bf Is $R^{p+q}$ always
conjugation by $f(0)\cdot(f(1))^{-1}$ on a $p\times q$-rectangle?} This is similar to
Kontsevich's periodicity. (Noncommutative determinants?)
\end{overprint}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Announcement}

There will be a {\bf workshop on Dynamical Algebraic Combinatorics} at the American Institute of Mathematics the week of {\bf March 23-27, 2015}, organized by Propp, Roby, Striker and Williams.

{\tt \red http://aimath.org/workshops/upcoming/dynalgcomb/}

Among the subjects of the workshop:

\begin{itemize}
\item everything touched upon in this talk
\item yes, that includes Young tableaux and Bender-Knuth, promotion, Lascoux-Sch\"utzenberger crystal operators
\item homomesies (unsurprisingly)
\item alternating sign matrices and gyration
\item probably cluster algebras
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Acknowledgments}

\begin{itemize}

\item \textbf{Tom Roby}: collaboration

\item \textbf{Pavlo Pylyavskyy, Gregg Musiker}: suggestions to mimic Volkov's proof of Zamolodchikov conjecture

\item \textbf{James Propp, David Einstein}: introducing birational rowmotion and conjecturing the rectangle results; helpful advice

\item \textbf{Nathan Williams}: bringing root systems into play

\item \textbf{Jessica Striker}: familiarizing the author with rowmotion

\item \textbf{Alexander Postnikov}: organizing a seminar where the author first met the problem

\item \textbf{David Einstein, Hugh Thomas}: corrections

\item \textbf{Sage and Sage-combinat}: computations

\item \textbf{FPSAC referees}: useful comments

\end{itemize}

\textbf{Thank you for listening!}

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Some references}

\begin{itemize}

\small{

\item Andries E. Brouwer and A. Schrijver, \textit{On the period of an operator, defined on antichains}, 1974. {\red \url{http://www.win.tue.nl/~aeb/preprints/zw24.pdf}}

\item David Einstein, James Propp, \textit{Combinatorial, piecewise-linear, and birational homomesy for products of two chains}, 2013. {\red \url{http://arxiv.org/abs/1310.5294}}

\item David Rush, XiaoLin Shi, \textit{On Orbits of Order Ideals of Minuscule Posets}, 2013. {\red \url{http://arxiv.org/abs/1108.5245}}

\item Jessica Striker, Nathan Williams, \textit{Promotion and Rowmotion}, 2012. {\red \url{http://arxiv.org/abs/1108.1172}}

\item Alexandre Yu. Volkov, \textit{On the Periodicity Conjecture for Y-systems}, 2007. ({Old version available at \red \url{http://arxiv.org/abs/hep-th/0606094}})

\item Nathan Williams, \textit{Cataland}, 2013. {\red \url{https://conservancy.umn.edu/bitstream/159973/1/Williams_umn_0130E_14358.pdf}}

\item See our paper {\red \url{http://mit.edu/~darij/www/algebra/skeletal.pdf}} for the full bibliography.

}

\end{itemize}

\end{frame}

\end{document}