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

\date{10 March 2014 \\ The Applied Algebra Seminar, York University, Toronto}

\begin{document}

\frame{\titlepage
\textbf{slides:} {\red \url{http://mit.edu/~darij/www/algebra/skeletal-slides-mar2014.pdf}} \\
\textbf{paper:} {\red \url{http://mit.edu/~darij/www/algebra/skeletal.pdf}}
}

\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) green.
\[
\xymatrixcolsep{1.5pc}
\xymatrix{
& \fullmoon \ar@{-}[ld] \ar@{-}[rd] & & \fullmoon \ar@{-}[ld] \ar@{-}[rd] & \\
\newmoon \ar@{-}[rd] & & \green \newmoon \ar@{-}[ld] \ar@{-}[rd] & & \green \newmoon \ar@{-}[ld] \\
& \newmoon & & \newmoon &
}
\] }}

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

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

\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:}

Let $S$ be the order ideal of the $2\times 3$-rectangle given by:
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]

\end{frame}

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

{\bf Example:}

$\mathbf{r}(S)$ is
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
\left(2,1\right) \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]

\end{frame}

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

{\bf Example:}

$\mathbf{r}^2(S)$ is
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & {\red \left(1,3\right)} \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]

\end{frame}

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

{\bf Example:}

$\mathbf{r}^3(S)$ is
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& {\red \left(2,2\right)} \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]

\end{frame}

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

{\bf Example:}

$\mathbf{r}^4(S)$ is
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & {\red \left(1,3\right)} \ar@{-}[ld]\\
\left(2,1\right) \ar@{-}[rd] & & {\red \left(1,2\right)} \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]

\end{frame}

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

{\bf Example:}

$\mathbf{r}^5(S)$ is
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& & \left(2,3\right) \ar@{-}[rd] \ar@{-}[ld] & \\
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & & \left(1,3\right) \ar@{-}[ld]\\
{\red \left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] & \\
& {\red \left(1,1\right)} & &
}
\]
which is precisely the $S$ we started with.

$\ord(\mathbf r) = p+q = 2+3 = 5$.

\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.

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 $\Delta$-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''.

\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
$S \bigtriangleup \left\{v\right\}$ if this is an order ideal, and $S$
otherwise. (``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 $S$:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\left(2,1\right)} \ar@{-}[rd] & & \left(1,2\right) \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\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 $\mathbf r(S)$:
\[
\xymatrixrowsep{1.5pc}
\xymatrix{
& \left(2,2\right) \ar@{-}[rd] \ar@{-}[ld] & \\
\left(2,1\right) \ar@{-}[rd] & & \red{\left(1,2\right)} \ar@{-}[ld] \\
& \red{\left(1,1\right)} &
}
\]
}

\pause \pause \pause \pause \pause 

\end{frame}

\begin{frame}
\frametitle{\ \ \ \ Goals}

\begin{itemize}

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

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

\item I will 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 $0$ to be less than every other element, and $1$ to be greater than every other element.

\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}
[c]{l}%
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
\end{array}
\right.
\end{equation}
for all $w \in \whP$.

\only<1>{\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}
}

\only<2>{\item Notice that this is a 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 equivalence.}

\end{itemize}

\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:
\[
\begin{array}{c|c}
\text{original labelling }f & 
\only<1>{\text{labelling }T_{(2,2)} f}
\only<2>{\text{labelling }T_{(2,1)}T_{(2,2)} f}
\only<3>{\text{labelling }T_{(1,2)}T_{(2,1)}T_{(2,2)} f}
\only<4>{\text{labelling }T_{(1,1)}T_{(1,2)}T_{(2,1)}T_{(2,2)} f = Rf} \\
\hline
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\phantom{yyy}
&
\phantom{yyy}
\only<1>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \red{\frac{b(x+y)}{z}} \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<2>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\red{\frac{bw(x+y)}{xz}} \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<3>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \red{\frac{bw(x+y)}{yz}} \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
}
\only<4>{
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \frac{bw(x+y)}{yz} \ar@{-}[ld] \\
& \red{\frac{ab}{z}} \ar@{-}[d] & \\
& a &
}
}
\end{array}
\]
We are using
\only<1>{$R = T_{(1,1)} \circ T_{(1,2)} \circ T_{(2,1)} \circ \red{T_{(2,2)}}$.}
\only<2>{$R = T_{(1,1)} \circ T_{(1,2)} \circ {\red{T_{(2,1)}}} \circ T_{(2,2)}$.}
\only<3>{$R = T_{(1,1)} \circ {\red{T_{(1,2)}}} \circ T_{(2,1)} \circ T_{(2,2)}$.}
\only<4>{$R = {\red{T_{(1,1)}}} \circ T_{(1,2)} \circ T_{(2,1)} \circ T_{(2,2)}$.}
\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
\[
\left(  \operatorname*{tlab}S\right)  \left(  v\right)  =\left\{
\begin{array}
[c]{c}%
1,\text{ if }v\notin S\cup\left\{  0\right\}  ;\\
0,\ \text{if }v\in S\cup\left\{  0\right\}
\end{array}
\right.  .
\]
\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 semirings, 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 $\phi^k=\id$, 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.

$R^0 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]

\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.

$R^1 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{b(x+y)}{z} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{bw(x+y)}{xz} \ar@{-}[rd] & & \frac{bw(x+y)}{yz} \ar@{-}[ld] \\
& \frac{ab}{z} \ar@{-}[d] & \\
& a &
}
\]

\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.

$R^2 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{bw(x+y)}{xy} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{ab}{y} \ar@{-}[rd] & & \frac{ab}{x} \ar@{-}[ld] \\
& \frac{az}{x+y} \ar@{-}[d] & \\
& a &
}
\]

\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.

$R^3 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& \frac{ab}{w} \ar@{-}[rd] \ar@{-}[ld] & \\
\frac{ayz}{w(x+y)} \ar@{-}[rd] & & \frac{axz}{w(x+y)} \ar@{-}[ld] \\
& \frac{xy}{aw(x+y)} \ar@{-}[d] & \\
& a &
}
\]

\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.

$R^4 f = $
\[
\xymatrixrowsep{0.9pc}\xymatrixcolsep{0.20pc}\xymatrix{
& b \ar@{-}[d] & \\
& z \ar@{-}[rd] \ar@{-}[ld] & \\
x \ar@{-}[rd] & & y \ar@{-}[ld] \\
& w \ar@{-}[d] & \\
& a &
}
\]

\pause So we are back where we started.
\[
\ord(R) = 4.
\]

\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 $\left(a_1,a_2,...,a_n\right)\in \left(\mathbb K\setminus 0\right)^n$ such that
\[
g\left(v\right) = a_i f\left(v\right)\ \ \ \ \ \text{for all }i\text{ and all }v\in P_i.
\]
\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{\ \ \ \ 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.

\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
\[
f\left(\left(p+1-i, q+1-k\right)\right) = \dfrac{f(0) f(1)}{\left(R^{i+k-1}f\right)\left(\left(i,k\right)\right)}.
\]

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

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

\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 $A_a$, $A_{a+1}$, ..., $A_{b-1}$, $A_c$, $A_{c+1}$, ..., $A_{d-1}$ 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
\[
\left(  \operatorname*{Grasp}\nolimits_{j}A\right)  \left(  \left(
i,k\right)  \right)   =\dfrac{\det\left(  A\left[  j+1:j+i\mid
j+i+k-1:j+p+k\right]  \right)  }{\det\left(  A\left[  j:j+i\mid
j+i+k:j+p+k\right]  \right)  }
\]
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:} We have $ \operatorname*{Grasp}\nolimits_{j}A =  \operatorname*{Grasp}\nolimits_{p+q+j}A$ for all $j$ and $A$.
\item \textbf{Claim 2:} We have $ 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}

\item
\only<1>{Claim 1 is immediate from the definitions.}
\only<2>{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 $A\in \mathbb{K}^{u\times v}$, $a\leq b$, $c\leq d$ and $b-a+d-c=u-2$.
}
\only<3>{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.}
\only<4>{Claim 4 follows by recalling $\ord(R) = \lcm\left(n+1, \ord\left(\overline R\right)\right)$.}
\only<5>{The reciprocity statement can be proven in a similar vein.}

\end{itemize}

\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 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 rectangular 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.

\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 those $P$ which 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).

\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

\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

\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}