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%% define your title in the usual way
\title{Combinatorics of $\mathcal{X}$-variables in Finite Type Cluster Algebras}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Melissa Sherman-Bennett\thanks{\href{mailto:m\_shermanbennett@berkeley.edu}{m\_shermanbennett@berkeley.edu}}\addressmark{1}}

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\address{\addressmark{1}Department of Mathematics, UC Berkeley, Berkeley, CA}

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\received{11/13/2017}

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%\revised{}

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\abstract{We compute the number of $\mathcal{X}$-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For non-exceptional types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{
%Nous calculons le nombre de $ \mathcal{X} $-variables (\'{e}galement appel\'{e}es coefficients) d'une alg\'{e}bre amass\'{e}e de type fini lorsque le demi-champ sous-jacent est le demi-champ universel. Pour les types non-exceptionnels, ces nombres r\'{e}sultent d'une bijection entre coefficients et quadrilat\'{e}res (avec un choix de diagonale) apparaissant dans les triangulations de certaines surfaces marqu\'{e}es. Nous conjecturons que des r\'{e}sultats similaires s'appliquent aux alg\'{e}bres amass\'{e}es \'{a} partir de surfaces marqu\'{e}es arbitraires, et obtenons des corollaires concernant la structure des alg\'{e}bres de type fini de type g\'{e}om\'{e}trique.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{cluster algebra, coefficients, finite type}

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\begin{document}

\maketitle
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\section{Introduction}

Cluster algebras were introduced by Fomin and Zelevinsky in the early 2000s \cite{FZ1}, with the intent of establishing a general algebraic structure for studying dual canonical bases of semisimple groups and total positivity. A cluster algebra, or equivalently its seed pattern, is determined by an initial set of cluster variables (which we call $\A$-variables) and coefficients (which we call $\X$-variables), along with some additional data. As the terminology suggests, in the original definitions, $\X$-variables were somewhat in the background, and $\A$-variables were the object of importance. This is reflected in much of the research on cluster algebras to date, which focuses largely on $\A$-variables and their dynamics. However, $\X$-variables are important in total positivity and $\X$-variables over the universal semifield have recently appeared in the context of scattering amplitudes in $\mc{N}=4$ Super Yang-Mills theory \cite{physics}. Moreover, in the setting of cluster varieties, introduced by Fock and Goncharov \cite{FG}, the $\A$- and $\X$-varieties (associated with $\A$- and $\X$-variables, respectively) are on equal footing. Fock and Goncharov conjectured that a duality holds between the two varieties \cite[Conjecture 4.3]{FG}, which was later shown to be true under fairly general assumptions \cite{GHKK}. This duality suggests that studying $\X$-variables could be fruitful both in its own right and in furthering our understanding of cluster algebras. 

A study along these lines was undertaken by Speyer and Thomas in the case of acyclic cluster algebras with principal coefficients \cite{ST}. Using methods from quiver representation theory, they found that the $\X$-variables are in bijection with roots of an associated root system and give a combinatorial description of which roots can appear in the same $\X$-cluster. Seven found that in this context, mutation of $\X$-seeds roughly corresponds to reflection across hyperplanes orthogonal to roots \cite{Sev}. However, the above results do not address $\X$-variables over the universal semifield, and the combinatorics obtained in the case of principal coefficients is quite different from what we obtain here.

We investigate the combinatorics of $\X$-variables for seed patterns of finite type, particularly in the case when the underlying semifield is the universal semifield. The combinatorics of $\A$-variables for finite type seed patterns is particularly rich, with connections to finite root systems \cite{FZ2} and triangulations of certain marked surfaces \cite[Chapter 5]{book}. Parker and Scherlis give a partial proof that in type $A$, $\X$-variables over the universal semifield are in bijection with the quadrilaterals of these triangulations \cite{parker, scherlis}. We prove this statement for all types.

\begin{thm} \label{quadbijection} Let $\mc{S}$ be an $\X$-seed pattern of non-exceptional type $Z_n$ over the universal semifield such that one $\X$-cluster consists of algebraically independent elements. Let $P$ be the marked polygon associated to type $Z_n$. Then the $\X$-variables of $\mc{S}$ are in bijection with the quadrilaterals (with a choice of diagonal) appearing in triangulations of $P$.
\end{thm}

As a corollary, we also compute the number of $\X$-variables in a finite type $\X$-seed pattern $\mc{S}_{sf}$ over the universal semifield, listed in the second row of the following table (the numbers for $A_n$ for $n \leq6$, $D_4$, and $E_6$ were also computed in \cite{parker}). For comparison, the third row gives the number of $\X$-variables in a finite type $\X$-seed pattern $\mc{S}_{pc}$ with principal coefficients, a corollary of the results in \cite{ST}. 

\begin{center}
\setlength\tabcolsep{3pt}
\begin{tabular}{| c | c | c| c | c | c| c| c| c| }
 \hline
   Type & $A_n$ & $B_n, C_n$ & $D_n$ & $E_6$ & $E_7$ & $E_8$ & $F_4$ & $G_2$ \\ \hline
$|\X(\mc{S}_{sf})|$ & $2\binom{n+3}{4}$ & $\frac{1}{3}n(n+1)(n^2+2)$ & $\frac{1}{3}n(n-1)(n^2+4n-6)$ & 770 & 2100 & 6240 & 196 & 16 \\ \hline
$|\X(\mc{S}_{pc})|$ & $n(n+1)$ & $2n^2$ & $2n(n-1)$ & $72$ & $126$ & $240$ & $48$ & $12$\\
\hline
\end{tabular}
\end{center}


\section{Seed Patterns}

We largely follow the conventions of \cite{FZ4}. % cluster variables= $\AA$-variables and coefficients=$\XX$-variables, to blend the conventions of FZ & FG.


\subsection{Seeds and Mutation}

We begin by fixing a \emph{semifield} $(\PP, \cdot, \oplus)$, a multiplicative abelian group $(\PP, \cdot)$ equipped with an (auxiliary) addition $\oplus$, a binary operation which is associative, commutative, and distributive with respect to multiplication.  

\begin{ex} Let $t_1, \dots, t_k$ be algebraically independent over $\QQ$. The \emph{universal semifield} $\QQ_{sf}(t_1, \dots, t_k)$ is the set of all rational functions in $t_1, \dots, t_k$ that can be written as subtraction-free expressions in $t_1, \dots, t_k$. This is a semifield with respect to the usual multiplication and addition of rational expressions. Note that any (subtraction-free) identity in $\QQ_{sf}(t_1, \dots, t_k)$ holds in an arbitrary semifield for any elements $u_1,  \dots, u_k$ \cite{FZ4}.
\end{ex}

%\begin{ex}The \emph{tropical semifield} Trop$(t_1, \dots, t_k)$ is the free multiplicative group generated by $t_1, \dots, t_k$, with auxiliary addition defined by 
%
%\[\prod_{i=1}^k t_i^{a_i} \oplus \prod_{i=1}^k t_i^{b_i}= \prod_{i=1}^k t_i^{\min(a_i, b_i)}.
%\]
%\end{ex} %looks like I don't need this

Let $\QQ\PP$ denote the field of fractions of the group ring $\ZZ\PP$. We fix an \emph{ambient field} $\mc{F}$, isomorphic to $\QQ\PP(t_1, \dots, t_n)$.

%some transition sentence?

\begin{defn} A \emph{labeled $\X$-seed} in $\PP$ is a pair $(\textbf{x}, B)$ where $\textbf{x}=(x_1, \dots, x_n)$ is a tuple of elements in $\PP$ and $B=(b_{ij})$ is a \emph{skew-symmetrizable} $n \times n$ integer matrix, that is there exists a diagonal integer matrix $D$ with positive diagonal entries such that $DB$ is skew-symmetric. %check

A \emph{labeled $\A$-seed} in $\mc{F}$ is a triple $(\textbf{a}, \textbf{x}, B)$ where $(\textbf{x}, B)$ is a labeled $\X$-seed in $\PP$ and $\textbf{a}=(a_1, \dots, a_n)$ is a tuple of elements of $\mc{F}$ which are algebraically independent over $\QQ\PP$ and generate $\mc{F}$. $\textbf{x}$ is the (labeled) $\X$\emph{-cluster}, $\textbf{a}$ the (labeled) $\A$\emph{-cluster}, and $B$ the \emph{exchange matrix} of the labeled seed $(\textbf{a}, \textbf{x}, B)$.
\end{defn}

The elements of an $\X$- (respectively $\A$-)cluster are called $\X$-
(respectively $\A$-)variables. In the language of Fomin and Zelevinsky,
the $\X$-cluster is the coefficient tuple, the $\A$-cluster is the
cluster, and the $\X$- and $\A$-variables are coefficients and cluster
variables, respectively. The notation here is chosen to parallel Fock and
Goncharov's $\A$- and $\X$- cluster varieties. Note that an $\X$-seed
consists only of an exchange matrix and an $\X$-cluster, but an $\A$-seed
consists of an exchange matrix, an $\A$-cluster \emph{and} an
$\X$-cluster. For simplicity, we use ``cluster'', ``seed'', etc.  without a prefix when a statement holds regardless of prefix. %shorten?

One moves from labeled seed to labeled seed by a process called \emph{mutation}.



%If $B$ is skew-symmetric, we can rephrase mutation in terms of quivers. A \emph{quiver} $Q$ is a set of vertices and a multiset of arrows between vertices. We will assume throughout that $Q$ has no loops (arrows from $i$ to $i$) or 2-cycles (an arrow from $i$ to $j$ and an arrow from $j$ to $i$). Given a quiver $Q$ with vertices $[n]$, one can associate to it a skew-symmetric adjacency matrix $B_Q=(b_{ij})$ where $b_{ij}$ is the number of arrows from $i$ to $j$. Likewise, for each skew-symmetric matrix $B$, one can create a quiver $Q$ such that $B$ is the adjacency matrix of $Q$.

%\begin{defn} Let $Q$ be a quiver on $[n]$ and $k \in [n]$. \emph{Quiver mutation} in direction $k$ takes $Q$ to the quiver $\mu_k(Q)$ with vertex set $[n]$ and arrows obtained from those of $Q$ by performing the following steps:
% \begin{itemize}	
%\item Add an arrow $i \to j$ for every path $i \to k \to j$ in $Q$
%\item Reverse all arrows incident to $k$
%\item Delete a maximal collection of 2-cycles.
%\end{itemize}
% \end{defn} %add figure as example

%It is easy to verify that matrix mutation in direction $k$ takes the adjacency matrix of $Q$ to the adjacency matrix of $\mu_k(Q)$, so one can think of skew-symmetric matrices and quivers interchangeably here. 

\begin{defn} Let $(\textbf{a},\textbf{x}, B)$ be a labeled $\A$-seed in $\mc{F}$. \emph{$\A$-seed mutation} in direction $k$, denoted $\mu_k$, takes $(\textbf{a},\textbf{x}, B)$ to the labeled $\A$-seed $(\textbf{a}',\textbf{x}', B')$ where 

\begin{itemize}
\item The entries $b_{ij}'$ of $B'$ are given by
\begin{equation} \label{matrixmut}
	b'_{ij}=\left \{ 
		\begin{array}{ll}
		-b_{ij} & \text{if } i=k \text{ or } j=k\\
		b_{ij}+b_{ik}|b_{kj}| & \text{if } b_{ik}b_{kj}>0\\
		b_{ij} & \text{else}.
		\end{array} \right. 
\end{equation}
\item
The $\A$-cluster $\textbf{a}'=(a_1', \dots, a_n')$ is obtained from $\textbf{a}$ by replacing the $k^\text{th}$ entry $a_k$ with an element $a_k' \in \mc{F}$ satisfying the \emph{exchange relation}

\begin{equation}\label{exchangerel}
a_k' a_k=\frac{x_k\prod\limits_{b_{ik}>0}a_i^{b_{ik}}+\prod\limits_{b_{ik}<0}a_i^{-b_{ik}}}{x_k \oplus 1}.
\end{equation}
\item The $\X$-cluster $\textbf{x}'=(x_1', \dots, x_n')$ is given by
\begin{equation}\label{xmut}
	x'_j= \left \{ 
		\begin{array}{ll}
		x_j^{-1} & \text{if } j=k\\
		x_j (x_k^{sgn(-b_{kj})} \oplus 1)^{-b_{kj}} & \text{else}
		\end{array}
	\right .
\end{equation}
where $sgn(x)=0$ for $x=0$ and $sgn(x)=|x|/x$ otherwise.
\end{itemize}

Similarly, the \emph{$\X$-seed mutation} $\mu_k$ in direction $k$ takes the labeled $\X$-seed $(\textbf{x}, B)$ to $\X$-seed $(\textbf{x}', B')$ and \emph{matrix mutation} takes $B$ to $B'$. Two skew-symmetrizable integer matrices are \emph{mutation equivalent} if some sequence of matrix mutations takes one to the other.

\end{defn}
 
Note that $\mu_k(\textbf{a},\textbf{x}, B)$ is indeed another labeled $\A$-seed, as $B'$ is skew-symmetrizable and $\textbf{a}'$ again consists of algebraically independent elements generating $\mc{F}$. One can check that $\mu_k$ is an involution. 

%It will become important later that mutation is a ``local'' process in
%the following sense. In a labeled $\A$-seed $(\textbf{a}, \textbf{x},
%B)$, we think of $a_i$ and $a_j$ (and $x_i$ and $x_j$) as ``adjacent'' if $b_{ij} \neq 0$. Matrix mutation in direction $k$ changes $b_{ij}$ only if $b_{ik}$ and $b_{jk}$ are both nonzero, so seed mutation $\mu_k$ changes adjacencies only among the neighbors of $a_k$ and $x_k$. Further, $\mu_k$ changes only $\X$-variables adjacent to $x_k$, and the $\A$-variable replacing $a_k$ depends only on the $\A$-variables adjacent to $a_k$.

%As long as the exchange matrix is skew-symmetric, we can phrase seed
%mutation in terms of quivers as well. One can think of a seed as a
%quiver $Q$ on $[n]$ with 2 sets of vertex labels, $a_i$'s and $x_i$'s.
%Mutation in direction $k$ changes the arrows among $k$ and its
%neighbors, the $a$-label of $k$ and the $x$-labels of $k$ and its
%neighbors (because if $b_{kj}=0$, $x_j'=x_j$). Note that in this
%context, the exchange relation governing $a_k'$ involves only the labels
%of neighbors of $k$; the first product in \eqref{exchangerel} is the
%product of labels at sources of arrows into $k$, and the second is the
%product of labels at targets of arrows out of $k$. With this picture in
%mind, it's clear that mutation is a ``local'' process, depending only on
%the neighborhood of the vertex where you mutate. Of course, this remains
%true even when the exchange matrix is not the adjacency matrix of a
%quiver; we can still think of variables $a_j$ and $a_k$ being
%``adjacent'' if $b_{jk}\neq 0$

\subsection{Seed Patterns and Exchange Graphs}

We organize all seeds obtainable from each other by a sequence of mutations in a seed pattern. Let $\TT_n$ denote the (infinite) $n$-regular tree with edges labeled with $1, \dots, n$ so that no vertex is in two edges with  the same label.

\begin{defn} A rank $n$ \emph{$\A$-seed pattern} (respectively, \emph{$\X$-seed pattern}) $\mc{S}$ is an assignment of labeled $\A$-seeds (respectively $\X$-seeds) $\Sigma_t$ to the vertices $t$ of $\TT_n$ so that if $t$ and $t'$ are connected by an edge labeled $k$, then $\Sigma_t=\mu_k(\Sigma_{t'})$.
\end{defn}
 
Since mutation is involutive, a seed pattern $\mc{S}$ is completely
determined by the choice of a single seed $\Sigma$; we write
$\mc{S}(\Sigma)$ for the seed pattern containing $\Sigma$. Note that in
the language of Fomin and Zelevinsky, an $\A$-seed pattern is a ``seed
pattern'' and an $\X$-seed pattern is a ``$Y$-pattern''.

Given an $\A$-seed pattern $\mc{S}(\textbf{a}, \textbf{x}, B)$, one can
obtain two $\X$-seed patterns. The first is
$\mc{S}|_{\X}:=\mc{S}(\textbf{x}, B)$, the $\X$-seed pattern in $\PP$
obtained by simply ignoring the $\A$-clusters of every seed. The second,
which we denote $\hat{\mc{S}}=\mc{S}(\hat{\textbf{x}}, B)$, is an $\X
$-seed pattern in $\mc{F}$ constructed in \cite[Proposition 3.9]{FZ4}.
One can think of $\hat{\mc{S}}$ as recording the ``exchange information'' of $\mc{S}(\textbf{a}, \textbf{x}, B)$; indeed, the $\X$-variables of $\hat{\mc{S}}$ are rational expressions whose numerators and denominators are, up to multiplication by an element of $\PP$, the two terms on the right hand side of an exchange relation of $\mc{S}$.

Two labeled seeds $\Sigma=(\textbf{a}, \textbf{x}, B)$ and $\Sigma'=(\textbf{a}', \textbf{x}', B')$ are equivalent (written as $\Sigma \sim \Sigma'$) if one can obtain $\Sigma'$ by simultaneously reindexing $\textbf{a}$, $\textbf{x}$, and the rows and columns of $B$. We define an analogous equivalence relation for $\X$-seeds, also denoted $\sim$.

An \emph{$\A$-seed} (respectively \emph{$\X$-seed}) is an equivalence class of labeled $\A$-seeds (respectively labeled $\X$-seeds) with respect to $\sim$. The seed represented by the labeled seed $\Sigma$ is denoted $[\Sigma]$. We mutate a seed $[\Sigma]$ by applying a mutation $\mu_k$ to $\Sigma$ and taking its equivalence class.

% Just as with labeled seeds, the (unlabeled) $\A$-cluster of an $\A$-seed $[(\textbf{a}, \textbf{x}, B)]$ is the unordered set consisting of entries of $\textbf{a}$; similarly for the $\X$-cluster. We mutate seeds $[\Sigma]$ by applying a mutation $\mu_k$ to a labeled representative $\Sigma$ and then taking the equivalence class of $\mu_k(\Sigma)$. $\Sigma \sim \Sigma'$ implies $\mu_k(\Sigma) \sim \mu_k(\Sigma')$, so this is well-defined.

\begin{defn} The \emph{exchange graph} of a seed pattern $\mc{S}$ is the ($n$-regular connected) graph whose vertices are the seeds in $\mathcal{S}$ and whose edges connect seeds related by a single mutation. Equivalently, the exchange graph is the graph one obtains by identifying the vertices $t, t'$ of $\TT_n$ such that $\Sigma_t \sim \Sigma_{t'}$.
\end{defn}

Exchange graphs were defined for $\A$-seed patterns in \cite{FZ2, FZ4}, but can equally be defined for $\X$-seed patterns. It is conjectured that the exchange graph of an $\A$-seed pattern $\mc{S}=\mc{S}(\textbf{a}, \textbf{x}, B)$ depends only on $B$ \cite[Conjecture 4.3]{FZ4}, meaning that $\mc{S}|_{\X}$ does not influence the combinatorics of $\mc{S}$. The exchange graphs of $\mc{S}|_{\X}$ and $\hat{\mc{S}}$ can be obtained by identifying some vertices of the exchange graph of $\mc{S}$, as passing to either $\X$-seed pattern preserves mutation and the equivalence of labeled seeds. It is not known in general when any pair of these exchange graphs is equal.

\section{Finite Type Seed Patterns}

We now restrict our attention to seed patterns of finite type.

\begin{defn} An $\A$-seed pattern is of \emph{finite type} if it has finitely many seeds. 
\end{defn}

Finite type seed patterns were classified completely in \cite{FZ2}; they correspond exactly to finite (reduced crystallographic) root systems, or equivalently, finite type Cartan matrices (see for example \cite[Chapter 5]{Bour}).

For a skew-symmetrizable integer matrix $B=(b_{ij})$, its \emph{Cartan counterpart} is the matrix $A(B)=(a_{ij})$ defined by $a_{ii}=2$ and $a_{ij}=-|b_{ij}|$ for $i \neq j$.


\begin{thm}[\protect{\cite[Theorems 1.5--1.7]{FZ2}}]
\begin{enumerate}[i.]
\item An $\A$-seed pattern is of finite type if and only if the Cartan counterpart of one of its exchange matrices is a finite type Cartan matrix. 
\item Suppose $B$, $B'$ are skew-symmetrizable integer matrices such that
$A(B)$, $A(B')$ are finite type Cartan matrices. Then $A(B)$ and $A(B')$
are of the same Cartan--Killing type if and only if $B$ and $B'$ are mutation equivalent (modulo simultaneous relabeling of rows and columns.).
\end{enumerate}
\end{thm}

In light of this theorem, we refer to a finite type $\A$-seed pattern as type $A_n$, $B_n$, etc.

%There are many further connections between an $\A$-seed pattern $\mc{S}$
%of type $Y$ and a root system $\Phi$ of the same type, which we briefly
%review here. There is a bijection between $\A$-variables of $\mc{S}$ and
%the almost positive roots (the positive roots and the negative simple
%roots) of $\Phi$; the property of two $\A$-variables appearing in the
%same $\A$-cluster can be translated into a purely root-theoretic
%statement about the associated roots. Additionally, the number of seeds
%in $\mc{S}$ is the Coxeter--Catalan number for type $Y$. %check if this is true for non-indecomposable types
%Note that these statements hold regardless of $\PP$ and the $\X$-clusters of $\mc{S}$.

\begin{defn} An $\X$-seed pattern $\mc{S}$ is of type $Z_n$ if the Cartan companion of one of its exchange matrices is a type $Z_n$ Cartan matrix.
\end{defn}

We will call such $\X$-seed patterns \emph{Dynkin type} (rather than finite type, since in fact not all $\X$-seed patterns with finitely many seeds are of this form).

%To rephrase, finite type $\X$-seed patterns are those that come from finite type $\A$-seed patterns, \emph{not} those that have finitely many seeds. Though finite type $\X$-seed patterns of course have only finitely many seeds, many other seed patterns do as well. In particular, if $B$ is a skew-symmetrizable matrix that is mutation equivalent to finitely many matrices, then the $\X$-seed pattern over the trivial semifield $\PP=\{1\}$ with $B$ as an exchange matrix will have finitely many seeds.

%Finally, we review the combinatorics of $\A$-seed patterns of non-exceptional type. For simply laced types, this is part of the far more general theory of $\A$-seed patterns from surfaces, developed in \cite{FST}. %also cite FT?

\subsection{Triangulations for types $A$ and $D$}
The material in this section is part of a more general theory of $\A$-seed patterns from surfaces, developed in \cite{FST}.

Let $P_n$ denote a convex $n$-gon and $P^\bullet_n$ denote a convex $n$-gon with a distinguished point $p$ (a \emph{puncture}) in the interior, with vertices labeled by $1, \dots, n$. For $P \in \{P_n, P^\bullet_n\}$, the vertices and puncture of $P$ are called \emph{marked points}. An \emph{arc} of $P$ is a non-self-intersecting curve $\gamma$ in $P$ such that the endpoints of $\gamma$ are distinct marked points, the relative interior of $\gamma$ is disjoint from $\partial P\cup \{p\}$, and $\gamma$ does not cut out an unpunctured digon. An arc incident to the puncture $p$ is a \emph{radius}. Arcs are considered up to isotopy. 

\begin{centering}
\begin{figure}
\includegraphics[width=\textwidth]{AD}
\caption{Triangulations of $P_8$ and $P_8^\bullet$ are shown in solid lines; the dashed arc is the flip of $\tau$. On the right, $q_T(\tau)=\{\alpha, \beta, \gamma, \delta\}$.}
\label{fig:AD}
\end{figure}
\end{centering}

A \emph{tagged arc} of $P$ is either an ordinary arc between two vertices
or a radius that is labeled either ``notched'' or ``plain.'' Two tagged arcs $\gamma$, $\gamma'$ are \emph{compatible} if their untagged versions do not cross (or to be precise, there are two noncrossing arcs isotopic to $\gamma$ and $\gamma'$) with the following modification: if $\gamma$ is a notched radius and $\gamma'$ is plain, they are compatible if and only if their untagged versions coincide.

A \emph{tagged triangulation} $T$ is a maximal collection of pairwise compatible tagged arcs. All tagged triangulations of $P$ consist of the same number of arcs. Given a tagged triangulation $T$, the \emph{quadrilateral} $q_{T}(\gamma)$ of an arc $\gamma$ in $T$ consists of the arcs of $T$ and boundary segments adjacent to $\gamma$ (see Figure \ref{fig:AD}). $\gamma$ is the \emph{diagonal} of its quadrilateral.

\begin{prop}[\cite{FST}] Let $T=\{\gamma_1, \dots, \gamma_n\}$ be a tagged triangulation of $P$. For all $k$, there exists a unique tagged arc $\gamma_k'\neq \gamma_k$ such that $\mu_k(T):=T \setminus \{\gamma_k\} \cup \{\gamma_k'\}$ is a tagged triangulation of $P$.
\end{prop}

The arc $\gamma'$ in the above proposition is called the $\emph{flip}$ of $\gamma$ with respect to $T$ (or with respect to $q_T(\gamma)$, since $\gamma$ and $\gamma'$ are exactly the two diagonals of $q_T(\gamma)$). %should this be in here?

We define the \emph{flip graph} of $P$ to be the graph whose vertices are tagged triangulations of $P$ and whose edges connect triangulations that can be obtained from each other by flipping a single arc. The flip graph of $P$ is connected. %needed?

We can encode a tagged triangulation $T$ in a skew-symmetric $n \times n$ integer matrix $B(T)$. The nonzero entries in $B(T)$ correspond to pairs of adjacent arcs; the sign of these entries records the relative orientation of the arcs (see \cite{FST} for details).

%Before defining the entries, we must decide on a convention for
%adjacencies in the case where $T$ contains the notched and plain
%versions of the same radius, say $\gamma_k$ and $\gamma_l$. In this
%case, $\gamma_k$ and $\gamma_l$ are enclosed by two arcs with the same
%endpoints, $\gamma$ and $\gamma'$. Let $v$ denote the shared endpoint of
%the arcs $\gamma_k, \gamma_l, \gamma, \gamma'$, and suppose if one
%travels clockwise from $v$ along the loop $\gamma \gamma'$, $\gamma$
%precedes $\gamma'$ (see Figure \textcolor{red}{[example of two of same
%radius]}). Then we introduce two ``formal'' triangles of $T$ whose arcs are, in clockwise order,$\gamma_k, \gamma, \gamma'$ and $\gamma_l, \gamma,  \gamma'$. One can think of the formal triangles as those that would result if the puncture $p$ was identified with $v$.
%
%For every triangle and formal triangle $\Delta$ of $T$, let $B^\Delta=(b_{ij}^\Delta)$ be given by
%\[b_{ij}^\Delta=\left \{ 
%		\begin{array}{ll}
%		1 & \text{ if }\gamma_i, \gamma_j \in \Delta \text{ and } \gamma_i \text{ immediately precedes } \gamma_j \text{ moving clockwise around } \Delta\\
%		-1 & \text{ if }\gamma_i, \gamma_j \in \Delta \text{ and } \gamma_j \text{ immediately precedes } \gamma_i \text{ moving clockwise around } \Delta\\
%		0 & \text{ if }\gamma_ i \text{ or } \gamma_j \text{ is not in } \Delta.
%		\end{array} \right. 
%\]
%
% Then $B(T):=\sum_\Delta B^\Delta$. \textcolor{red}{[Insert example, with figure?]}
 
Flips of arcs are related to matrix mutation in the following way: for a tagged triangulation $T=\{\gamma_1, \dots, \gamma_n\}$ of $P$, $\mu_k(B(T))=B(\mu_k(T))$, or, in words, flipping $\gamma_k$ changes $B(T)$ by mutation in direction $k$.

As the following theorem shows, these triangulations entirely encode the combinatorics of type $A$ and $D$ $\A$-seed patterns. 

\begin{thm}[\cite{FST}] \label{ADtriangulation} Let $P=P_{n+3}$ (resp. $P=P_n^\bullet$). Consider an $\A$-seed pattern $\mc{S}$ such that some exchange matrix is $B(T_0)$ for some triangulation $T_0$ of $P$. Then $\mc{S}$ is type $A_n$ (resp. $D_n$) and there is a bijection $\gamma \mapsto a_\gamma$ between arcs of $P$ and $\A$-variables of $\mc{S}$. Further, if $\Sigma=(\textbf{a}, \textbf{x}, B)$ is a seed of $\mc{S}$, there is a unique triangulation $T$ such that $\textbf{a}=\{a_{\gamma}\}_{\gamma \in T}$ and $B=B(T)$. Finally, mutation in direction $k$ takes the seed corresponding to $T$ to the seed corresponding to $\mu_k(T)$, implying that the exchange graph of $\mc{S}$ is isomorphic to the flip graph of $P$.
\end{thm}


\subsection{Triangulations for types $B$ and $C$}

To obtain triangulations whose adjacency matrices are exchange matrices
of type $B_n$ and $C_n$ $\A$-seed patterns, we ``fold'' triangulations of $P_{2n+2}$ and $P_{n+1}^\bullet$. This is part of a larger theory of folded cluster algebras (see \cite{book}).

 Let $G=\ZZ/2\ZZ$. We write $P^{~G}_{2n}$ for $P_{2n}$ equipped with the $G$-action taking vertex $i$ to vertex $i':=i+n$ (with labels considered modulo $2n$). This induces an action of $G$ on the arcs of $P_{2n}$. The triangulations of $P^{~G}_{2n}$ are the triangulations of $P_{2n}$ fixed by the $G$-action, commonly called \emph{centrally symmetric} triangulations. 
 %It is not too difficult to see that, if $\gamma$ is an arc of a centrally symmetric triangulation, flipping all arcs in the orbit of $\gamma$ results in another centrally symmetric triangulation. As the arcs in an orbit are not adjacent, this triangulation does not depend on the order in which the arcs were flipped.
 
\begin{centering}
\begin{figure}
\includegraphics[width=\textwidth]{BC}
\caption{Triangulations of $P_8^G$ and $P_8^{\bullet ~G}$ are shown in solid lines. The orbit $[\tau]$ of $\tau$ is $\{\tau, \tau'\}$ and the flip of $[\tau]$ is dashed. On the right, $q_T([\tau])=\{\alpha, \beta, \tau, \tau'\}$.}
\label{fig:BC}
\end{figure}
\end{centering}

We write $P_{n}^{\bullet \ G}$ for $P_{n}^\bullet$ equipped with the $G$-action switching the notched and plain version of a radius. Again, the triangulations of $P_{n}^{\bullet \ G}$ are the triangulations of $P_{n}^{\bullet}$ fixed under the $G$-action, which are exactly those triangulations containing both the notched and plain versions of the same radius. 

Let $P \in \{P_{2n}, P_{n}^\bullet\}$, and let $T$ be a triangulation of $P^G$. For $\gamma \in T$, let $[\gamma]$ denote the $G$-orbit of $\gamma$. The \emph{quadrilateral} of $[\gamma]$, denoted $q_T([\gamma])$, is the $G$-orbit of $q_T(\gamma)$, or, in other words, all of the arcs and boundary segments adjacent to arcs in $[\gamma]$. Flipping the arcs in $[\gamma]$ results in another triangulation of $P^G$ (which does not depend on the order of arc flips, since arcs in the same orbit are pairwise not adjacent). We define the flip graph of $P^{G}$ in direct analogy to that of $P$; again, it is connected.

We associate to each triangulation $T$ of $P^G$ a skew-symmetrizable integer matrix $B^G(T)$, whose rows and columns are labeled by $G$-orbits of arcs of $T$. $B^G(T)$ is obtained from the usual signed adjacency matrix $B(T)$ of $T$ (see \cite[Section 4.4]{book} for details); again, if two $G$-orbits consist of pairwise non-adjacent arcs, the corresponding matrix entry is 0.% Let $T=\{\gamma_1, \dots, \gamma_m\}$, and let $I, J$ be the indices of arcs in the $G$-orbit of $\gamma_i$ and $\gamma_j$. Then
%
%\begin{equation}b^G_{IJ}(T)= \sum_{i \in I} b_{ij}(T)
%\end{equation}
%
%where $B(T)=(b_{ij}(T))$ is the usual signed adjacency matrix of $T$. The entries of $B^G(T)$ are well-defined: for $g \in G$, if $g(\gamma_i)=\gamma_i'$ and $g(\gamma_j)=\gamma_j'$, then $b_{ij}=b_{i'j'}$, so the value of $b_{IJ}^G$ does not change if a different element of $J$ is used to compute $(2.4)$. 
%
%\textcolor{red}{[insert examples with Cartan companion of type $B_n$ and $C_n$?]}.

Just as with usual triangulations, (orbit of) arc flips and matrix mutation interact nicely: if $T$ is a triangulation containing arc $\gamma$, flipping the arcs in $[\gamma]$ corresponds to mutating $B^G(T)$ in the direction labeled by $[\gamma]$. Further, we have the following theorem.

\begin{thm}[\cite{book}] \label{BCtriangulation} Let $P=P_{n+1}^\bullet$ (resp. $P =P_{2n+2}$). Consider an $\A$-seed pattern $\mc{S}$ such that some exchange matrix is $B^G(T_0)$ for some triangulation $T_0$ of $P^G$. Then $\mc{S}$ is type $B_n$ (resp. $C_n$) and there is a bijection $[\gamma] \mapsto a_{[\gamma]}$ between arcs of $P$ and $\A$-variables of $\mc{S}$. Further, if $\Sigma=(\textbf{a}, \textbf{x}, B)$ is a seed of $\mc{S}$, there is a unique triangulation $T$ such that $\textbf{a}=\{a_{[\gamma]}\}_{\gamma \in T}$ and $B=B^G(T)$. Finally, mutation in direction $K$ takes the seed corresponding to $T$ to the seed corresponding to $\mu_K(T)$, implying that the exchange graph of $\mc{S}$ is isomorphic to the flip graph of $P^G$.
\end{thm}

\section{Dynkin type $\X$-seed patterns}

Let $\mc{S}$ be an $\X$-seed pattern of type $Z_n$ ($Z\in \{A, B, C,
D\}$) over an arbitrary semifield $\PP$, and let $\X(\mc{S})$ denote the
set of $\X$-variables of $\mc{S}$. Let $P$ be the surface whose
triangulations encapsulate the combinatorics of type $Z_n$ $\A$-seed
patterns ($P=P_{n+3}$ for $Z=A$, $P=P_n^\bullet$ for $Z=D$,
$P=P^{~G}_{2n+2}$ for $Z=C$, $P=P_{n+1}^{\bullet ~ G}$ for $Z=B$). In
this section, we relate the $\X$-variables of $\mc{S}$ to the
triangulations of $P$, and show a bijection between $\X{\mc{S}}$ and
quadrilaterals (with a choice of diagonal) of $P$ in the case when $\PP$
is the universal semifield. Note that in what follows, ``arc'' should
usually be understood to mean ``orbit of arc'' if $\mc{S}$ is type $B$ or $C$. 

 First, notice that Theorems \ref{ADtriangulation} and \ref{BCtriangulation} imply that one can associate to each triangulation of $P$ a seed of $\mc{S}$ such that mutation of seeds corresponds to flips of arcs. Indeed, consider any $\A$-seed pattern $\mc{R}$ with $\mc{R}|_{\X}=\mc{S}$; if the triangulation $T$ corresponds to the $\A$-seed $(\textbf{a}, \textbf{x}, B)$ with arc $\gamma_k$ corresponding to $\A$-variable $a_k$, then we associate to $T$ the $\X$-seed $(\textbf{x}, B)$ and to the arc $\gamma_k$ the $\X$-variable $x_k$. We write $\Sigma_T$ to indicate this association. Note that \emph{a priori} two distinct triangulations may be associated to the same $\X$-seed, and an $\X$-variable may be associated to a number of different arcs. Further, an arc may be associated to different $\X$-variables in different triangulations. 
 
 The next observation follows immediately from the definition of seed mutation.
 
 \begin{rmk} Consider a seed $(\textbf{x}, B)$. For $j \neq k$, if $b_{jk}=0$, then mutating at $k$ will not change $x_j$. Further, $b'_{ji}=b_{ji}$ and $b'_{ij}=b_{ij}$ for all $i$, since the skew-symmetrizability of $B$ implies $b_{kj}=0$ as well. Thus, if $k_1, \dots, k_t$ are indices such that $b_{k_{s}j}=0$, then the mutation sequence $\mu_{k_t}\circ\cdots \circ \mu_{k_2} \circ \mu_{k_1}$ leaves $x_j$ unchanged.
\end{rmk}

In other words, let $x$ be an $\X$-variable of $\Sigma_T$ corresponding to arc $\gamma \in T$. Any sequence of flips of arcs \emph{not} in $q_T(\gamma) \cup \{\gamma\}$ will result in a triangulation $T'$ such that $\gamma \in T'$ and the $\X$-variable of $\Sigma_{T'}$ corresponding to $\gamma$ will be $x$. This, in combination with results on the connectivity of certain subgraphs of the flip graph of $P$ \cite{FST}, gives the following proposition.

\begin{prop} \label{surjection} Let $Q'=\{q_T(\gamma) \cup \{\gamma\} | ~T \text{ a triangulation of }P, \gamma \in T\}$ be the set of quadrilaterals (with a choice of diagonal) of $P$. Then there is a surjection $f:Q' \to \X(\mc{S})$. 
\end{prop}

We remark that this proposition in fact holds in the generality of $\X$-seed patterns from surfaces, though we do not need that here. 

We have the immediate corollary:

\begin{cor} Let $\mf{q}(P)=|\{q_T(\gamma) | ~T \text{ a triangulation of }P, \gamma \in T\}|$ denote the number of quadrilaterals of $P$. Then $|\X(\mc{S})|\leq 2\mf{q}(P)$.
\end{cor}

The above statements hold regardless of the choice of $\PP$ and initial
$\X$-cluster. Because $|\X(\mc{S})|$ can easily be 1, it is clear that we
must make additional assumptions to determine anything further about
$|\X(\mc{S})|$. If we fix an exchange matrix $B$ and allow $\PP$ and the
$\X$-cluster of the seed $(\textbf{x}, B)$ to vary, $\mc{S}(\textbf{x},
B)$ will have the largest number of $\X$-variables when
$\PP=\QQ_{sf}(t_1, \dots, t_n)$ and $\textbf{x}$ consists of elements
that are algebraically independent over $\QQ(t_1, \dots, t_n)$. Indeed,
let $\mc{S}_{sf}$ be such a seed pattern, and $\mc{S}$ an arbitrary seed
pattern containing the exchange matrix $B$. The $\X$-variables of
$\mc{S}$ can be obtained from the $\X$-variables of $\mc{S}_{sf}$ by
replacing ``$+$'' with ``$\oplus$'' and evaluating at the appropriate elements of $\PP$, so we have $|\X(\mc{S})| \leq |\X(\mc{S}_{sf})|$. 

In light of this observation, we now focus on $\X(\mc{S}_{sf})$. To show that the surjection $f$ of Proposition \ref{surjection} is a bijection for $\mc{S}_{sf}$, it suffices to show that $f$ is injective for some $\X$-seed pattern of type $Z_n$. 

To do this, we use the ``geometric'' $\A$-seed pattern of type $Z_n$ given in \cite{FZ2, book}, which we denote by $\mc{R}(Z_n)$. Briefly, the $\A$-variables are certain elements of $\CC[\text{Mat}_{2, m}]$, the algebra of polynomial functions on the space of $2 \times m$ complex matrices, where $m$ depends on $Z$ and $n$. In $\mc{R}(A_n)$ for example, $m=n+3$ and the $\A$-variables are exactly the Pl\"{u}cker coordinates; the $\X$-variables are quotients of Pl\"{u}cker coordinates.


\begin{prop} Consider the $\A$-seed pattern $\mc{R}=\mc{R}(Z_n)$. Then the surjection $f:Q' \to \X(\hat{\mc{R}})$ of Proposition \ref{surjection} is injective, where $\hat{\mc{R}}$ is given in \cite[Proposition 3.9]{FZ4}.
\end{prop}

This proposition is proved by showing that the $\X$-variables in $\hat{\mc{R}}$ associated to different quadrilaterals have different values on specific elements of $\text{Mat}_{2, m}$.

Theorem \ref{quadbijection} follows as a corollary, as does the number of $\X$-variables in $\mc{S}_{sf}$.

\begin{cor} $|\X(\mc{S}_{sf})|= 2\mf{q}(P)$.
\end{cor}

\subsection{Quadrilateral counts}

\begin{prop}\label{quadnumber} The number of quadrilaterals of each surface $P$ is

\begin{center}
\begin{tabular}{| c | c | c| c | }
 \hline
   $P$ & $P_{n+3}$ & $P_{2n+2}^{~G}, P_{n+1}^{\bullet G}$ & $P_{n}^\bullet$  \\ \hline
  $\mf{q}(P)$ & $\binom{n+3}{4}$ & $\frac{1}{6}n(n+1)(n^2+2)$ & $\frac{1}{6}n(n-1)(n^2+4n-6)$ \\ \hline
\end{tabular}
\end{center}
\end{prop}

For $P \neq P_{2n+2}^{~G}$, $\mf{q}(P)$ follow from a fairly straightforward inspection of the triangulations of the appropriate surfaces. For $P=P_{2n+2}^{~G}$, $\mf{q}(P)$ follows from a bijection between quadrilaterals not appearing in centrally symmetric triangulations of $P_{2n+2}$ and quadrilaterals of $P_{2n+2}$ that appear in centrally symmetric triangulations and are not fixed by $G$-action.

% \begin{prop} Let $Q$ be the set of quadrilaterals of $P_{2n+2}$, $Q_1$ be the set of quadrilaterals whose vertices are contained in $\{i, \dots, i+n+1\}$ for some $i$ (with vertex labels considered mod $2n+2$), and $Q_2$ the subset of $Q\setminus Q_1$ of quadrilaterals whose vertices are not $\{i, j, i+n+1, j+n+1\}$. There is a bijection $Q_1 \to Q_2$, and thus $|Q_1|=\binom{2n+2}{4}-\binom{n+1}{2}$.
% \end{prop}
% 
% Note that the quadrilaterals of centrally symmetric triangulations of $P_{2n+2}$ are exactly those in $Q_1$ or those with vertices equal to $\{i, j, i+n+1, j+n+1\}$ for some $i, j$. The latter quadrilaterals are fixed by the action of $G$; the quadrilaterals in $Q_1$ have $G$-orbits of size 2. So we have as a corollary
% 
%  \[\mf{q}(C_n)=\frac{1}{2}|Q_1|+ \binom{n+1}{2}\]
%
%  
%  which is the value given in Proposition \ref{quadnumber}.

\subsection{Exceptional types}

Let $Z \in \{E_6, E_7, E_8, F_4, G_2\}$ and let $\mc{S}_{sf}$ be a type $Z$ $\X$-seed pattern over $\QQ_{sf}$ with one (equivalently every) $\X$-cluster consisting of algebraically independent elements. $\X(\mc{S}_{sf})$ was computed using a computer algebra system (Mathematica), by generating all possible $\X$-seeds via mutation.

\section{Corollaries and Conjectures}

We make a few remarks regarding the implications of these results to the larger theory of $\X$-seed patterns and cluster algebras, and conjectural extensions.

The bijection between quadrilaterals and $\X$-variables for $\mc{S}_{sf}$ implies

\begin{cor}
The exchange graph of $\mc{S}_{sf}$ in types $A$, $B$, $C$, and $D$ coincides with the exchange graph of any $\A$-seed pattern of the same type.
\end{cor}

Recall that the diagonals of a quadrilateral can be uniquely associated to a pair of $\A$-variables. These pairs are precisely those variables that appear together on the left hand side of an exchange relation; such $\A$-variables are called \emph{exchangeable}. Composing the bijection from ordered pairs of exchangeable $\A$-variables to quadrilaterals with a choice of diagonal with the bijection of Theorem \ref{quadbijection} gives the following corollary for classical types. It was checked by computer for exceptional types.

\begin{cor}\label{expairbijection} Let $\mc{R}$ be an $\A$-seed pattern of finite type. There is a bijection between ordered pairs of exchangeable $\A$-variables in $\mc{R}$ and $\X(\mc{S}_{sf})$.
\end{cor}

If $\mc{R}$ is a finite type $\A$-seed pattern over the tropical semifield, the numbers computed here are the number of $\X$-variables in $\hat{\mc{R}}$ if one (equivalently, every) extended exchange matrix of $\mc{R}$ is full rank. Recall that in this setting, the $\X$-variables of $\hat{\mc{R}}$ exactly record the two terms on the right hand side of an exchange relation. In the bijection of Corollary 5.2, the pairs of exchangeable $\A$-variables are mapped exactly to the $\X$-variables recording the exchange relation that the $\A$-variables satisfy. This implies

 \begin{cor}\label{uniqueexchange} Let $\mc{R}$ be an $\A$-seed pattern of type $A$, $B$, $C$, or $D$ over the tropical semifield with one full rank extended exchange matrix. Then the two terms of the right hand side of an exchange relation \eqref{exchangerel} uniquely determines the left.
 \end{cor}
 
 \begin{conj} Let $\mc{S}$ be an $\X$-seed pattern from a marked surface $(S, M)$. Then the map from $\{q_T(\gamma) \cup \{\gamma\}| ~ T \text{ a triangulation of } (S, M), \gamma \in T\}$ and $\X(\mc{S}_{sf})$, which sends $q \cup \{\gamma\}$ to $x_{q, \gamma}$, is a bijection.
 \end{conj}

%We also remark that Corollary \ref{uniqueexchange} gives rise to the following question regarding the structure of $\A$-seeds: for which $\A$-seed patterns does an exchangeable pair of $\A$-variables uniquely determine an exchange relation?

Lastly, in the original development of finite type $\A$-seed patterns, seed patterns were connected to root systems of the same type. In particular, there is a bijection between $\A$-variables and almost positive roots (positive roots and negative simple roots). Two variables are exchangeable if and only if the corresponding roots $\alpha, \beta$ have $(\alpha|| \beta)=(\beta|| \alpha)=1$, where $(-||-)$ is the \emph{compatibility degree} \cite{FZ2}. This, combined with Corollary \ref{expairbijection}, give a root theoretic interpretation of the number of $\X$-coordinates of $\mc{S}_{sf}$.

\begin{cor} \label{cor:rootinterp}For $\mc{S}_{sf}$ of finite type, $|\X(\mc{S}_{sf})|$ is the number of pairs of roots $(\alpha, \beta)$ such that $(\alpha|| \beta)=(\beta|| \alpha)=1$ in the root system of the same type.
\end{cor}

\acknowledgements{The author would like to thank Lauren Williams for her helpful comments on drafts of this abstract, and Dan Parker and Adam Scherlis for making their undergraduate theses available. The author would also like to thank Nathan Reading for computing the number of pairs of exchangeable $\A$-variables for seed patterns of exceptional types, thereby extending Corollaries \ref{expairbijection} and \ref{cor:rootinterp} to exceptional types.}

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