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% * <alexander.garver@gmail.com> 2017-10-27T13:07:17.454Z:
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\title{Semistable subcategories for tiling algebras}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Monica
Garcia\thanks{\href{mailto:monica.garcia@matem.unam.mx}{monica.garcia@matem.unam.mx}}\addressmark{1}
\and Alexander
Garver\addressmark{2}\thanks{\href{mailto:alexander.garver@lacim.ca}{alexander.garver@lacim.ca}}}

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\address{\addressmark{1}Departamento de Matem\'aticas y F\'isica, Universidad Aut\'onoma de Aguascalientes \\ \addressmark{2}Laboratoire de Combinatoire et d'Informatique Math\'ematique, Universit\'e du Qu\'ebec \`a Montr\'eal}

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\abstract{Semistable subcategories were introduced in the context of Mumford's GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls  and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding lattice of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Sim{\~o}es and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Our work recovers that of Ingalls and Thomas in Dynkin type A.}%\lipsum[1]

%% put your French abstract here, or comment this out if you don't have one
\resume{Les sous-cat\'egories semi-stables ont \'et\'e introduites dans le contexte du GIT de Mumford et interpr\'et\'ees par King en termes de th\'eorie des repr\'esentations des alg\`ebres de dimension finie. Ingalls et Thomas ont montr\'e plus tard que pour les alg\`ebres de dimension finie de type Dynkin et affines, l'ensemble ordonn\'e des sous-cat\'egories semi-stables est isomorphe au treillis des partitions non-crois\'ees correspondant. Nous montrons que les sous-cat\'egories semi-stables d\'efinies par des alg\`ebres de pavage, introduites par Sim{\~o}es et Parsons, sont en bijection avec des partitions arborescentes non-crois\'ees, introduites par le second auteur et McConville. Nous retrouvons des r\'esultats  d'Ingalls et Thomas pour Dynkin de type A.}%\lipsum[2]

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\keywords{quiver, stability condition, noncrossing partition, wide subcategory}

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\begin{document}

\maketitle
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\section{Introduction}
Mumford's geometric invariant theory (GIT) provides a technique for taking the quotient of an algebraic variety by certain types of group actions in such a way that the resulting quotient is again an algebraic variety. Given a variety $V$ and a reductive algebraic group $G$ acting linearly on $V$, one replaces $V$ by its ``semistable points'' and then forms the GIT quotient $V\!/\!\!/\!G$, which is an algebraic variety. 

In \cite{king1994moduli}, King interpreted this notion of semistable points in terms of representation theory of algebras as follows. Let $\Lambda = \Bbbk Q/I$ be the path algebra of a \textbf{quiver} $Q$ (i.e., a directed graph) modulo an admissible ideal $I$ and $\Bbbk$ is an algebraically closed field. Recall that the \textbf{path algebra} $\Bbbk Q$ consists of formal $\Bbbk$-linear combinations of paths in $Q$, and its multiplication is given by composition of paths. For such algebras, any $\Lambda$-module $M$ may be regarded as a \textbf{representation} of $Q$ (i.e., an assignment of a finite dimensional $\Bbbk$-vector space $M_i$ to each vertex of $Q$ and a $\Bbbk$-linear map to each arrow of $Q$). A representation $M$ of $Q$ naturally defines a \textbf{dimension vector}, denoted by $\textbf{dim}(M) := (\dim_\Bbbk M_i)_{i=1}^n \in \mathbb{Z}^n_{\ge 0},$ where $n$ will henceforth denote the number of vertices of $Q$. 

Now let $V=\text{mod}(\Lambda,\textbf{d})$, the variety of finitely generated $\Lambda$-modules with dimension vector $\textbf{d} = (d_1, \ldots, d_n),$  and let $G = \prod_{i=1}^n \text{GL}_{d_i}(\Bbbk)$ act by base change at each vertex of $Q$. In \cite{king1994moduli}, King showed that the semistable points of $V$, which from now on we call \textbf{semistable representations} (resp., \textbf{stable representations}), are the representations $M$ where there exists a linear map $\theta \in \text{Hom}(\mathbb{Z}^n, \mathbb{Z})$ satisfying 

$\bullet$ $\theta(\textbf{dim}(M)) = 0$, and

$\bullet$ for any subrepresentation $N \subset M$, one has
$\theta(\textbf{dim}(N)) \le 0$ (resp., $\theta(\textbf{dim}(N)) < 0$).

\noindent We refer to such linear maps $\theta \in \text{Hom}(\mathbb{Z}^n,\mathbb{Z})$ as \textbf{stability conditions} on $\text{mod}(\Lambda)$, the category of finitely generated $\Lambda$-modules. Any choice of stability condition $\theta$ defines a subcategory $\theta^{ss}$ of $\text{mod}(\Lambda)$ consisting of the \textbf{$\theta$-semistable representations}. We refer to $\theta^{ss}$ as a \textbf{semistable subcategory}. Note that two different stability conditions may define the same semistable subcategory.

We study the poset of all semistable subcategories of $\text{mod}(\Lambda)$ ordered by inclusion, denoted $\Lambda^{ss}$. There are close connections between the theory of semistable subcategories and the combinatorics of Coxeter groups. If $\Lambda = \Bbbk Q$ where $Q$ is an orientation of a simply-laced Dynkin or extended Dynkin diagram, it follows from \cite[Theorem 1.1]{ingalls2009noncrossing} that $\Lambda^{ss}$ is isomorphic to the poset of noncrossing partitions associated with $Q$. 


Other important examples of algebras $\Lambda$ include cluster-tilted algebras \cite{bakke2007cluster}, which appear in the context of cluster algebras, and also preprojective algebras. In the latter case, in \cite{thomas2017stability} it is shown that $\Lambda^{ss}$ is isomorphic to the shard intersection order of the Coxeter arrangement associated with $Q$ (see \cite{reading2011noncrossing} for more on the shard intersection order). %\textcolor{red}{cite Buan--Marsh--Reiten}, and... 

%\lipsum[3]

%\begin{equation}\label{eqn:eq1}
%\int_0^\infty \exp(-x)dx = 1.
%\end{equation}

The purpose of this work is to combinatorially classify the semistable
subcategories for the class of \textbf{tiling algebras}, introduced in
\cite{simoes2017endomorphism} to study endomorphism algebras of maximal
rigid objects in some negative Calabi--Yau categories. Following \cite{garver_oriented_rep_thy}, these algebras, denoted $\Lambda_T$, are defined by the data of a tree $T$ embedded in the disk $D^2$ whose interior vertices have degree at least 3 (see Figure~\ref{tree_quiver}). Examples of tiling algebras are given by the cluster-tilted algebras of cluster type $A$; the trees defining these algebras are those whose interior vertices are of degree 3.

\begin{figure}
$$\begin{array}{ccccccccc}\includegraphics[scale=1.1]{QuivTreeEx1.pdf} & & & \includegraphics[scale=1.1]{QuivTreeEx2.pdf} \\ (a) & & & (b)
\end{array}$$
\caption{We show a tree $T$ in (a) and the quiver $Q_T$ it defines in (b). The associated tiling algebra is $\Lambda_T = \Bbbk Q_T/I_I$ where $I_T = \langle \alpha_2\alpha_1, \alpha_3\alpha_2, \alpha_1\alpha_3, \alpha_5\alpha_4, \alpha_6\alpha_5, \alpha_4\alpha_6, \alpha_8\alpha_7 \rangle$.}
\label{tree_quiver}
\end{figure}

The tree $T$ defines a simplicial complex of noncrossing arcs on $T$ called the noncrossing complex, denoted by $\Delta^{NC}(T)$ (see Section~\ref{Sec_nc_complex}). Each facet of $\Delta^{NC}(T)$ consists of \textbf{red arcs}, \textbf{green arcs}, and \textbf{boundary arcs}. In \cite{manneville2017geometric}, it is shown that if $\delta$ is a green or red arc in a facet of $\Delta^{NC}(T)$, it gives rise to a \textbf{g-vector}, denoted $\textbf{g}(\delta) \in \mathbb{Z}^n$. Additionally, in \cite{garver_oriented_rep_thy}, it is shown that the facets of $\Delta^{NC}(T)$ are in bijection with \textbf{wide subcategories} of $\text{mod}(\Lambda_T).$ With these facts in mind, we arrive at our main theorem.

\begin{thm}\label{thm:thm1}
Let $\mathcal{W} \subset \text{mod}(\Lambda_T)$ be a wide subcategory, and let $\mathcal{F}_\mathcal{W}$ be the corresponding facet of ${\Delta}^{NC}(T)$ and $\mathcal{F}_\mathcal{W}^{gr}$ the set of green arcs. Then the \textbf{Kreweras stability condition} defined as
$$\begin{array}{rcl}
\theta_{\mathcal{F}_\mathcal{W}} : \mathbb{Z}^{n} & \longrightarrow &\mathbb{Z} \\
\textbf{dim}(M) & \longmapsto & \sum_{\delta \in \mathcal{F}_\mathcal{W}^{gr}} \langle \mathbf{g}(\delta), \textbf{dim}(M) \rangle, 
\end{array}$$
where $M \in \text{mod}(\Lambda_T)$ and $\langle -,-\rangle$ is the standard Euclidean inner product, satisfies $\theta^{ss}_{\mathcal{F}_\mathcal{W}} = \mathcal{W}$. Conversely, any semistable subcategory of $\text{mod}(\Lambda_T)$ is a wide subcategory of $\Lambda_T$.
\end{thm}

The paper is organized as follows. In Section~\ref{Sec_nc_complex}, we review the noncrossing complex of arcs on a tree. In Section~\ref{Section_cg_vec}, we associate $\textbf{g}$- and $\textbf{c}$-vectors to each facet of this complex, which are essential to our construction of semistable subcategories. In Section~\ref{sec_til_alg}, we define the tiling algebras that we will study. In Section~\ref{Sec_noncrossing_tree_part}, we define noncrossing tree partitions, which will classify the semistable subcategories of $\text{mod}(\Lambda_T)$. We also sketch an important step in the proof of Theorem~\ref{thm:thm1}. Lastly, in Section~\ref{Sec_additiona_stuff}, we propose a natural extension of our work to general gentle algebras. %\textcolor{red}{more?}


\section{Noncrossing complex}\label{Sec_nc_complex}

A \textbf{tree} $T=(V_T,E_T)$ is a finite connected acyclic graph. Any tree may be embedded in the disk $D^2$ in such a way that a vertex is on the boundary if and only if it is a leaf.  We will assume that any tree is accompanied by such an embedding in $D^2$. We say two trees $T$ and $T^\prime$ are \textbf{equivalent} if there is an ambient isotopy between the spaces $D^2\backslash T$ and $D^2\backslash T^\prime$. We consider trees up to equivalence. Additionally, we assume that the \textbf{interior vertices} of any tree $T$ (i.e., the nonleaf vertices of $T$) have degree at least 3. 

We say the closure of a connected component of $D^2\backslash T$ is a \textbf{face} of $T$. A \textbf{corner} $(v,F)$ of $T$ is a pair consisting of an interior vertex $v$ of $T$ and a face $F$ of $T$ that contains $v$.

\begin{figure}
$$\includegraphics[scale=1.05]{FlipEx1prime.pdf}$$
\caption{Both facets of this noncrossing complex contain 5 arcs. The boundary arcs are shown in gold. The marked corners of arcs are indicated by black dots. The faces are $F_1, F_2, F_3, F_4.$}
\label{nc_complex}
\end{figure}

An \textbf{acyclic path} supported by a tree $T$ is a sequence $(v_0, v_1,...,v_t)$ of pairwise distinct vertices of $T$ such that $v_i$ and $v_j$ are adjacent if and only if $|i-j|=1$. We will refer to $v_0$ and $v_t$ as the \textbf{endpoints} of the acyclic path. Since $T$ is acyclic, any acyclic path is determined by its endpoints, and we can therefore write $(v_0,v_1,...,v_t)=[v_0,v_t]$. 

An \textbf{arc} $\delta=(v_0, v_1,...,v_t)$ is an acyclic path such that its endpoints are leaves and any two edges $(v_{i-1},v_i)$ and $(v_i,v_{i+1})$ are incident to a common face. We say $\delta$ \textbf{contains a corner} $(v,F)$ if $v = v_i$ for some $i = 0,1, \ldots, t$ and $(v_{i-1},v_i)$ and $(v_i,v_{i+1})$ are incident to $F$. We also note that $\delta$ divides $D^2$ into two \textbf{regions} composed of disjoint subsets of faces of $T$. We let $\text{Reg}(\delta,F)$ denote the region defined by $\delta$ which contains $F$. We say that two arcs $\delta$ and $\delta^\prime$ are \textbf{crossing} if given any regions $\text{R}_\delta$ and $\text{R}_{\delta^\prime}$ defined by $\delta$ and $\delta^\prime$, respectively, then $\text{R}_\delta \not \subset \text{R}_{\delta^\prime}$ or $\text{R}_{\delta^\prime} \not \subset \text{R}_{\delta}.$ Otherwise, we say $\delta$ and $\delta^\prime$ are \textbf{noncrossing}.

Define the \textbf{noncrossing complex} of $T$, denoted $\Delta^{NC}(T)$, to be the abstract simplicial complex of noncrossing arcs of $T$. By \cite[Corollary 3.6]{garver2016oriented}, this is a \textbf{pure complex} (i.e., any two facets have the same cardinality). We will primarily work with the facets of $\Delta^{NC}(T)$. 

Let $\mathcal{F}$ be any facet of $\Delta^{NC}(T)$. The arcs of $\mathcal{F}$ containing a corner $(v,F)$ are linearly ordered: two arcs $\delta, \gamma \in \mathcal{F}$ containing $(v,F)$ satisfy $\delta \le_{(v,F)} \gamma$ if and only if $\text{Reg}(\delta,F) \subset \text{Reg}(\gamma, F).$ That such arcs are linearly ordered follows from the fact that they are pairwise noncrossing. We say that an arc $\delta$ of $\mathcal{F}$ is \textbf{marked} at corner $(v,F)$ if $\delta$ contains $(v,F)$ and it is the maximal such arc with respect to $\le_{(v,F)}$. In \cite[Proposition 3.5]{garver2016oriented}, it is shown that every $\delta \in \mathcal{F}$ is marked at either one or two corners. In the latter case, the two corners at which $\delta$ is marked belong to different regions defined by $\delta$. We refer to the arcs marked at a single corner as \textbf{boundary arcs}, and we denote the set of boundary arcs of $\mathcal{F}$ by $\mathcal{F}^\partial$. We show an example of the noncrossing complex in Figure~\ref{nc_complex}.

The arcs of $\mathcal{F}$ that are not boundary arcs come with extra data of a color as follows. A \textbf{flag} is a triple $(v,e,F)$ of a vertex $v$ incident to an edge $e$, which is incident to the face $F$. We say a flag is \textbf{green} if by orienting $e$ away from $v$, face $F$ is to the left of $e$. Otherwise, we say $(v,e,F)$ is \textbf{red}. Let $(v,F)$ and $(u,G)$ be the two corners in which $\delta \in \mathcal{F}$ is marked, and let $e$ and $e'$ be edges of $T$ contained in $[v,u]$ where the former is incident to $v$ and the latter is incident to $u$. Both $(v,e,F)$ and $(u,e',G)$ have to be of the same color, as $F$ and $G$ belong to different regions determined by $\delta$. We say $\delta$ is a \textbf{green arc} if it is marked at corners belonging to green flags, otherwise we say it is a \textbf{red arc}. Define $\mathcal{F}^{gr}$ (resp., $\mathcal{F}^{red}$) to be the set of green (resp., red) arcs of $\mathcal{F}$. Observe that $\mathcal{F}=\mathcal{F}^{red} \sqcup \mathcal{F}^{gr} \sqcup \mathcal{F}^\partial$. We show examples of red and green arcs in Figures~\ref{nc_complex} and \ref{nc_to_ncp}.


\section{Facets and their \textbf{c}- and \textbf{g}-vectors}\label{Section_cg_vec}

In this section, we show how to associate a family of vectors in $\mathbb{Z}^n$ to each facet of the noncrossing complex where $n$ denotes the number of edges of $T$ connecting two interior vertices of $T$. We let $\text{Int}(E_T)$ denote the set of such edges of $T$ and $\{\textbf{h}_e\}_{e \in \text{Int}(E_T)}$ the canonical basis of $\mathbb{Z}^{|\text{Int}(E_T)|} \cong \mathbb{Z}^n$. The definitions we present in this section are reformulations of the definitions presented in \cite{manneville2017geometric}.

Now, fix a facet $\mathcal{F} \in \Delta^{NC}(T)$ and a red or green arc $\gamma = (v_0, v_1 \ldots, v_t) \in \mathcal{F}$. By choosing an orientation of $\gamma$, we define $\textbf{g}(\gamma):= \sum_{e \in \text{Int}(E_T)}g^e_{\gamma}\textbf{h}_e \in \mathbb{Z}^n$ where for each $e = (v_i,v_{i+1}) \in \text{Int}(E_T)$ we set
\[
g^{e}_\gamma := 
\begin{cases}
1 & \textnormal{if } \gamma \textnormal{ turns left at } v_i \textnormal{ and right at } v_{i+1},\\
-1 & \textnormal{if } \gamma \textnormal{ turns right at } v_i \textnormal{ and left at } v_{i+1}, \\
0  &  \textnormal{if } \gamma \textnormal{ does not change direction from } v_{i}\\ 
 &  \textnormal{to } v_{i+1} \textnormal{ or if } e \textnormal{ is not an edge in } \gamma, \\
\end{cases}
\]
and we refer to $\textbf{g}(\gamma)$ as the $\textbf{g}$\textbf{-vector} of $\gamma$ (see Figure~\ref{g_vec_ex}). Observe that $\textbf{g}(\gamma)$ is independent of the choice of orientation of $\gamma$. We define the \textbf{zigzag} of $\gamma$ to be the set $Z_\gamma = Z_\gamma^+ \sqcup Z_\gamma^- \subset \text{Int}(E_T)$ of edges $e$ of $T$ such that $g_\gamma^e \neq 0$, where $Z_\gamma^+$ (resp., $Z_\gamma^-$) is the set of edges such that $g_\gamma^e = 1$ (resp., $g_\gamma^e = -1$). We also let $G(\mathcal{F}):= \{\textbf{g}(\gamma)\}_{\gamma\in \mathcal{F}^{red}\sqcup \mathcal{F}^{gr}}$.

\begin{figure}[h]
\includegraphics[width=.7\textwidth]{gcoef.pdf}
\centering
\caption{\footnotesize{Different values for $g^e_\gamma$}}
\label{g_vec_ex}
\end{figure}

Next, we let $s_{\gamma,\mathcal{F}} = [v,u]$ denote the acyclic path where $(u,F)$ and $(v,G)$ are the corners at which $\gamma$ is marked in $\mathcal{F}$. We define the $\textbf{c}$\textbf{-vector} of $\gamma$ with respect to $\mathcal{F}$ to be $\textbf{c}_\mathcal{F}(\gamma) := \sum_{e\in s_{\gamma,\mathcal{F}}}\textbf{h}_e \in \mathbb{Z}^n$ (resp., $\textbf{c}_\mathcal{F}(\gamma) := -\sum_{e\in s_{\gamma,\mathcal{F}}}\textbf{h}_e \in \mathbb{Z}^n$) if $\gamma$ is green (resp., red). Note that the \textbf{c}-vector of $\gamma$ depends on the choice of facet $\mathcal{F}$ containing $\gamma$, whereas the \textbf{g}-vector $\textbf{g}(\gamma)$ is intrinsic to $\gamma$. We also let $C(\mathcal{F}) := \{\textbf{c}_\mathcal{F}(\gamma)\}_{\gamma\in \mathcal{F}^{red}\sqcup \mathcal{F}^{gr}}$. As the following proposition shows, the \textbf{c}- and \textbf{g}-vectors defined by a given facet are dual bases of $\mathbb{R}^n$.

\begin{prop}\cite[Proposition 20]{manneville2017geometric}
For any $\gamma, \delta \in \mathcal{F}$ we have $\langle \textbf{g}(\delta), \textbf{c}_\mathcal{F}(\gamma) \rangle \in \{0,1\}$ and equals 1 if and only if $\gamma = \delta$.
\end{prop}

\begin{ex}\label{a2_vectors_ex}
Consider the tree in Figure~\ref{A2_vecs} where $Int(E_T)=\{e_1,e_2\}$. The \textbf{g}- and \textbf{c}- vectors associated to the facet in this figure are as follows:

\begin{center}
\begin{minipage}[c]{.4\textwidth}
\begin{tabular}{l l}
$\mathbf{g}(\gamma)=(-1,0)$ & $\mathbf{c_\mathcal{F}}(\gamma)= (-1, -1)$\\
$\mathbf{g}(\delta)=(-1,1)$ & $\mathbf{c_\mathcal{F}}(\delta)= (0, 1).$\\
\end{tabular}
\end{minipage}
\end{center}  
\end{ex}


We end this section with a lemma that we interpret representation theoretically in the next section. For $s_{\gamma, \mathcal{F}} = (v_0,\ldots, v_t)$, let $C_{s_{{\gamma},\mathcal{F}}}$ denote the set of acyclic paths $(v_i,\ldots,v_j)$ such that

$\bullet$ if $i>0$ then $s$ turns right at $v_i$, and

$\bullet$ if $j<l$ then $s$ turns left at $v_j$.

\noindent {One has that $C_{s_{{\gamma},\mathcal{F}}}\backslash\{s_{\gamma,\mathcal{F}}\} $ is non-empty if and only if $s_{\gamma, \mathcal{F}}$ contains at least two edges.}

\begin{lem}\label{lem1}
Let $\mathcal{F}$ be a facet of $\Delta^{NC}(T)$ with at least one green arc, and let $\gamma \in \mathcal{F}^{red}$ be a red arc such that $s_{\gamma, \mathcal{F}}$ contains at least two edges of $T$. Then there exists a green arc $\mu \in \mathcal{F}^{gr}$ such that  $|Z_\mu^- \cap \{\text{edges of $t$}\}|=|Z_\mu^+ \cap \{\text{edges of $t$}\}|+1$ for any $t \in C_{s_{{\gamma},\mathcal{F}}}\backslash \{s_{\gamma,\mathcal{F}}\}$. Moreover, for any green arc $\mu$, $|Z_\mu^- \cap \{\text{edges of $t$}\}| \geq |Z_\mu^+ \cap \{\text{edges of $t$}\}|$.
\end{lem}

\begin{figure}
$$\includegraphics[scale=1.5]{A2_tree1.pdf}$$
\caption{\footnotesize{The tree and facet $\mathcal{F}$ of $\Delta^{NC}(T)$ from Example~\ref{a2_vectors_ex}.}}
\label{A2_vecs}
\end{figure}

\section{Tiling algebras}\label{sec_til_alg}

We now recall how a tree gives rise to a finite dimensional algebra. Given a tree $T$, let $Q_T$ be the quiver whose vertex set is $\text{Int}(E_T)$ and where $e, e' \in \text{Int}(E_T)$ are connected by an arrow in $Q_T$ if they meet in a corner of $T$. In this case, 
$e \: \overset{\alpha}{\rightarrow} \: e'$
if and only if $e'$ is counter-clockwise from $e$. We define $I_T \subset \Bbbk Q_T$ to be the ideal generated by the relations $\alpha \beta$ where $\alpha : e_2 \rightarrow e_3$ defines the corner $(v,F)$ and $\beta : e_1 \rightarrow e_2$ defines the corner $(v,G)$. 

We define the \textbf{tiling algebra} of $T$ to be $\Lambda_T := \Bbbk Q_T/I_T$ where $\Bbbk$ is an algebraically closed field. We invite the reader to check that $\text{dim}_\Bbbk\Lambda_T = 3$ (resp., 25) when $T$ is the tree from Figure~\ref{A2_vecs} (resp., Figure~\ref{tree_quiver}).

%and $\Bbbk Q$ denotes the $\Bbbk$-algebra spanned by all paths of length $l \geq 0$ in $Q_T$ , with the multiplication of two paths given by its concatenation.

The category of finitely generated left modules over $\Lambda_T$ is equivalent to category of finite dimensional representations of $Q_T$ over $\Bbbk$ that are \textbf{compatible} with the relations from  $I_T$ (i.e., a representation $V=((V_i)_{i \in {Q_T}_0}, (\varphi_\alpha)_{\alpha \in {Q_T}_1})$ of $Q_T$ where  $\varphi_{\alpha}\varphi_{\beta} = 0$ for all $\alpha\beta\in I_T$).\footnote{For a general finite dimensional $\Bbbk$-algebra $\Lambda = \Bbbk Q/I$ where $I$ is an admissible ideal, one can also equivalently describe modules over $\Lambda$ as representations of $Q$ compatible with $I$.} We also know that the indecomposable $\Lambda_T$-modules are \textbf{string modules}, denoted $M(w)$, which was first observed in \cite[Proposition 3.2]{simoes2017endomorphism}. A \textbf{string} $w = w_1\stackrel{\alpha_1}{\longleftrightarrow} w_2 \stackrel{\alpha_2}{\longleftrightarrow} \cdots \stackrel{\alpha_{k-1}}{\longleftrightarrow} w_k$ is an irredundant walk in $Q_T$ where for any $i \in \{1,\ldots, k-1\}$ if $\alpha_i\alpha_{i+1}\neq 0$ in $\Bbbk Q_T$, then $\alpha_i\alpha_{i+1} \not \in I_T$. In other words, a string is an irredundant walk in $Q_T$ that obeys the relations in $I_T$. In the setting of tiling algebras, all strings are supported on connected acyclic subgraphs of $Q_T$, but this is not the case in general. The string module $M(w)$ is the representation of $Q_T$ obtained by assigning the vector space $\Bbbk$ to each vertex in the string $w$ and identity morphisms to each arrow in $w$.\footnote{For simplicity, we have given the definition of $M(w)$ only in the generality of tiling algebras.} 

Using these facts, we obtain that the indecomposable $\Lambda_T$-modules are parameterized by \textbf{segments} of $T$ (i.e., acyclic paths $s=(v_0, v_1,...,v_t)$ whose endpoints are interior vertices of $T$ and any two edges $(v_{i-1},v_i)$ and $(v_i,v_{i+1})$  are incident to a common face) \cite[Corollary 4.3]{garver_oriented_rep_thy}. We show an example of this bijection in Figure~\ref{ind_to_segs}. Using this bijection, one obtains the following lemma.

\begin{lem}\label{lem2}
Given a facet $\mathcal{F}$ of $\Delta^{NC}(T)$ and $\gamma \in \mathcal{F}$, the set map $C_{s_{{\gamma},\mathcal{F}}} \to \text{mod}(\Lambda_T)$ defined by $t \mapsto M(w(t))$ induces a bijection from $C_{s_{{\gamma},\mathcal{F}}}$ to the indecomposable submodules of $M(w(s_{\gamma, \mathcal{F}})).$
\end{lem}

%\textcolor{violet}{
%It follows from this and Lemma~\ref{lem1} that for fixed facet $\mathcal{F}$, any segment $s_{\gamma , \mathcal{F}}$ corresponding to a red arc $\gamma \in \mathcal{F}$ satisfies that $\theta_\mathcal{F}(M(w(s_{\gamma, \mathcal{F}})))=0$ and $\theta_\mathcal{F}(M(w(t))) < 0$ for any indecomposable submodule $M(w(t))$ of $M(w(s_{\gamma, \mathcal{F}}))$. That is, $M(w(s_{\gamma, \mathcal{F}}))$ is an stable object of the stability condition $\theta_\mathcal{F}$ introduced in Theorem~\ref{thm:thm1}. On the other hand, if $s_{\gamma , \mathcal{F}}$ corresponds to a green arc, $\theta_\mathcal{F}(M(w(s_{\gamma, \mathcal{F}})))=1$, which means that no indecomposable module coming from a green arc is semistable.
%}

\begin{figure}
%$$\includegraphics[scale=1]{}$$
$$\begin{array}{cccc} \includegraphics[scale=1.1]{SegEx2.pdf} & \includegraphics[scale=1.1]{StringEx2.pdf} & \raisebox{1.1in}{\begin{xy} 0;<1pt,0pt>:<0pt,-1pt>:: 
(0,0) *+{0} ="0",
(30,0) *+{\Bbbk} ="1",
(15,30) *+{\Bbbk} ="2",
(0,60) *+{\Bbbk} ="3",
(30,75) *+{0} ="4",
(60,0) *+{\Bbbk} ="5",
(75,30) *+{\Bbbk} ="6",
(90,60) *+{\Bbbk} ="7",
(65,75) *+{0} ="8",
"1", {\ar_0"0"},
"0", {\ar_0"2"},
"2", {\ar_1"1"},
"1", {\ar^1"5"},
"2", {\ar_1"3"},
"4", {\ar_0"2"},
"3", {\ar_0"4"},
"5", {\ar^1"6"},
"7", {\ar_1"6"},
"8", {\ar_0"7"},
\end{xy}}  \\ {segment \ s = [5,8]} & {string \ w(s)} &  {string \ module \ M(w(s))} \end{array}$$
\caption{A segment $s = [5,8]$, the string $w(s)$ that it defines, and the corresponding string module $M(w(s))$.}
\label{ind_to_segs}
\end{figure}

It follows from this and Lemma~\ref{lem1} that for fixed facet $\mathcal{F}$, any red arc $\gamma \in \mathcal{F}$ satisfies $\theta_\mathcal{F}(M(w(s_{\gamma, \mathcal{F}})))=0$ and $\theta_\mathcal{F}(M(w(t))) < 0$ for any indecomposable submodule $M(w(t))$ of $M(w(s_{\gamma, \mathcal{F}}))$. That is, $M(w(s_{\gamma, \mathcal{F}}))$ is a $\theta_\mathcal{F}$-stable object. On the other hand, any green arc $\gamma \in \mathcal{F}$ satisfies $\theta_\mathcal{F}(M(w(s_{\gamma, \mathcal{F}})))=1$, which means that no indecomposable module coming from a green arc is $\theta_\mathcal{F}$-semistable.


%\textcolor{blue}{Moreover, as $\Lambda_T$ is in particular a string algebra, its indecomposable modules can be parametrized in terms of string modules $M(\omega)$, that is modules whose associated representation is supported by strings in $Q_T$. A \textbf{string} in $Q_T$ is a connected subgraph of $Q_T$ such that it obeys the relations in $I_T$, a subgraph of $Q_T$ with no cycles.}



\section{Noncrossing tree partitions}\label{Sec_noncrossing_tree_part}
Now let $V^{\circ}$ denote the set of interior vertices of $T$, choose an $\epsilon$ so that the balls of radius $\epsilon$ around each vertex in $V^\circ$ do not intersect each other and are contained in $D^2$. For each corner $(v,F)$, fix a point $z(v,F)$ in the interior of $F$ and such that $d(z(v,F),v)=\epsilon$. Let  $(v,e,F)$ and $(u,e',G)$  be two green (resp., red) flags, a \textbf{green} (resp., \textbf{red}) admissible curve for $[v,u]$ is a simple curve $\sigma:[0,1] \rightarrow D^2\backslash V$ for which $\sigma(0)=z(v,F)$, $\sigma(1)=z(u,G)$ and $\sigma([0,1])$ crosses an edge $h$ of $T$ only if $h$ is an edge in $[v,u]$.\footnote{Up to coloring-preserving isotopy relative to $z(v,F)$ and $z(u,G)$, there is a unique green (resp., red) admissible curve for $[v,u]$.} Two segments are \textbf{noncrossing} if they admit admissible curves that do not intersect each other, otherwise they are \textbf{crossing}. A segment is   \textbf{green} (resp., \textbf{red}) if it is represented by a green (resp., red) admissible curve. For $B\subseteq V^\circ$, let Seg$_r(B)$ be the set of inclusion-minimal red segments whose endpoints lie in $B$. A \textbf{noncrossing tree partition} $\textbf{B}=\{{B}_1,{B}_2,...,{B}_\ell\}$ is a set partition of $V^\circ$ such that any two segments of Seg$_r(\textrm{\textbf{B}})=\cup_{i=1}^{l} \textnormal{Seg}_r(B_i)$ are noncrossing and each block of \textbf{B} is \textbf{segment-connected} (i.e., for any two vertices in $B_i$ there exists a sequence of segments in $\text{Seg}_r(B_i)$ that joins them). Let NCP($T$) be the poset of noncrossing tree partitions of $T$ ordered by refinement. 


Returning to tiling algebras, a full, additive subcategory $\mathcal{W} \subset \textnormal{mod}(\Lambda_T)$ is a \textbf{wide subcategory} if it is abelian and for any short exact sequence $0\to X \to Z \to Y \to 0$ with $X,Y \in \mathcal{W}$, one has that $Z \in \mathcal{W}$. Let wide$(\Lambda_T)$ denote the poset of wide subcategories of mod$(\Lambda_T)$ ordered by inclusion. The intersection of two wide subcategories is a wide subcategory, and the zero subcategory (resp., mod($\Lambda_T$)) is the bottom (resp., top) element in wide$(\Lambda_T)$. As  $\Lambda_T$ is representation-finite, the poset wide$(\Lambda_T)$ is a lattice. In \cite[Theorem 7.1]{garver_oriented_rep_thy}, the second author and McConville obtained a poset isomorphism between the lattice of noncrossing tree partitions and the lattice of wide subcategories given by $$\begin{array}{rcl}\phi : \textnormal{NCP}(T) &\longrightarrow & \textnormal{wide}(\Lambda_T) \\
\textnormal{\textbf{B}} & \longmapsto &\textnormal{add}\left( \oplus_{w(s)} M(w(s)) \ | \ s \in \overline{\text{Seg}(\textnormal{\textbf{B}})} \right)\end{array}$$
where $\overline{\text{Seg}(\textnormal{\textbf{B}})} \subset \text{Seg}(T)$ is the smallest set containing $\text{Seg}(\textrm{\textbf{B}})$ such that $s = (v_0, v_1, \ldots, v_k),$ $t = (v_k, v_{k+1} \ldots, v_\ell) \in \text{Seg}(\textrm{\textbf{B}})$ and ${s\circ t} := (v_0, v_1 \ldots, v_k, v_{k+1}, \ldots, v_\ell) \in \text{Seg}(T)$ implies $s\circ t \in \overline{\text{Seg}(\textnormal{\textbf{B}})}$. Here $\text{add}(M)$ where $M \in \text{mod}(\Lambda_T)$ is defined as the smallest full, additive subcategory of $\text{mod}(\Lambda_T)$ that contains $M$ and is closed under direct summands. We show in Figure~\ref{bij_wide_nctp} an example of this isomorphism.

\begin{figure}
$$\includegraphics[scale=0.8]{NC_Set_Part.pdf}
\hspace{2mm} \includegraphics[scale=1.15]{A2_wide.pdf}$$
\caption{The lattice of wide subcategories of mod$(\Lambda_T)$ and its corresponding lattice of noncrossing tree partitions, which we show realized as sets of red admissible curves.}
\label{bij_wide_nctp}
\end{figure}

Now fix a facet $\mathcal{F} \in \Delta^{NC}(T)$. For each $\delta \in \mathcal{F}\backslash\mathcal{F}^\partial$, let $\sigma$ be an admissible curve for $s_{\delta,\mathcal{F}} = [v,u]$ where $v$ and $u$ are the two vertices corresponding to the corners $(v,F)$ and $(u,G)$ where $\delta$ is marked so that $\sigma$ has the same color as $\delta$. We define $\psi_r(\mathcal{F})$ (resp., $\psi_g(\mathcal{F})$) to be the set partition associated to set of red (resp., green) admissible curves only (see Figure~\ref{nc_to_ncp}). The map $\psi_r$ is a bijection between the facets of $\Delta^{NC}(T)$ and $\text{NCP}(T)$. In fact, it induces a bijection $\text{Kr}: \text{NCP}(T) \to \text{NCP}(T)$ defined by $\text{Kr}(\psi_r(\mathcal{F})):= \psi_g(\mathcal{F})$. We refer to $\text{Kr}(\textbf{B})$ with $\textbf{B} \in \text{NCP}(T)$ as the \textbf{Kreweras complement} of \textbf{B}. We remark that using the map $\phi$, we can regard the Kreweras complementation map as a cyclic action on wide subcategories of $\text{mod}(\Lambda_T)$.

In {\cite[Theorem 5.11]{garver2016oriented}}, it is shown that given a facet $\mathcal{F}$, the union of the sets $S_r = \{ \textnormal{Red curves associated to } \mathcal{F} \}$ and $S_g = \{ \textnormal{Green curves associated to  } \mathcal{F} \}$ define a new tree $\mathcal{T}_\mathcal{F}$, that we call the \textbf{red-green tree}, whose vertex set is $V^o$ and whose edge set is $S_r \sqcup S_g$. The red-green tree $\mathcal{T}_\mathcal{F}$ allows us to evaluate $\theta_\mathcal{F}$ on any indecomposable module.

% associated to neither red nor green arcs

%In \cite[Theorem 5.10]{garver_oriented_rep_thy}, it is shown that given a facet $\mathcal{F}$, the union of the sets $S_r = \{ \textnormal{Red curves associated to } \mathcal{F} \}$ and $S_g = \{ \textnormal{Green curves associated to  } \mathcal{F} \}$ define a new tree $\mathcal{T}_\mathcal{F}$, that we call the \textbf{red-green tree}, whose vertices are the interior vertices of $T$ all pairwise connected by a sequence of edges in $S_r \cup S_g$. 


%Now fix a facet $\mathcal{F} \in \Delta^{NC}(T)$, for each $\delta \in \mathcal{F}\backslash\mathcal{F}^\partial$ we assign an admissible curve $\sigma$ to the segment $[v,u]$ where $v$ and $u$ are the two vertices corresponding to the two corners $(v,F)$ and $(u,G)$ where $\delta$ is marked. The admissible curve $\sigma$ has the same color as that of $\delta$. We define $\psi(\mathcal{F})$ to be the set partition associated to set of red admissible curves only (see Figure~\ref{nc_to_ncp}). This process also defines a bijection in NCP($T$) by taking the partition \textbf{B} associated to the set of red admissible curves defined by $\mathcal{F}$ , and sending it to the partition associated to the set of green admissible curves. We will call this new partition the \textbf{Kreweras complement} of \textbf{B}, and we denote it by $\text{Kr}(\textbf{B})$. In \cite[Theorem 5.10]{garver_oriented_rep_thy}, it is shown that given a facet $\mathcal{F}$, the union of the sets $S_r = \{ \textnormal{Red curves associated to } \mathcal{F} \}$ and $S_g = \{ \textnormal{Green curves associated to  } \mathcal{F} \}$ define a new tree $\mathcal{T}_\mathcal{F}$, that we call the \textbf{red-green tree}, whose vertices are the interior vertices of $T$ all pairwise connected by a sequence of edges in $S_r \cup S_g$. 






%At the end of Section~\ref{sec_til_alg} we arrived to the conclusion that the indecomposable modules associate to red arcs are stable objects, whereas the ones associated to green arcs are not semistable. 



 \begin{lem}\label{basis}
For any facet $\mathcal{F}$, the set $\{ \textbf{dim}(M(w(s_{\gamma, \mathcal{F}}))) \ | \ \gamma \in \mathcal{F} \} $ is obtained by taking absolute values of the entries of all $\textbf{c}$-vectors in $C(\mathcal{F})$. Thus, this set is a basis of $\mathbb{Z}^{|Int(E_T)|}$. \footnote{We can rephrase this lemma by saying that this set generates the Grothendieck group of mod$(\Lambda_T)$.}
 \end{lem}

{With Lemma~\ref{basis} in mind, we show how to express the dimension vector of any indecomposable $\Lambda_T$-module as a linear combination of elements of $\{\textbf{dim}(M(w(s_{\gamma, \mathcal{F}}))) | \ \gamma \in \mathcal{F}\}$. Let $s=[v,u]$ be a segment in $T$ and $\varsigma=(\sigma_1,...,\sigma_\ell)$ the shortest sequence of admissible curves in $\mathcal{T}_\mathcal{F}$ joining $u$ and $v$ where $\sigma_i$ and $\sigma_j$ with $i < j$ share an endpoint only if $j = i+1$. By abuse of notation, we identify admissible curves with the segments associated to them. Writing $s = (v_0, v_1, \ldots, v_k)$ with $v = v_0$ and $u = v_k$, let $(v_0, v_{i_1}, \ldots, v_{i_r})$ with $u = v_{i_r}$ denote the vertices in $s$ that are endpoints of the curves in $\varsigma$. }

Next, consider the subsequence $(\sigma_1, \ldots, \sigma_{i_1})$ of $\varsigma$ where $\sigma_{i_1}$ is the first admissible curve in $\varsigma$ that has $v_{i_1}$ as an endpoint, and suppose $\sigma_i=[e,g]$ and $\sigma_{i+1}=[f,g]$ are two admissible curves in this sequence with a common subsegment. As shown in \cite[Section 5]{garver_oriented_rep_thy}, they must be different colors and agree along the shorter of the two segments. Without loss of generality suppose that $[f,g]$ is the shorter one. Construct a new admissible curve $\sigma^\prime$ for the segment $[e,f]$. The color of $\sigma^\prime$ is that of the admissible curve with longest associated segment. Note that $\textbf{dim}(M(w(\sigma')))= \textbf{dim}(M(w(\sigma_{i})))-\textbf{dim}(M(w(\sigma_{i+1})))$. Replace $(\sigma_1, \ldots, \sigma_{i_1})$ with the sequence of  admissible curves $(\sigma_1,...,\sigma_{i-1}, \sigma', \sigma_{i+2},..., \sigma_{i_1})$ that also joins $u$ with $v$. 

Alternatively, suppose $\sigma_i$ and $\sigma_{i+1}$ are two admissible curves of the same color. Replace $(\sigma_1, \ldots, \sigma_{i_1})$ with the sequence $(\sigma_1,...,\sigma_{i-1}, \sigma^{\prime\prime}, \sigma_{i+2},..., \sigma_{i_1})$ where $\sigma^{\prime\prime}$ is the admissible curve with the same color as $\sigma_i$ and whose segment satisfies $\textbf{dim}(M(w(\sigma^{\prime\prime})))= \textbf{dim}(M(w(\sigma_i)))+\textbf{dim}(M(w(\sigma_{i+1})))$.

We repeatedly apply these operations to obtain a sequence consisting of a single admissible curve $\sigma^\prime_1$ connecting $v_0$ and $v_{i_1}$. Now apply the same process to the remaining sequences $(\sigma_{i_j},\ldots, \sigma_{i_{j+1}})$ to obtain admissible curves $\sigma^\prime_{j+1}$ connecting $v_{i_j}$ and $v_{i_{j+1}}$.



\begin{definition}
We define the \textbf{simple red-green path} of $s$ in $\mathcal{T}_\mathcal{F}$ to be the  resulting sequence $\varsigma_\mathcal{F}(s)=(\sigma^\prime_{1},...,\sigma^\prime_{r})$ of red and green admissible curves where $\sigma^\prime_j$ is an admissible curve for $[v_{i_{j-1}}, v_{i_j}]$. An example of this construction is shown in Figure~\ref{ex_rgpath}. 
\end{definition}

%We now show how to express the dimension vector of any indecomposable module of $\Lambda_T$ as a linear combination of elements in $ \{ dim(M(w(s_{\gamma, \mathcal{F}}))) \ | \ \gamma \in \mathcal{F} \} $. Let $s=[v,u]$ be a segment in $T$ and let $\varsigma=(\sigma_1,...,\sigma_l)$ be the shortest sequence of admissible curves in $\mathcal{T}_\mathcal{F}$ joining $u$ and $v$. By abuse of notation, we identify each admissible curve $\sigma_k$ with the segments associated to them. Suppose $\sigma_i=[e,g]$ and $\sigma_{i+1}=[f,g]$ are two admissible curves in $\varsigma$ that have a common subsegment. As shown in \cite[Section 5]{garver_oriented_rep_thy}, they must be different colors and agree along the shorter of the two segments. Without loss of generality suppose that $[f,g]$ is the shortest one.


%We can construct a new admissible curve $\sigma'$ for the segment $[e,f]$ with $\sigma'(0)=\sigma_i (0)$ and $\sigma'(1)=\sigma_{i+1}(0)$ such that does not intersect any other $\sigma_k$ in $\varsigma$. \textcolor{red}{Is it actually important that $\sigma^\prime$ does not crossthe other curves?} We color $\sigma'$ as the admissible curve with longest associated segment. Note that $dim(M(w(\sigma')))= dim(M(w(\sigma_{i})))-dim(M(w(\sigma_{i+1})))$. Replace $\varsigma$ with the sequence of  admissible curves $(\sigma_1,...,\sigma_{i-1}, \sigma', \sigma_{i+2},..., \sigma_l)$ that also joins $u$ with $v$. We repeat this process until there are no two admissible curves in the sequence with a common subsegment. \textcolor{red}{Note that the dimension vectors of the new admissible curves are in terms of the dimension vectors of the original admissible curves in $\mathcal{T}_\mathcal{F}$.} \textcolor{red}{Is it clear that the resulting sequence is independent of the replacements made?}

%We now show how to express the dimension vector of any indecomposable module of $\Lambda_T$ as a linear combination of elements in $ \{ dim(M(w(s_{\gamma, \mathcal{F}}))) \ | \ \gamma \in \mathcal{F} \} $. Let $s=[v,u]$ be a segment in $T$ and let $\varsigma=(\sigma_1,...,\sigma_l)$ be the shortest sequence of admissible curves in $\mathcal{T}_\mathcal{F}$ joining $u$ and $v$. By abuse of notation, we identify each admissible curve $\sigma_k$ with the segments associated to them. Suppose $\sigma_i=[e,g]$ and $\sigma_{i+1}=[f,g]$ are two admissible curves in $\varsigma$ that have a common subsegment. As shown in \cite[Section 5]{garver_oriented_rep_thy}, they must be different colors and agree along the shorter of the two segments. Without loss of generality suppose that $[f,g]$ is the shortest one.

%We can construct a new admissible curve $\sigma'$ for the segment $[e,f]$ with $\sigma'(0)=\sigma_i (0)$ and $\sigma'(1)=\sigma_{i+1}(0)$ such that does not intersect any other $\sigma_k$ in $\varsigma$. \textcolor{red}{Is it actually important that $\sigma^\prime$ does not crossthe other curves?} We color $\sigma'$ as the admissible curve with longest associated segment. Note that $dim(M(w(\sigma')))= dim(M(w(\sigma_{i})))-dim(M(w(\sigma_{i+1})))$. Replace $\varsigma$ with the sequence of  admissible curves $(\sigma_1,...,\sigma_{i-1}, \sigma', \sigma_{i+2},..., \sigma_l)$ that also joins $u$ with $v$. We repeat this process until there are no two admissible curves in the sequence with a common subsegment. \textcolor{red}{Note that the dimension vectors of the new admissible curves are in terms of the dimension vectors of the original admissible curves in $\mathcal{T}_\mathcal{F}$.} \textcolor{red}{Is it clear that the resulting sequence is independent of the replacements made?}


\begin{figure}
$$\includegraphics[scale=1.1]{Facet1.pdf}$$
\caption{We show the red-green tree corresponding to a facet of $\Delta^{NC}(T)$. We do not show the boundary arcs of this facet. Here $\psi_r(\mathcal{F}) = \{\{1,3,4\}, \{2,8\}, \{5,6,7,9\}, \{10\}\}$ and $\text{Kr}(\psi_r(\mathcal{F})) = \psi_g(\mathcal{F}) = \{\{1\}, \{2,4\}, \{3\}, \{5,8\}, \{6\}, \{7,10\}, \{9\}\}$.}
\label{nc_to_ncp}
\end{figure}

%Finally, if $\sigma_i$ and $\sigma_{i+1}$ are two adjacent admissible curves of the same color, replace $\varsigma$ with the sequence \textcolor{red}{(different notation) $(\sigma_1,...,\sigma_{i-1}, \sigma'', \sigma_{i+2},..., \sigma_l)$} where $\sigma''$ is the admissible curve for the segment satisfying $dim(M(w(\sigma'')))= dim(M(w(\sigma_i)))+dim(M(w(\sigma_{i+1})))$. We define the \textbf{simple red-green path} associated to $s$ in $\mathcal{T}_\mathcal{F}$ to be the resulting sequence $\varsigma_\mathcal{F}(s)=(\sigma'_{1},...,\sigma'_{r})$ of alternating red and green noncrossing admissible curves that do not meet in any segment that is produced after this process. An example of this construction is shown in Figure~\ref{ex_rgpath}. 

%Finally, if $\sigma_i$ and $\sigma_{i+1}$ are two adjacent admissible curves of the same color, replace $\varsigma$ with the sequence \textcolor{red}{(different notation) $(\sigma_1,...,\sigma_{i-1}, \sigma'', \sigma_{i+2},..., \sigma_l)$} where $\sigma''$ is the admissible curve for the segment satisfying $dim(M(w(\sigma'')))= dim(M(w(\sigma_i)))+dim(M(w(\sigma_{i+1})))$. We define the \textbf{simple red-green path} associated to $s$ in $\mathcal{T}_\mathcal{F}$ to be the resulting sequence $\varsigma_\mathcal{F}(s)=(\sigma'_{1},...,\sigma'_{r})$ of alternating red and green noncrossing admissible curves that do not meet in any segment that is produced after this process. An example of this construction is shown in Figure~\ref{ex_rgpath}. 






\begin{prop}\label{stab_outF}
Let $\mathcal{F}$ be a facet and let $s \in \text{Seg}(T)$. Then $\theta_\mathcal{F}(M(w(\sigma')))\leq 0$ (resp., $\theta_\mathcal{F}(M(w(\sigma'))) > 0$) for any red (resp., green) admissible curve $\sigma'$ in $\varsigma_\mathcal{F}(s)$.
\end{prop}

{We can now show that if $\theta_\mathcal{F}(M(w(s)))=0$ and $s \neq s_{\gamma,\mathcal{F}}$ for any arc $\gamma \in \mathcal{F}$, then its simple red-green path must contain at least two curves of different color. In addition, we can show that any segment $t = [v_{i_{j-1}}, v_{i_j}] \circ \cdots \circ [v_{i_{k-1}}, v_{i_k}]$ associated to a maximal subsequence of $\varsigma_\mathcal{F}(s)$ of consecutive green admissible curves $(\sigma_j^\prime,..., \sigma_k^\prime )$ has $t \in C_s$. That is, $M(w(t))$ is a submodule of $M(w(s))$. By Proposition~\ref{stab_outF}, $M(w(s))$ is not a semistable representation since $\theta_\mathcal{F}(M(w(t))) = \sum_{i=1}^l \theta_\mathcal{F}(M(w(\sigma_i'))) > 0$. This proves Theorem~\ref{thm:thm1}.} %\textcolor{red}{I added the maximal stack notion}

\begin{remark}\label{ingallsthomas}
If $T$ is a tree all of whose interior vertices have degree 3 and has no subconfiguration of the form shown in Figure~\ref{bad_subtree}, then $Q_T$ is a type $A$ Dynkin quiver. Theorem~\ref{thm:thm1} therefore recovers Ingalls' and Thomas' bijection in \cite[Theorem 1]{ingalls2009noncrossing} between wide subcategories and semistable subcategories in type $A$.
\end{remark}

\begin{remark}
The assertion in Theorem~\ref{thm:thm1} that all wide subcategories of $\text{mod}(\Lambda_T)$ are semistable can be deduced from \cite[Theorem 1.2]{yurikusa}. Our work differs from that of Yurikusa in that we have found a canonical choice of stability condition realizing a wide subcategory as a semistable subcategory.
\end{remark}

%We can now show that if $\theta_\mathcal{F}(M(w(s)))=0$ for a segment $s$ associated neither to a red nor a green arc of $\mathcal{F}$, then its simple red-green path must contain at least two curves of different color. In addition, we can show that any segment with green admissible curve $\sigma'$ in $\varsigma_\mathcal{F}(s)$ satisfies that $\sigma' \in C_s$, that is, $M(w(\sigma'))$ is a submodule of $M(w(s))$. By Proposition~\ref{stab_outF}, $M(w(s))$ is not a stable object since $\theta_\mathcal{F}(M(w(\sigma'))) > 0$. This proves Theorem~\ref{thm:thm1}.

%This gives us that if $\theta_\mathcal{F}(M(w(s)))=0$ for a segment $s$ associated neither to a red nor a green arc of $\mathcal{F}$, then its simple red-green path must contain at least two curves of different color. In addition, we have that any segment associated to a green admissible curve $\sigma'$ in $\varsigma_\mathcal{F}(s)$ satisfies that $\sigma' \in C_s$, that is, $M(w(\sigma'))$ is a submodule of $M(w(s))$. By Proposition~\ref{stab_outF}, $M(w(s))$ is not a stable object since $\theta_\mathcal{F}(M(w(\sigma'))) > 0$. This proves Theorem~\ref{thm:thm1}.

\begin{figure}
$$\includegraphics[scale=1.15]{colapsingcurves2.pdf}$$
\caption{The simple red-green path for the segment $s=[c,e]$ is $(\sigma^{\prime}_1, \sigma^\prime_2)$. Here $\textnormal{\textbf{dim}}(M(w(s)))=\textnormal{\textbf{dim}}(M(w(\sigma_1')))+\textnormal{\textbf{dim}}(M(w(\sigma_2')))= [\textnormal{\textbf{dim}}(M(w(\sigma_2))) - \textnormal{\textbf{dim}}(M(w(\sigma_1)))] + [\textnormal{\textbf{dim}}(M(w(\sigma_4))) - \textnormal{\textbf{dim}}(M(w(\sigma_3)))]$. }
\label{ex_rgpath}
\vspace{-1.5em}
\end{figure}


From Theorem~\ref{thm:thm1} and \cite[Theorem 6.23]{garver_oriented_rep_thy} it follows that the poset of semistable subcategories of mod$(\Lambda_T)$ is isomorphic to the lattice of noncrossing tree partitions of $T$ as the following corollary shows. In particular, we obtain a combinatorial classification of the semistable subcategories of mod$(\Lambda_T)$.

\begin{cor}
For any tree $T$, the map $\phi: \text{NCP}(T)\to \Lambda^{ss}_T$ is an isomorphism of posets.
\end{cor}

\begin{figure}[h]
$$\includegraphics[scale=1]{A3_cyclic.pdf}$$
\caption{The forbidden subconfigurations from Remark~\ref{ingallsthomas}.}
\label{bad_subtree}
\vspace{-1.5em}
\end{figure}

\section{Additional questions}\label{Sec_additiona_stuff}

A crucial step in proving Theorem~\ref{thm:thm1} was the use of the combinatorics of the red-green tree $\mathcal{T}_\mathcal{F}$ to evaluate $\theta_\mathcal{F}$ on any indecomposable $\Lambda_T$-module. A second crucial step in the proof was the fact that for any facet $\mathcal{F}$ of the noncrossing complex, the $\textbf{g}$-vectors in $G(\mathcal{F})$ and the $\textbf{c}$-vectors in $C(\mathcal{F})$ are dual bases with respect to $\langle -, - \rangle$. This fact has already been established for general \textbf{gentle algebras} of which tiling algebras are examples (see \cite[Proposition 4.16]{palu2017non}). There the role of the noncrossing complex is played by the \textbf{blossoming complex}. However, we are not aware of a notion of the red-green tree associated to a facet of this complex. We propose the following problem.

%For general finite dimensional algebras $\Lambda = \Bbbk Q/I$, one also has a notion of $\textbf{c}$-vectors and $\textbf{g}$-vectors. The description of the combinatorics of $C(\mathcal{F})$ We are not aware of an analogous description of 

\begin{prob}
Find a combinatorial description of the Kreweras complement $\text{Kr}:\text{wide}(\Lambda) \to \text{wide}(\Lambda)$ where $\Lambda$ is a gentle algebra, and use this to determine when the Kreweras stability condition $\theta_{\mathcal{F}_\mathcal{W}}$ satisfies $\theta^{ss}_{\mathcal{F}_\mathcal{W}} = \mathcal{W}$ for all facets $\mathcal{F}_\mathcal{W}$ of the blossoming complex.
\end{prob}


%\lipsum[10]

%\begin{figure}
%\centering
%\includegraphics[width=0.5\textwidth]{plot}
%\caption{A plot of a function.}
%\label{fig:plot}
%\end{figure}

%\subsection{A subsection}

%\lipsum[5]

%\begin{align*}
%1 + 1 + 1 &= 2 + 1 \\ &= 3.
%\end{align*}

%\subsubsection{A subsubsection}

%\lipsum[17]

%Here is a citation~\cite{greenwade93}. And a reference to equation~\eqref{eqn:eq1}. And a reference to Figure~\ref{fig:plot}.

\acknowledgements{This project began at a Mitacs Globalink Research Internship at Universit\'e du Qu\'ebec \`a Montr\'eal. M. Garcia was supported by Mitacs Globalink and the project CONACyT-238754. A. Garver was supported by NSERC and the Canada Research Chairs Program.}%\lipsum[20]}

%\acknowledgements{This project is the result of a Mitacs Globalink Research Internship hosted by Universit\'e du Qu\'ebec \`a Montr\'eal. M. Garcia received support from Mitacs Globalink and the project CONACyT-238754. A. Garver received support from NSERC and the Canada Research Chairs Program.}

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