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\title{Toric Schubert varieties and directed Dynkin diagrams}
\alttitle{Variétés de Schubert toriques et diagrammes de Dynkin dirigés}

\author{\firstname{Eunjeong} \lastname{Lee}\CDRorcid{0000-0002-2089-0878}}
\address{Department of Mathematics, Chungbuk National University, Cheongju 28644, Republic of Korea}
\email{eunjeong.lee@chungbuk.ac.kr}

\author{\firstname{Mikiya} \lastname{Masuda}\CDRorcid{0000-0002-4484-0090}}
\address{Osaka Central Advanced Mathematics Institute (OCAMI), Osaka Metropolitan University, Sumiyoshi-ku, Sugimoto, 558-8585, Osaka, Japan}
\email{mikiyamsd@gmail.com}

\author{\firstname{Seonjeong} \lastname{Park}\CDRorcid{0000-0003-1958-0087}}
\address{Department of Mathematics Education, Jeonju University, Jeonju 55069, Republic of Korea}
\email{seonjeongpark@jj.ac.kr}

\keywords{\kwd{Schubert varieties} \kwd{toric varieties} \kwd{Bott manifolds} \kwd{cohomological rigidity}}
\altkeywords{\kwd{Variétés de Schubert} \kwd{variétés toriques} \kwd{variétés de Bott} \kwd{rigidité cohomologique}}

\thanks{Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.~RS-2022-00165641, No.~RS-2023-00239947). Masuda was supported in part by the HSE University Basic Research Program.  Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [NRF-2020R1A2C1A01011045] and (RS-2026-25490927). This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849)}
\CDRGrant[National Research Foundation of Korea (NRF)]{RS-2022-00165641}
\CDRGrant[National Research Foundation of Korea (NRF)]{RS-2023-00239947}
\CDRGrant[National Research Foundation of Korea (NRF)]{NRF-2020R1A2C1A01011045}
\CDRGrant[National Research Foundation of Korea (NRF)]{RS-2026-25490927}
\CDRGrant[Osaka City University Advanced Mathematical Institute]{JPMXP0619217849}

\subjclass{14M25, 14M15, 57S12, 05A05}

\begin{abstract}
A flag variety is a smooth projective homogeneous variety $G/B$, where $G$ is a simple algebraic group over the complex numbers and $B$ is a Borel subgroup of $G$. A Schubert variety $X_w$ is a subvariety of $G/B$ indexed by an element~$w$ of the Weyl group of $G$. It is called toric if it is a toric variety with respect to the action of the maximal torus $T \subset B$. In this paper, we associate an edge-labeled digraph~$\mathcal{G}_w$ to a toric Schubert variety $X_w$ and classify toric Schubert varieties up to isomorphism. We also give a simple criterion for when a toric Schubert variety $X_w$ is (weak) Fano in terms of~$\mathcal{G}_w$. Finally, we discuss whether toric Schubert varieties can be distinguished by their integral cohomology rings up to isomorphism and show that this holds when $G$ is simply-laced.
\end{abstract}

\begin{altabstract}
Une variété de drapeaux est une variété homogène $G/B$, où $G$ est un groupe algébrique simple sur $\mathbb{C}$ et $B$ un sous-groupe de Borel de $G$. Une variété de Schubert $X_w$ est une sous-variété de $G/B$ indexée par un élément $w$ du groupe de Weyl de $G$. On dit qu’elle est torique si elle est une variété torique pour l’action du tore maximal $T\subset B$. Dans cet article, nous associons à une variété de Schubert torique $X_w$ un digraphe à arêtes étiquetées $\mathcal{G}_w$ et nous classifions les variétés de Schubert toriques à isomorphisme près. Nous donnons également un critère simple pour déterminer quand $X_w$ est (faiblement) Fano en termes de $\mathcal{G}_w$. Enfin, nous discutons si les variétés de Schubert toriques peuvent être distinguées par leurs anneaux de cohomologie entière (à isomorphisme près) et montrons que c’est le cas lorsque $G$ est de type simplement lacé.
\end{altabstract}

\COI{The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.}

\dateposted{2026-07-27}
\begin{document}
%\input{CR-pagedemetas}
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\maketitle

\section*{Introduction}

A flag variety is a smooth projective homogeneous variety $G/B$, where $G$ is a simple algebraic group over the complex numbers $\C$ and $B$ is a Borel subgroup of $G$. Let $T$ be the maximal torus of $G$ in $B$ and $W$ the Weyl group of $G$. The Bruhat decomposition $G/B=\bigsqcup_{w\in W}BwB/B$ provides a cell decomposition of~$G/B$. Indeed, $BwB/B$ is isomorphic to an affine space $\C^{\ell(w)}$ where $\ell(w)$ denotes the Coxeter length of~$w$. The Schubert variety~$X_w$ is the closure of the cell $BwB/B$ in the flag variety~$G/B$, and $X_w$ is called toric if it is a toric variety with respect to the torus $T$. It is known that $X_w$ is toric if and only if $w$ is a product of \emph{distinct} simple reflections (see~\cite{Fan98, Karu13Schubert}). This implies that a toric Schubert variety~$X_w$ is smooth; in fact, $X_w$ is a Bott manifold that is the total space of an iterated $\C P^1$-bundle over a point.

In this paper, we associate to an edge-labeled digraph $\mathcal{G}_w$ a toric Schubert variety $X_w$ and prove the following.

\begin{theo}[Theorem~\ref{theo:graph_determines_Xw}] \label{theo:main1}
Let $W$ and $W'$ be the Weyl groups of simple algebraic groups~$G$ and~$G'$, respectively. Then toric Schubert varieties $X_w$ ($w\in W$) and $X_{w'}$ ($w'\in W'$) are isomorphic as varieties if and only if\/ $\G_{w}$ and\/ $\G_{w'}$ are isomorphic as edge-labeled digraphs.
\end{theo}

The edge-labeled digraph $\mathcal{G}_w$ is defined as follows. Take a reduced expression $s_{i_1}s_{i_2} \dotsm s_{i_m}$ of~$w$ as a product of simple reflections $s_i$ in $W$, where $i_1,i_2,\dots,i_m$ are mutually distinct. Then the vertex set of~$\mathcal{G}_w$ is $\{i_1,\dots,i_m\}$, and there is a directed edge $(i_j,i_k)$ between two vertices in $\mathcal{G}_w$ if and only if $j>k$ and the $(i_j,i_k)$ entry $c_{i_j,i_k}$ in the Cartan matrix of $G$ is nonzero. We label the directed edge by the integer $-c_{i_j,i_k}$, which is $1$, $2$, or $3$. Therefore, the underlying graph of $\mathcal{G}_w$ is an induced subgraph of the Dynkin diagram of $G$, with a multiple edge (if any) replaced by a single edge. All directed edges of $\mathcal{G}_w$ are labeled $1$, except when the underlying edge of the Dynkin diagram is multiple and the directed edge is oriented in the same direction as the arrow on the Dynkin diagram. In that case, the label is $2$ for a double edge, and $3$ for a triple edge. If the orientation is opposite to the Dynkin-diagram arrow, then its label is $1$. (See, for instance, Table~\ref{table_F4}.) Here, we notice that $\mathcal{G}_w$ is independent of the choice of reduced expression of $w$ (see Remark~\ref{rmk_graph_is_well-defined}).

A smooth projective variety $X$ is called Fano (respectively, weak Fano) if the anticanonical divisor $-K_X$ is ample (respectively, nef and big). In birational geometry, the Fano and weak Fano conditions encode the positivity of the anticanonical divisor $-K_{X}$. Such positivity properties play a fundamental role in the birational classification of projective varieties and single out particularly rigid and well-behaved families. The next theorem shows that the edge-labeled digraph $\mathcal{G}_w$ is also useful for determining whether the toric Schubert variety $X_w$ is Fano or weak Fano.

\begin{theorem}[Theorem~\ref{thm_Fano_weak_Fano}] \label{theo:main2}
A toric Schubert variety $X_w$ is Fano (respectively, weak Fano) if and only if every vertex of\/ $\mathcal{G}_w$ has indegree at most $1$ (respectively, $2$).
\end{theorem}

An element of the Weyl group $W$ is called a Coxeter element if it is a product of the simple reflections, each appearing exactly once. When $w\in W$ is a Coxeter element, the underlying graph of $\mathcal{G}_w$ agrees with the Dynkin diagram of $G$ with a multiple edge (if any) replaced by a single edge. We enumerate the isomorphism classes of (weak) Fano toric Schubert varieties~$X_w$ for Coxeter elements $w$ in each Lie type (see Table~\ref{table_cardinality_iso_classes_Coxeter}).

We also discuss whether a toric Schubert variety $X_w$ is determined by its integral cohomology ring~$H^*(X_w;\Z)$, in other words, whether $\G_w$ can be recovered from $H^*(X_w;\Z)$.

\begin{theorem}[Theorem~\ref{theo:recover}] \label{theo:main3}
The edge-labeled digraph $\mathcal{G}_w$ can be recovered (up to isomorphism) from the cohomology ring $H^*(X_w;\Z)$ if all edge labels of\/ $\G_w$ are $1$.
\end{theorem}

Combining Theorem~\ref{theo:main3} with Theorem~\ref{theo:main1}, we obtain the following corollary.

\begin{corollary}
Let $W$ and $W'$ be the Weyl groups of simple algebraic groups $G$ and $G'$ of simply-laced type (i.e.\ type $\dyn{A}$, $\dyn{D}$, or $\dyn{E}$). Then toric Schubert varieties $X_w$ ($w\in W$) and $X_{w'}$ ($w'\in W'$) are isomorphic as varieties if and only if $H^{\ast}(X_w;\Z)$ and $H^{\ast}(X_{w'};\Z)$ are isomorphic as graded rings.
\end{corollary}

\begin{remark}\label{rema:1}
The assumption in Theorem~\ref{theo:main3} cannot be weakened as is seen in the following examples with simple reflections numbered as in Table~\ref{table_finite}.
\begin{enumerate}[leftmargin=*]
\item When $G$ is of type $\dyn{G}_2$, there are two Coxeter elements $w,w'$ and both $\mathcal{G}_w$ and $\mathcal{G}_{w'}$ consist of two vertices with a directed edge but one has label $1$ while the other has label $3$. However, the cohomology rings $H^*(X_w;\Z)$ and $H^*(X_{w'};\Z)$ are isomorphic. Indeed, if $F_a$ denotes the Hirzebruch surface indexed by a nonnegative integer $a$, then $X_w$ and $X_{w'}$ are $F_1$ and $F_3$. As is well-known, $F_a$ and $F_{a'}$ are isomorphic if and only if $a=a'$, and $H^*(F_a;\Z)$ and $H^*(F_{a'};\Z)$ are isomorphic as graded rings (more strongly, $F_a$ and $F_{a'}$ are diffeomorphic) if and only if $a\equiv a'\pmod{2}$.
\item When $G$ is of type $\dyn{C}_3$, $\G_{s_1s_3}$ consists of two vertices with no edge and $\G_{s_2s_3}$ consists of two vertices and a directed edge with label $2$. On the other hand, $X_{s_1s_3}$ and $X_{s_2s_3}$ are isomorphic to $F_0$~and~$F_2$ respectively, so their integral cohomology rings are isomorphic. We obtain a similar observation when $G$ is of type $\dyn{B}_3$ or $\dyn{F}_4$. The toric Schubert variety $X_{s_3s_2}$ in type $\dyn{B}_3$~or~$\dyn{F}_4$ is isomorphic to the Hirzebruch surface $F_2$ while $X_{s_1s_3}$ is isomorphic to $F_0$.
\item When $G$ is of type $\dyn{F}_4$, the cohomology rings of toric Schubert varieties $X_{s_3s_2s_1s_4}$ and $X_{s_2s_1s_4s_3}$ are isomorphic but $X_{s_3s_2s_1s_4}$ and $X_{s_2s_1s_4s_3}$ are not isomorphic as varieties (see Remark~\ref{rmk_F4}).
\end{enumerate}
\end{remark}

As mentioned in Remark~\ref{rema:1}, not all toric Schubert varieties are distinguished as varieties by their integral cohomology rings. However, they are distinguished by their integral cohomology rings as smooth manifolds. Indeed, toric Schubert varieties are Bott manifolds, and it is known that Bott manifolds are distinguished by their integral cohomology rings up to diffeomorphism~(see~\cite{Choi_et}). Related to this, it is asked and studied in~\cite{richmond2021isomorphism} whether smooth (not necessarily toric) Schubert varieties are distinguished by their integral cohomology rings up to diffeomorphism or homeomorphism.

This paper is organized as follows. As mentioned above, toric Schubert varieties are Bott manifolds. In Section~\ref{sec:general}, we review fans of Bott manifolds and recall a criterion for determining when they are Fano or weak Fano. In Section~\ref{section_toric_Schubert_directed_graphs}, we associate the edge-labeled digraph $\mathcal{G}_w$ to a toric Schubert variety $X_w$ and prove Theorems~\ref{theo:main1} and~\ref{theo:main2}. Using Theorem~\ref{theo:main1}, we enumerate the isomorphism classes of (Fano or weak Fano) toric Schubert varieties in Section~\ref{sect:enumeration}. In Section~\ref{section_cohomology_ring_ADE}, we prove Theorem~\ref{theo:main3} using a presentation of $H^*(X_w;\Z)$ as a graded ring with generators and relations.

\section{Preliminaries: toric Schubert varieties and Bott manifolds}\label{sec:general}

In this section, we recall the classification of toric Schubert varieties and describe them in terms of Bott manifolds. To do so, we first review Bott manifolds, which are smooth projective toric varieties. Moreover, we recall criteria for the Fano and weak Fano properties of Bott manifolds.

\subsection{Toric Schubert varieties}

Let $G$ be a simple algebraic group over $\C$ of rank $r$, let $B$ be a Borel subgroup, and let $T$ be a maximal torus of $G$ contained in $B$. We denote by $W$ the Weyl group of $G$. A \emph{flag variety} is the homogeneous space $G/B$, which is a smooth projective variety. When $G$ is of type $A$ with rank $n-1$, the flag variety $G/B$ is diffeomorphic to
\[
\flag(n) \colonequals \braces[\big]{\parens[\big]{\{0\} \subset V_1 \subset V_2 \subset \dotsb \subset V_{n} = \C^{n}} \st[\big] \dim_{\C} V_i = i
\ \text{for all $i =1,\dots,n$}},
\]
where each $V_i$ is a linear subspace of $\C^{n}$. Moreover, the Weyl group is the symmetric group $\Sn$ on the set $[n] \colonequals \{1,\dots,n\}$.

The Weyl group $W$ of $G$ is generated by simple reflections $s_i$ for $i=1,\dots,r$, so each element $w \in W$ can be expressed as a product of generators:
\[
w = s_{i_1} s_{i_2} \dotsm s_{i_m}.
\]
If $m$ is minimal among all such expressions for $w$, then $m$ is called the (Coxeter) \emph{length} of $w$ and we write $\ell(w) = m$. Moreover, we call the word $s_{i_1} s_{i_2} \dotsm s_{i_m}$ a \emph{reduced decomposition} for $w$. An expression $s_{i_1} s_{i_2} \dotsm s_{i_m}$ provides a string $(i_1,i_2,\dots,i_m)$ in $[r]^{m}$, and we call it a \emph{word}. A word $(i_1,i_2,\dots,i_m)$ is \emph{reduced} if the corresponding expression $s_{i_1} s_{i_2} \dotsm s_{i_m}$ is reduced. In Table~\ref{table_finite}, we provide Dynkin diagrams for finite types. In this manuscript, we use the ordering on the simple roots as in the table following~\cite{Humphreys78Lie}.

\begin{table}[t]
\caption{Dynkin diagrams for finite types.}\label{table_finite}
\begin{tabular}{@{}c@{\kern 2em}l@{}}
\toprule
$\Phi$ & Dynkin diagram \\
\midrule
$\dyn{A}_r$ ($r \geq 1$) &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}
\node[Dnode] (1) {} node[below = 0cm of 1] {$1$};
\node[Dnode] (2) [right = of 1] {} node[below=0cm of 2] {$2$};
\node[Dnode] (3) [right = of 2] {} node[below=0cm of 3] {$3$};
\node[Dnode] (4) [right =of 3] {} node[below=0cm of 4] {$r-1$};
\node[Dnode] (5) [right =of 4] {} node[below=0cm of 5] {$r$};

\draw (1)--(2)--(3) (4)--(5);
\draw[dotted] (3)--(4);
\end{tikzpicture} \\[2ex]
$\dyn{B}_r$ ($r \geq 2$) &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} node[below = 0cm of 1] {$1$};
\node[Dnode] (2) [right = of 1] {} node[below = 0cm of 2] {$2$};
\node[Dnode] (3) [right = of 2] {} node[below = 0cm of 3] {$r-2$};
\node[Dnode] (4) [right =of 3] {} node[below = 0cm of 4] {$r-1$};
\node[Dnode] (5) [right =of 4] {} node[below = 0cm of 5] {$r$};

\draw (1)--(2) (3)--(4);
\draw [dotted] (2)--(3);
\draw[double line] (4)--(5);
\end{tikzpicture} \\[2ex]
$\dyn{C}_r$ ($r \geq 3$) &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} node[below = 0cm of 1] {$1$};
\node[Dnode] (2) [right = of 1] {} node[below = 0cm of 2] {$2$};
\node[Dnode] (3) [right = of 2] {} node[below = 0cm of 3] {$r-2$};
\node[Dnode] (4) [right =of 3] {} node[below = 0cm of 4] {$r-1$};
\node[Dnode] (5) [right =of 4] {} node[below = 0cm of 5] {$r$};

\draw (1)--(2) (3)--(4);
\draw [dotted] (2)--(3);
\draw[double line] (5)--(4);
\end{tikzpicture} \\[2ex]
$\dyn{D}_r$ ($r \geq 4$) &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} node[below = 0cm of 1] {$1$};
\node[Dnode] (2) [right = of 1] {} node[below = 0cm of 2] {$2$};
\node[Dnode] (3) [right = of 2] {} node[below = 0cm of 3] {$r-3$};
\node[Dnode] (4) [right =of 3] {} node[below = 0cm of 4] {$r-2$};

\node[Dnode] (5) [above right= 0.3cm and 1cm of 4] {} node[right = 0cm of 5] {$r-1$};
\node[Dnode] (6) [below right= 0.3cm and 1cm of 4] {} node[right = 0cm of 6] {$r$};

\draw(1)--(2) (3)--(4)--(5) (4)--(6);
\draw[dotted] (2)--(3);
\end{tikzpicture} \\[2ex]
$\dyn{E}_6$ &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} ;
\node[Dnode] (3) [right=of 1] {} ;
\node[Dnode] (4) [right=of 3] {} ;
\node[Dnode] (2) [above=of 4] {} node[right = 0cm of 2] {$2$};
\node[Dnode] (5) [right=of 4] {} ;
\node[Dnode] (6) [right=of 5] {} ;

\foreach \x in {1,3,4,5,6}{
\node[below = 0cm of \x] {$\x$};}

\draw(1)--(3)--(4)--(5)--(6) (2)--(4);
\end{tikzpicture} \\[2ex]
$\dyn{E}_7$ &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} ;
\node[Dnode] (3) [right=of 1] {} ;
\node[Dnode] (4) [right=of 3] {} ;
\node[Dnode] (2) [above=of 4] {} node[right = 0cm of 2] {$2$};
\node[Dnode] (5) [right=of 4] {} ;
\node[Dnode] (6) [right=of 5] {} ;
\node[Dnode] (7) [right=of 6] {} ;

\foreach \x in {1,3,4,5,6,7}{
\node[below = 0cm of \x] {$\x$};}

\draw(1)--(3)--(4)--(5)--(6)--(7) (2)--(4);
\end{tikzpicture} \\[2ex]
$\dyn{E}_8$ &
\begin{tikzpicture}[scale=.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {} ;
\node[Dnode] (3) [right=of 1] {} ;
\node[Dnode] (4) [right=of 3] {} ;
\node[Dnode] (2) [above=of 4] {} node[right = 0cm of 2] {$2$};
\node[Dnode] (5) [right=of 4] {} ;
\node[Dnode] (6) [right=of 5] {} ;
\node[Dnode] (7) [right=of 6] {};
\node[Dnode] (8) [right=of 7] {};

\foreach \x in {1,3,4,5,6,7,8}{
\node[below = 0cm of \x] {$\x$};}

\draw(1)--(3)--(4)--(5)--(6)--(7)--(8) (2)--(4);
\end{tikzpicture} \\[2ex]
$\dyn{F}_4$ &
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {};
\node[Dnode] (2) [right = of 1] {};
\node[Dnode] (3) [right = of 2] {};
\node[Dnode] (4) [right =of 3] {};

\foreach \x in {1,...,4}{
\node[below = 0cm of \x] {$\x$};}

\draw (1)--(2) (3)--(4);
\draw[double line] (2)-- (3);
\end{tikzpicture} \\[2ex]
$\dyn{G}_2$ &
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) {};
\node[Dnode] (2) [right = of 1] {};

\foreach \x in {1,2}{
\node[below = 0cm of \x] {$\x$};}

\draw[triple line] (2)--(1);
\draw (1)--(2);
\end{tikzpicture}\\
\bottomrule
\end{tabular}
\end{table}

The left multiplication of $T$ on $G$ induces an action of $T$ on $G/B$. Then there is a bijective correspondence between the $T$-fixed point set $(G/B)^T$ and the Weyl group $W$ of $G$, and we denote by $wB$ the fixed point in $G/B$ corresponding to $w \in W$. For $w \in W$, the \emph{Schubert variety} $X_w$ is a subvariety of $G/B$ defined by the (Zariski) closure of $BwB/B \subset G/B$. A Schubert variety is $T$-invariant and has complex dimension $\ell(w)$.

Considering a reduced decomposition for $w \in W$, one can decide whether the Schubert variety~$X_w$ is toric or not with respect to the torus action $T$ as follows.

\begin{theorem}[{see~\cite{Fan98, Karu13Schubert}}]\label{thm_toric_Schubert_distinct}
For $w \in W$, the following statements are equivalent:
\begin{enumerate}
\item $X_w$ is a toric variety (with respect to the $T$-action);
\item $X_w$ is a smooth toric variety (with respect to the $T$-action);
\item some (equivalently, any) reduced expression for $w$ consists of distinct simple reflections.
\end{enumerate}
\end{theorem}

\goodbreak

\begin{example}\label{example_toric_A2}
There are six elements in the Weyl group $\mathfrak{S}_3$ of type $\dyn{A}_2$. For each $w \in \mathfrak{S}_3$, we display whether the corresponding Schubert variety $X_w$ is toric or not and the length $\ell(w) = \dim_{\C} X_w$ in the following table:
{\small
\[
\begin{tabular}{@{}c@{\kern 2em}*{6}{W{c}{3em}}@{}}
\toprule
$w$ & $e$ & $s_1$ & $s_2$ & $s_1s_2$ & $s_2s_1$ & $s_1s_2s_1$ \\
\midrule
$X_w$ is toric & yes & yes & yes & yes & yes & no \\
$\ell(w)$ & $0$ & $1$ & $1$ & $2$ & $2$ & $3$ \\
\bottomrule
\end{tabular}
\]}
\end{example}

\begin{example}\label{example_toric_B2}
The Weyl group $W_{\dyn{B}_2}$ of type $\dyn{B}_2$ is given by $W_{\dyn{B}_2} = \angles[\big]{s_1, \: s_2 \bigmid (s_1)^2 = (s_2)^2 = (s_1s_2)^4 = e}$. There are eight elements in $W_{\dyn{B}_2}$, and five of them produce toric Schubert varieties:
{\small
\[
\begin{tabular}{@{}c@{\kern 2em}*{8}{W{c}{4em}}@{}}
\toprule
$w$ & $e$ & $s_1$ & $s_2$ & $s_1s_2$ & $s_2s_1$ & $s_1s_2s_1$ & $s_2s_1s_2$ & $s_1s_2s_1s_2$ \\
\midrule
$X_w$ is toric & yes & yes & yes & yes & yes & no & no & no \\
$\ell(w)$ & $0$ & $1$ & $1$ & $2$ & $2$ & $3$ & $3$ & $4$ \\
\bottomrule
\end{tabular}
\]}
\end{example}

When $G$ is of type $\dyn{A}_{n-1}$, the fan of a toric Schubert variety $X_w$ is the same as the normal fan of a polytope
\[
\Q_{w} \colonequals \Conv\braces[\big]{\parens[\big]{v^{-1}(1),\dots,v^{-1}(n)} \in \R^n \st[\big] v \leq w}.
\]
Here, we compare two elements $v,w \in W$ with respect to the \emph{Bruhat order}, that is, if $w = s_{i_1} \dotsm s_{i_{k}}$ is a reduced expression, then $v \leq w$ if and only if there exists a reduced expression $v = s_{i_{j_1}} s_{i_{j_2}} \dotsm s_{i_{j_q}}$ with $1 \leq j_1 < \dots < j_q \leq k$. To consider the fan of a toric Schubert variety $X_w$ in general Lie types, we recall~\cite[Section~3.7]{GK94Bott} and~\cite[Section~4.3]{LMP_torus_orbit_closures_book_chapter}.

\begin{theorem}[{\cite[Theorem~4.23]{LMP_torus_orbit_closures_book_chapter}}]\label{thm_char_matrix_of_toric_Schubert}
Let $w = s_{i_1} \dotsm s_{i_m}$ be a reduced decomposition for $w \in W$. Assume that $i_1,\dots,i_m$ are distinct. Then the fan of the toric Schubert variety $X_w$ is isomorphic to the fan in\/ $\R^m$ whose primitive ray generators are the\/ $2m$ column vectors of the following matrix. A subset of these column vectors spans a cone if and only if it does not contain both the $k$-th column of the left $m\times m$ submatrix and the $k$-th column of the right $m\times m$ submatrix for any $k=1,\dots,m$:
\begin{equation}\label{equation_ray_vectors}
{\left[
\begin{tabular}{cccc|cccc}
$1$ & $0$ & $\cdots$ & $0$ & $-1$ & $0$ & $\cdots$ & $0$ \\
$0$ & $1$ & $\cdots$ & $0$ & & $-1$ & $\cdots$ & $0$ \\
\raisebox{.2\height}{$\vdots$} & & \raisebox{.2\height}{$\ddots$} & & & \raisebox{1.3\height}{$-c_{i_j,i_k}$} & \raisebox{.2\height}{$\ddots$} & \\
$0$ & $0$ & $\cdots$ & $1$ & & & & $-1$
\end{tabular}
\right]},
\end{equation}
where the $(j,k)$ entry for $m \geq j >k \geq 1$ in the right submatrix above is $-c_{i_j,i_k}$ and the integers $c_{p,q}$ are Cartan integers. Therefore, if we denote the\/ $2m$ ray vectors in the above matrix from left to right by\/ $\mathbf{v}_{i_1},\dots,\mathbf{v}_{i_m},\mathbf{w}_{i_1},\dots,\mathbf{w}_{i_m}$, then
\begin{equation}\label{equation_ray_vectors2}
\mathbf{v}_{i_k}+\mathbf{w}_{i_k}=\sum_{j>k}(-c_{i_j,i_k})\mathbf{v}_{i_j}
\end{equation}
for $k=1,\dots,m$.
\end{theorem}

We briefly explain how to obtain Theorem~\ref{thm_char_matrix_of_toric_Schubert}. Suppose that $w = s_{i_1} \dotsm s_{i_m}$ is a reduced decomposition for $w \in W$ consisting of distinct simple reflections. Then the Schubert variety $X_w$ is isomorphic to the \emph{Bott--Samelson variety} corresponding to the word $(i_1,\dots,i_m)$. Here, a Bott--Samelson variety is a smooth projective variety that can be understood as the total space of an iterated $\C P^1$-bundle whose bundle structure is decided by the decomposition $(i_1,\dots,i_m)$. Moreover, such a Bott--Samelson variety is also toric and its fan structure is described in the paper by Grossberg and Karshon (see~\cite[Section~3.7]{GK94Bott}).

We call the right $m \times m$ submatrix in~\eqref{equation_ray_vectors} the \emph{reduced characteristic matrix}. Recall that the Cartan integers $c_{i,j}$ can be read directly from the Dynkin diagram as follows:
\[
\setlength{\tabcolsep}{2em}
\begin{tabular}{ccc}
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode, label=below:{$i$}] (2) {};
\node[Dnode, label=below:{$j$}] (3) [right=of 2] {};

\draw (2)-- (3);
\end{tikzpicture}&
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode, label=below:{$i$}] (2) {};
\node[Dnode, label=below:{$j$}] (3) [right=of 2] {};

\draw[double line] (2)--(3);
\end{tikzpicture}&
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode, label=below:{$i$}] (2) {};
\node[Dnode, label=below:{$j$}] (3) [right=of 2] {};

\draw[triple line] (2)--(3);
\draw (2)--(3);
\end{tikzpicture}
\\
$c_{i,j} = -1$ & $c_{i,j} = -2$ & $c_{i,j} = -3$
\\
$c_{j,i} = -1$ & $c_{j,i} = -1$ & $c_{j,i} = -1$
\end{tabular}
\]
Here, we note that an arrow on the Dynkin diagram represents the relative lengths of two simple roots. Indeed, an arrow points to the shorter of two roots. (Hence, one can also consider an arrow as an inequality comparing the length of two roots.) In the above diagrams with multiple edges, we have $\norm{\alpha_{i}} > \norm{\alpha_{j}}$ for two simple roots $\alpha_i$ and $\alpha_j$ associated with the vertices $i$ and $j$ in the Dynkin diagram.

\begin{example}\label{example_char_matrix_s31452}
Suppose that $G$ is of type $\dyn{A}_5$. Let $w = s_3s_1s_4s_5s_2$. Then the reduced characteristic matrix in~\eqref{equation_ray_vectors} is given as follows:
\[
{\begin{bmatrix}
-1 & 0 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & 0 \\
1 & 0 & -1 & 0 & 0 \\
0 & 0 & 1 & -1 & 0 \\
1 & 1 & 0 & 0 & -1
\end{bmatrix}}.
\]
Since $(i_1,i_2,i_3,i_4,i_5)=(3,1,4,5,2)$, the column vectors above are $\mathbf{w}_3, \mathbf{w}_1, \mathbf{w}_4, \mathbf{w}_5, \mathbf{w}_2$ from the left. Therefore we have
\[
\mathbf{v}_3+\mathbf{w}_3=\mathbf{v}_4+\mathbf{v}_2,
\qquad \mathbf{v}_1+\mathbf{w}_1=\mathbf{v}_2,
\qquad \mathbf{v}_4+\mathbf{w}_4=\mathbf{v}_5,
\qquad \mathbf{v}_5+\mathbf{w}_5=\mathbf{0},
\qquad \mathbf{v}_2+\mathbf{w}_2=\mathbf{0}.
\]
\end{example}

\begin{example}
Suppose that $G$ is of type $\dyn{B}_2$. As computed in Example~\ref{example_toric_B2}, there are five toric Schubert varieties. Considering toric Schubert varieties of dimension~$2$, we obtain the reduced characteristic matrices in~\eqref{equation_ray_vectors} as follows:
\[
\setlength{\tabcolsep}{2em}
\begin{tabular}{cc}
$s_1s_2$ & $s_2s_1$ \\
$
\begin{bmatrix}
-1 & 0 \\ 1 & -1
\end{bmatrix} $ & $
\begin{bmatrix}
-1 & 0 \\ 2 & -1
\end{bmatrix} $
\end{tabular}
\]
For $w = s_1s_2$, the $(2,1)$-entry of the reduced characteristic matrix is $-c_{i_2,i_1} = -c_{2,1} = -(-1)=1$. Moreover, for $w = s_2s_1$, the $(2,1)$-entry of the reduced characteristic matrix is $-c_{i_2,i_1} = -c_{1,2} = -(-2)=2$. Therefore,
\[
\mathbf{v}_1+\mathbf{w}_1=\mathbf{v}_2,
\qquad \mathbf{v}_2+\mathbf{w}_2=\mathbf{0}
\]
in the former case and
\[
\mathbf{v}_2+\mathbf{w}_2=2\mathbf{v}_1,
\qquad \mathbf{v}_1+\mathbf{w}_1=\mathbf{0}
\]
in the latter case.
\end{example}

\subsection{Fano or weak Fano toric varieties}

There is a combinatorial way to determine whether a smooth projective toric variety is Fano or weak Fano. We say that a smooth projective variety $X$ is Fano (respectively, weak Fano) if the anticanonical divisor $-K_{X}$ is ample (respectively, nef and big). For a fan $\Sigma$, a subset $R$ of the primitive ray vectors is called a \emph{primitive collection} of~$\Sigma$ if
\[
\Cone(R) \notin \Sigma
\quad \text{but}
\quad \Cone(R \setminus \{\mathbf{u}\}) \in \Sigma
\quad \text{for every $\mathbf{u} \in R$}.
\]
We denote by $\PC(\Sigma)$ the set of primitive collections of $\Sigma$.
\begin{example}
For the toric Schubert variety $X_w$ in Theorem~\ref{thm_char_matrix_of_toric_Schubert}, the primitive ray vectors of the fan~$\Sigma$ of $X_w$ are the column vectors $C \coloneqq \{\mathbf{v}_{i_1},\dots,\mathbf{v}_{i_m},\mathbf{w}_{i_1},\dots,\mathbf{w}_{i_m}\}$ in~\eqref{equation_ray_vectors}. Then
\begin{equation}\label{equation_PC_of_Bott}
\PC(\Sigma) = \braces[\big]{\{\mathbf{v}_{i_k},\mathbf{w}_{i_k}\} \st[\big] k =1,\dots,m}
\end{equation}
because a subset of $C$ forms a cone if and only if it does not contain both $\mathbf{v}_{i_k}$ and $\mathbf{w}_{i_k}$ for each $k$ as mentioned in Theorem~\ref{thm_char_matrix_of_toric_Schubert}.
\end{example}

For a primitive collection $R = \{\mathbf{u}'_1, \dots,\mathbf{u}'_{\ell}\}$, we get either $\mathbf{u}'_1 + \dotsb +\mathbf{u}'_{\ell} = \mathbf{0}$ or there exists a unique cone~$\sigma$ of positive dimension such that $\mathbf{u}'_1 + \dotsb +\mathbf{u}'_{\ell}$ is in the interior of $\sigma$. If the sum $\mathbf{u}'_1 + \dotsb +\mathbf{u}'_{\ell}$ is in the interior of $\sigma$, then
\begin{equation}\label{eq:primitive}
\mathbf{u}'_1 + \dotsb +\mathbf{u}'_{\ell}= a_1 \mathbf{u}_1 + \dotsb + a_{s} \mathbf{u}_{s},
\end{equation}
where $\mathbf{u}_1,\dots,\mathbf{u}_{s}$ are the primitive generators of $\sigma$ and $a_1,\dots,a_{s}$ are positive integers. We call~\eqref{eq:primitive} a \emph{primitive relation}, and the \emph{degree} $\deg R$ of a primitive collection $R$ is defined to be
\begin{equation}\label{eq:degree}
\deg R \colonequals
\begin{cases}
	\ell
	& \text{if $\mathbf{u}'_1 + \dotsb +\mathbf{u}'_{\ell} = \mathbf{0}$},
\\	\ell - (a_1+ \dotsb +a_s)
	& \text{otherwise}.
\end{cases}
\end{equation}
Batyrev~\cite{Batyrev} provided a criterion for a smooth projective toric variety to be Fano or weak Fano.
\begin{proposition}[{\cite[Proposition~2.3.6]{Batyrev}}]\label{prop:batyrev}
A smooth projective toric variety $X$ is Fano (respectively, weak Fano) if and only if $\deg R>0$ (respectively, $\deg R\geq 0$) for every primitive collection $R$ of the fan $\Sigma$ of $X$.
\end{proposition}

A smooth compact toric variety is called a \emph{Bott manifold} if it is isomorphic to the total space of a Bott tower that is an iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is the projectivization of the Whitney sum of two complex line bundles. It is known that a smooth compact toric variety $X$ is a Bott manifold if and only if the fan $\Sigma$ of $X$ has primitive collections $\PC(\Sigma)$ of the form in~\eqref{equation_PC_of_Bott}. Theorem~\ref{thm_char_matrix_of_toric_Schubert} says that a toric Schubert variety is a Bott manifold. We refer the reader to~\cite{GK94Bott} for Bott towers, and to~\cite{MasudaPanov08} for details on Bott manifolds.

\section{Toric Schubert varieties and directed graphs}\label{section_toric_Schubert_directed_graphs}

In this section, we associate an edge-labeled digraph $\G_w$ to a toric Schubert variety $X_w$ and prove that two toric Schubert varieties are isomorphic as varieties if and only if the associated edge-labeled digraphs are isomorphic (Theorem~\ref{theo:graph_determines_Xw}). We also give a simple criterion for determining when $X_w$ is Fano or weak Fano in terms of $\G_w$.

As before, let $W$ be the Weyl group of a simple algebraic group $G$ of rank $r$ and $s_1,\dots,s_r$ the simple reflections in $W$. Suppose that a Schubert variety $X_w$ ($w\in W$) is toric. Then simple reflections in a reduced decomposition $w=s_{i_1} \dotsm s_{i_m}$ for $w$ are mutually distinct by Theorem~\ref{thm_toric_Schubert_distinct}.

\begin{definition}
For a reduced expression $w=s_{i_1} \dotsm s_{i_m}$ with distinct simple reflections, we define an edge-labeled digraph $\G_{w}$ as follows:
\begin{itemize}
\item $V(\G_{w}) = \{i_1,\dots,i_m\}$;
\item $(i_j,i_k) \in E(\G_{w})$ if and only if $c_{i_j,i_k} \neq 0$ for $1 \leq k < j \leq m$.
\end{itemize}
Note that $1\le -c_{i_j, i_k}\le 3$ if $c_{i_j,i_k} \neq 0$. We assign the positive integer $-c_{i_j, i_k}$ to the directed edge~$(i_j,i_k) \in E(\G_{w})$. When we draw\/ $\G_w$, we omit the label $1$ for simplicity.
\end{definition}

\begin{example}\label{exam:digraph_Gi}
\mbox{}
\begin{enumerate}[leftmargin=*]
\item \label{example_3.2_1}
Let $G$ be of type $\dyn{A}$. For $w=s_2s_1s_3s_4=s_2s_3s_1s_4=s_2s_3s_4s_1$ and $w'=s_3s_4s_2s_1=s_3s_2s_4s_1=s_3s_2s_1s_4$, we have
\[
\mathcal G_{w} =
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\node[node] (4) at (5.2,0) {$4$};
\draw[->] (1) to (2);
\draw[<-] (2) to (3);
\draw[<-] (3) to (4);
\end{tikzpicture}
\qquad \mathcal G_{w'} =
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\node[node] (4) at (5.2,0) {$4$};
\draw[->] (1) to (2);
\draw[->] (2) to (3);
\draw[<-] (3) to (4);
\end{tikzpicture}
\]
For $w''=s_1s_2s_4s_5=s_1s_4s_2s_5=s_1s_4s_5s_2=s_4s_1s_2s_5=s_4s_1s_5s_2=s_4s_5s_1s_2$, $\mathcal G_{w''}$ is not connected and is given by:
\[
\mathcal G_{w''} =
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$4$};
\node[node] (4) at (5.2,0) {$5$};
\draw[<-] (1) to (2);
\draw[<-] (3) to (4);
\end{tikzpicture}
\]
\item \label{example_3.2_2}
Let $G$ be of type $\dyn{C}_3$. For $w=s_1s_2s_3$ and $w' =s_3s_2s_1$, we have
\[
\G_{w} =
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\draw[<-] (1) to (2);
\draw[<-] (2) to node[above, midway] {$2$} (3) ;
\end{tikzpicture}
\qquad \G_{w'} =
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\draw[->] (1) to (2);
\draw[->] (2) to (3) ;
\end{tikzpicture}
\]
\item Let $G$ be of type $\dyn{D}_4$. For $w=s_2s_1s_3s_4=s_2s_3s_1s_4=s_2s_3s_4s_1=s_2s_1s_4s_3=s_2s_4s_1s_3=s_2s_4s_3s_1$, we have
\[
\G_{w} =
\begin{tikzpicture}
[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (0,0) {$2$};
\node[node] (2) at (60:1.4) {$3$};
\node[node] (3) at (-60:1.4) {$4$};
\node[node] (4) at (180:1.4) {$1$};

\draw[<-] (1) to (2) ;
\draw[<-] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture}
\]
\end{enumerate}
\end{example}

We say that two edge-labeled digraphs are isomorphic if there is a bijection between their vertices preserving directed edges and labels on the edges. For example, $\G_w$ and $\G_{w'}$ in Example~\ref{exam:digraph_Gi}\eqref{example_3.2_1} are isomorphic but those in Example~\ref{exam:digraph_Gi}\eqref{example_3.2_2} are not isomorphic as edge-labeled digraphs although they are isomorphic as digraphs.

\begin{remark}\label{rmk_graph_is_well-defined}
Suppose that $c_{{i_k},i_{k+1}}=0$ for some $k\in [m-1]$, i.e.\ $s_{i_k}$ and $s_{i_{k+1}}$ commute. Then replacing the factor $s_{i_k}s_{i_{k+1}}$ in the reduced decomposition $w=s_{i_1} \dotsm s_{i_m}$ by $s_{i_{k+1}}s_{i_k}$ is called a \emph{$2$-move}. Since $w$ is toric, every reduced decomposition uses each simple reflection at most once, so no braid moves of length $\geq 3$ can occur. Therefore, by the Word Property~\cite[Theorem~3.3.1]{BB05Combinatorics}, any two reduced decompositions are related by a sequence of $2$-moves. This implies that the edge-labeled digraph $\G_w$ does not depend on the choice of the reduced decomposition for $w$.
\end{remark}

Let $G$ be a simple algebraic group. We say an edge-labeled digraph $\G$ is an \emph{induced subgraph} of the Dynkin diagram of $G$ if the underlying graph of $\G$ is an induced subgraph of the Dynkin diagram of $G$, and the edge labels match with the Cartan integers, that is, for each directed edge $(p,q) \in E(\G)$, the label $-c_{p,q}$ is assigned. For instance, in type $C_3$, the first two graphs are induced while the last is not:
\[
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\draw[<-] (1) to (2);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (2) at (2,0) {$2$};
\node[node] (3) at (3.4,0) {$3$};
\draw[<-] (2) to node[above, midway] {$2$} (3) ;
\end{tikzpicture}
\qquad
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\draw[<-] (1) to node[above, midway] {$2$} (2);
\draw[<-] (2) to node[above, midway] {$2$} (3) ;
\end{tikzpicture}
\]

\begin{proposition}
Let $G$ be a simple algebraic group. An edge-labeled digraph $\G$ is an induced subgraph of the Dynkin diagram of $G$ if and only if $\G = \G_w$ for some $w \in W$ defining a toric Schubert variety. In particular, the underlying graph of\/ $\G$ is the Dynkin diagram of $G$ if and only if\/ $\G = \G_w$ for a Coxeter element $w$.
\end{proposition}

\begin{proof}
Suppose that $w \in W$ defines a toric Schubert variety. By definition, $\G_w$ is an induced subgraph of the Dynkin diagram of $G$.

For the opposite direction, suppose that $\G$ is an induced subgraph of the Dynkin diagram of~$G$ with the induced labeling of vertices. We construct a word for $w$ by iteratively removing sinks.

Recall that a vertex $i$ is a \emph{sink} if there is no directed edge of $\G$ emanating from $i$. Let $i_1$ be the smallest sink of $\G$. Let $\G_1$ be the induced subgraph obtained from $\G$ by deleting $i_1$, and let $i_2$ be the smallest sink of $\G_1$. Continuing this process, we obtain a sequence $i_1,\dots,i_m$, where $m=\card[\big]{V(\G)}$. Set
\[
w \colonequals s_{i_1}s_{i_2} \dotsm s_{i_m}.
\]
We claim that $\G_w=\G$.

First, by construction, we have $V(\G_w)=\{i_1,\dots,i_m\}=V(\G)$. Next we compare directed edges. Fix $1\le k<j\le m$. By definition of $\G_w$, there is a directed edge $(i_j,i_k)$ in $\G_w$ if and only if $c_{i_j,i_k}\neq 0$, i.e., the vertices $i_j$ and $i_k$ are adjacent in the Dynkin diagram of $G$.

On the other hand, since $\G$ is an induced subgraph of the Dynkin diagram, adjacency in the Dynkin diagram is equivalent to adjacency in the underlying graph of $\G$. Moreover, because $i_k$ is a sink in $\G_{k-1}$, there is no directed edge in $\G_{k-1}$ leaving $i_k$. In particular, for every $j>k$ with $c_{i_j,i_k}\neq 0$, the only possible orientation of the edge between $i_j$ and $i_k$ in $\G_{k-1}$ is toward $i_k$, i.e., it must be the directed edge $(i_j,i_k)$. Since $\G$ is obtained from $\G_{k-1}$ by adding back the previously removed vertices $i_1,\dots,i_{k-1}$, the orientation of the edge between $i_j$ and $i_k$ remains $(i_j,i_k)$ in $\G$ as well.

Therefore, for every pair $1\le k<j\le m$ we have:
\[
c_{i_j,i_k}\neq 0
\quad \iff \quad
\text{$i_j$ and $i_k$ are adjacent in the Dynkin diagram}
\quad \iff \quad
(i_j,i_k)\in E(\G).
\]
This is exactly the edge rule for $\G_w$, and the labels also agree because both $\G$ and $\G_w$ label the directed edge $(i_j,i_k)$ by $-c_{i_j,i_k}$. Hence $\G_w=\G$.

Finally, the underlying graph of $\G$ is the Dynkin diagram of $G$ if and only if $V(\G)$ is the full set of vertices of the Dynkin diagram, i.e., $\{i_1,\dots,i_m\}=\{1,\dots,r\}$. In that case $w=s_{i_1} \dotsm s_{i_r}$ is a product of all simple reflections, hence a Coxeter element. Conversely, if $w$ is a Coxeter element defining a toric Schubert variety, then $V(\G_w)$ contains all simple reflections, so the underlying graph of~$\G_w$ is the Dynkin diagram of $G$.
\end{proof}

The edge-labeled digraph $\G_w$ has the same information as the primitive relations in~\eqref{equation_ray_vectors2}. Indeed, $\G_w$ can be obtained from the primitive relations in~\eqref{equation_ray_vectors2} if we take a directed edge $(i_j,i_k)$ whenever $c_{i_j,i_k}$ in~\eqref{equation_ray_vectors2} is nonzero and put the label $-c_{i_j,i_k}$ on it. Conversely, it is clear that the primitive relations in~\eqref{equation_ray_vectors2} can be obtained from $\G_w$ through this correspondence.

\begin{theorem}\label{theo:graph_determines_Xw}
Let $W$ and $W'$ be the Weyl groups of simple algebraic groups $G$ and $G'$, respectively. Then the toric Schubert varieties $X_w$ ($w\in W$) and $X_{w'}$ ($w'\in W'$) are isomorphic as varieties if and only if\/ $\G_{w}$ and\/ $\G_{w'}$ are isomorphic as edge-labeled digraphs.
\end{theorem}

\begin{proof}
Note that two smooth complete toric varieties are isomorphic as varieties if and only if they are also isomorphic as toric varieties (see~\cite{Berchtold03}). This is because the automorphism group $\operatorname{Aut}(X)$ of a smooth complete toric variety $X$ is linear algebraic and the algebraic torus acting on~$X$ is a maximal torus of $X$ (see~\cite[Section~3.4]{Oda88} and references therein). Hence, it is enough to prove that $X_w$ and $X_{w'}$ are isomorphic as toric varieties if and only if $\mathcal{G}_w$ and $\mathcal{G}_{w'}$ are isomorphic as edge-labeled digraphs.

First, we prove the ``only if'' part. Suppose that $X_w$ and $X_{w'}$ are isomorphic as varieties. Then there is an isomorphism $\varphi$ from the fan $\Sigma$ of $X_w$ to the fan $\Sigma'$ of $X_{w'}$. Let
\begin{equation}\label{eq:ray_vectors_vw}
\{\mathbf v_{1},\dots,\mathbf v_{m}, \mathbf w_{1},\dots,\mathbf w_{m}\}
\qquad \text{and} \qquad
\{\mathbf v'_{1},\dots,\mathbf v'_{m}, \mathbf w'_{1},\dots,\mathbf w'_{m}\}
\end{equation}
be the ray generators (i.e., column vectors of the matrix in~\eqref{equation_ray_vectors}) of the fans $\Sigma$ and $\Sigma'$, respectively. (Within this proof, for notational simplicity, we index the vectors by $1,\dots,m$ instead of using $i_1,\dots,i_m$.) Since the isomorphism $\varphi$ preserves the primitive collections, there exists a bijection $f \colon [m] \to [m]$ such that
\begin{equation}\label{eq_sum_vi_wi_and_image_f}
\braces[\big]{\varphi(\mathbf v_{k}), \: \varphi(\mathbf w_{k})} = \braces[\big]{\mathbf v'_{f(k)}, \: \mathbf w'_{f(k)}}
\qquad (\forall k\in [m]).
\end{equation}
Therefore,
\begin{equation}\label{eq:sum_vw1}
\varphi(\mathbf v_{k} + \mathbf w_{k}) = \varphi(\mathbf v_{k})+\varphi(\mathbf w_{k})= \mathbf v'_{f(k)} + \mathbf w'_{f(k)}=\sum_{p>f(k)}\parens[\big]{-c_{i'_{p},i'_{f(k)}}}\mathbf v'_{p}
\end{equation}
while sending the identity in~\eqref{equation_ray_vectors2} by $\varphi$, we obtain
\begin{equation}\label{eq:sum_vw2}
\varphi(\mathbf v_{k} + \mathbf w_{k}) =\sum_{j > k} \parens{-c_{i_j,i_k}} \varphi(\mathbf v_{j}).
\end{equation}
It follows from~\eqref{eq:sum_vw1} and~\eqref{eq:sum_vw2} that
\begin{equation}\label{eq:sum_vw3}
\sum_{p>f(k)}\parens[\big]{-c_{i'_{p},i'_{f(k)}}}\mathbf v'_{p} =\sum_{j > k} \parens{-c_{i_j,i_k}} \varphi(\mathbf v_{j}).
\end{equation}
Here, $\varphi(\mathbf v_{j})=\mathbf v'_{f(j)}$ or $\mathbf w'_{f(j)}$ by~\eqref{eq_sum_vi_wi_and_image_f}.

When $c_{i_j,i_k}\not=0$, we claim that the latter case does not occur. Assume on the contrary that there exist indices such that $c_{i_j, i_k} \neq 0$ and $\varphi(\mathbf v_j) = \mathbf{w}_{f(j)}'$. Take $j$ such that $f(j)$ is minimal among such indices. Comparing the $f(j)$-th entries of both sides in~\eqref{eq:sum_vw3}, we obtain a contradiction. This is because the vector $\mathbf w'_{f(j)}$ has $-1$ on the $f(j)$-th entry and $-c_{i_j, i_k} \neq 0$, so the vector on the right-hand side of~\eqref{eq:sum_vw3} has a negative component while all components of the vector on the left-hand side of~\eqref{eq:sum_vw3} are nonnegative. Thus~\eqref{eq:sum_vw3} implies
\[
\sum_{p > f(k)} \parens[\big]{-c_{i'_{p},i'_{f(k)}}}\mathbf v'_{p} = \sum_{j > k} \parens{-c_{i_j,i_k}} \mathbf{v}'_{f(j)}.
\]
Since $\mathbf v'_{1},\dots,\mathbf v'_{m}$ are linearly independent, we obtain
\[
\braces[\big]{f(j) \st[\big] j > k, \ c_{i_j, i_k} \neq 0} = \braces[\big]{p \st[\big] p > f(k), \ c_{i'_p, i'_{f(k)}} \neq 0}.
\]
Therefore, if we have an edge $(i_j, i_k)$ in $\mathcal{G}_w$ with label $-c_{i_j, i_k}$, then we have $f(j) > f(k)$ with $c_{i_{f(j)}', i_{f(k)}'} = c_{i_j, i_k}$. Hence, we have an edge $(i_{f(j)}', i_{f(k)}')$ with label $-c_{i_j, i_k}$ in the graph $\mathcal{G}_{w'}$. Therefore, the bijection $f\colon [m] \to [m]$ provides an isomorphism from $\G_w$ to $\G_{w'}$ as edge-labeled digraphs.

Now we prove the ``if'' part. Suppose that $\G_w$ and $\G_{w'}$ are isomorphic as edge-labeled digraphs. Then there is a bijection $f\colon [m] \to [m]$ which satisfies
\begin{equation}\label{eq:cij_f}
c_{i_j,i_k} = c_{i_{f(j)}',i_{f(k)}'}
\quad \text{for all $1 \leq k < j \leq m$}.
\end{equation}
We shall observe that the linear automorphism $\varphi$ of $\R^m$ defined by $\varphi(\mathbf{v}_{j})=\mathbf{v}'_{f(j)}$ for $j\in [m]$ provides an isomorphism from the fan $\Sigma$ to the fan $\Sigma'$. First we note
\begin{equation}\label{eq:varphi(w)}
\varphi(\mathbf{w}_{k})=\mathbf{w}'_{f(k)}
\quad (\forall k\in [m]).
\end{equation}
Indeed, \eqref{eq:sum_vw3} holds by~\eqref{eq:cij_f} and it follows from~\eqref{equation_ray_vectors2} and~\eqref{eq:sum_vw3} that we have
\[
\begin{split}
\varphi(\mathbf{v}_{k})+\varphi(\mathbf{w}_{k})
	& = \varphi(\mathbf{v}_{k}+\mathbf{w}_{k})
\\	& = \sum_{j > k} \parens{-c_{i_j,i_k}} \varphi(\mathbf v_{j})
\\	& = \sum_{f(j)>f(k)}\parens[\big]{-c_{i'_{f(j)},i'_{f(k)}}}\mathbf v'_{f(j)}
\\	& = \mathbf v'_{f(k)} + \mathbf w'_{f(k)}.
\end{split}
\]
Since $\varphi(\mathbf{v}_{k})=\mathbf{v}'_{f(k)}$ by definition of $\varphi$, the identity above implies~\eqref{eq:varphi(w)} as we desired. Any maximal cone in $\Sigma$ is spanned by $\mathbf{u}_{1},\dots,\mathbf{u}_{m}$ where $\mathbf{u}$ denotes either $\mathbf{v}$ or $\mathbf{w}$, and the same is true for~$\Sigma'$. Therefore, $\varphi$ sends maximal cones in $\Sigma$ to those in $\Sigma'$ bijectively. This means that $\varphi$ is an isomorphism from $\Sigma$ to $\Sigma'$. Hence $X_w$ and $X_{w'}$ are isomorphic as toric varieties.
\end{proof}

The edge-labeled digraph $\G_{w}$ encodes all the geometrical information of the toric Schubert variety $X_w$ by Theorem~\ref{theo:graph_determines_Xw}. We shall give a simple criterion for when $X_w$ is (weak) Fano in terms of~$\G_w$. For each vertex $i_k$ of $\G_{w}$ (where $k \in [m]$), we define the \emph{indegree} $\indeg(i_k)$ of $i_k$ to be the sum of labels of the edges going into $i_k$:
\begin{equation}\label{eq:outdegree}
\indeg(i_k) \coloneqq \sum_{(i_j,i_k) \in E(\G_w)} (-c_{i_j,i_k})= \sum_{j>k} (-c_{i_j,i_k}).
\end{equation}
Similarly, the \emph{outdegree} $\outdeg(i_k)$ of $i_k$ is defined to be the sum of labels of the edges going out $i_k$:
\[
\outdeg(i_k) \coloneqq \sum_{(i_k, i_j) \in E(\G_w)} (-c_{i_j, i_k}) = \sum_{j < k} (-c_{i_j, i_k}).
\]
The \emph{degree} of $i_k$ is defined to be the sum of the indegree and outdegree:
\[
\deg(i_k) = \indeg(i_k) + \outdeg(i_k).
\]

\begin{theorem}\label{thm_Fano_weak_Fano}
A toric Schubert variety $X_w$ is Fano (respectively, weak Fano) if and only if every vertex of\/ $\G_w$ has indegree at most $1$ (respectively, $2$).
\end{theorem}

\begin{proof}
Let $\{\mathbf v_{i_1},\dots,\mathbf{v}_{i_m},\mathbf{w}_{i_1},\dots,\mathbf{w}_{i_m}\}$ be the column vectors of the matrix in~\eqref{equation_ray_vectors}. Then the primitive collections of the fan of $X_w$ are $\{\mathbf v_{i_k}, \mathbf w_{i_k} \}$ for $k=1,\dots,m$ and the primitive relations are
\[
\mathbf v_{i_k} + \mathbf w_{i_k} = \sum_{j > k} (-c_{i_j,i_k})\mathbf{v}_{i_j}.
\]
Therefore, it follows from~\eqref{eq:degree} that the degree of the primitive collection $\{\mathbf{v}_{i_k},\mathbf{w}_{i_k}\}$ is given by
\[
\deg \{\mathbf v_{i_k}, \mathbf w_{i_k}\} = 2 - \sum_{j>k}(-c_{i_j,i_k})=2-\indeg(i_k),
\]
where the latter equality follows from~\eqref{eq:outdegree}. Thus, the theorem follows from Proposition~\ref{prop:batyrev}.
\end{proof}

In type $\dyn{A}$, the edge-labeled digraph $\G_w$ is a union of directed path graphs whose all edges have label $1$. Therefore, we have the following.

\begin{corollary}
In Type $\dyn{A}$, any toric Schubert variety $X_w$ is weak Fano. Moreover, it is Fano if and only if\/ $\G_w$ has no sink other than leaves.
\end{corollary}

Using the graph $\G_w$, one can also determine whether the toric variety $X_w$ decomposes as a product of lower-dimensional toric Schubert varieties.

\begin{proposition}\label{prop_decomposition}
Let $w \in W$. Suppose that a reduced expression of $w$ consists of distinct simple reflections, so that $X_w$ is a toric Schubert variety. Let\/ $\G_1,\dots,\G_k$ be the connected components of the edge-labeled digraph $\G_w$. Then $X_w$ is isomorphic to the product of the toric Schubert varieties corresponding to\/ $\G_1,\dots,\G_k$.
\end{proposition}

\begin{proof}
Let $w=s_{i_1} \dotsm s_{i_m}$ be a reduced expression with mutually distinct simple reflections. Write $V(\G_w)=V(\G_1)\sqcup\dots\sqcup V(\G_k)$ and set $m_\nu\colonequals \card[\big]{V(\G_\nu)}$ for $1 \leq \nu \leq k$. To prove the proposition, recall from~\cite[Proposition~3.1.14]{CLS11Toric} that the fan of a product of toric varieties is the product of the fans of the factors.

As a first step, we use the connected components of $\G_w$ to identify the factors. If $p\in V(\G_\mu)$ and $q\in V(\G_\nu)$ with $\mu\neq \nu$, then there is no edge between $p$ and $q$ in the underlying graph of $\G_w$. Since $\G_w$ is an induced subgraph of the Dynkin diagram, this implies $c_{p,q}=c_{q,p}=0$, hence $s_p$ and $s_q$ commute. Therefore, by applying a sequence of $2$-moves, we may reorder the reduced expression so that it is a concatenation
\begin{equation}\label{eq_product_ordering}
w = w^{(1)} \dotsm w^{(k)},
\end{equation}
where each $w^{(\nu)}$ is the product of the simple reflections whose indices lie in $V(\G_\nu)$. Notice that a choice of reduced decomposition does not change $X_w$ and it does not change the fan up to a lattice automorphism.

By Theorem~\ref{thm_char_matrix_of_toric_Schubert}, the fan of $X_w$ is described by the matrix in~\eqref{equation_ray_vectors}. The right $m\times m$ block of that matrix has $(j,k)$-entry $-c_{i_j,i_k}$ for $j>k$. Considering the reduced decomposition in~\eqref{eq_product_ordering}, the index set $\{1,\dots,m\}$ is partitioned into consecutive blocks of sizes $m_1,\dots,m_k$, and for $\mu\neq \nu$ we have $c_{p,q}=0$ whenever $p\in V(\G_\mu)$ and $q\in V(\G_\nu)$. Hence, all off-block entries in the right block vanish, so the matrix in~\eqref{equation_ray_vectors} becomes block diagonal with $k$ blocks, each block being exactly the matrix associated to $w^{(\nu)}$. Consequently, after applying an appropriate lattice isomorphism, the fan $\Sigma_w$ of $X_w$ is the product fan
\[
\Sigma_w \cong \Sigma_{w^{(1)}} \times \dots \times \Sigma_{w^{(k)}}
\]
under the identification $\R^m \cong \R^{m_1}\oplus \dots \oplus \R^{m_k}$. Therefore, by~\cite[Proposition~3.1.14]{CLS11Toric}, we obtain a toric isomorphism $X_w \cong X_{w^{(1)}}\times\dots\times X_{w^{(k)}}$. Since $\G_{w^{(\nu)}}=\G_\nu$ for each $\nu$ by construction, the factors are precisely the toric Schubert varieties corresponding to $\G_1,\dots,\G_k$.
\end{proof}

\section{Enumeration of toric Schubert varieties} \label{sect:enumeration}

As discussed in Section~\ref{section_toric_Schubert_directed_graphs}, the graph $\G_{w}$ encodes fruitful geometric information about a toric Schubert variety~$X_w$. In this section, we enumerate the isomorphism classes of Fano or weak Fano toric Schubert varieties using $\G_w$ in Proposition~\ref{cor_enumeration_isom_Coxeter_Schubert}.

Let $G$ be a simple algebraic group of rank $r$ with Cartan matrix $C = (c_{i,j})_{1 \leq i, j \leq r}$. A bijection $\theta \colon [r] \to [r]$ is called a \emph{Dynkin diagram automorphism} if
\[
c_{\theta(i),\theta(j)} = c_{i,j}
\]
holds for all $i,j$. For each Lie type, we recall from~\cite[Section~12.2]{Humphreys78Lie} the group $\Gamma$ of Dynkin diagram automorphisms in Table~\ref{table_D_diagram_auto}. Note that except for $D_4$, the only nontrivial automorphism group is~$\Z_2$. When a generator $\theta\in\Gamma$ is realized by conjugation in $W$, i.e., when there exists $x\in W$ such that $x s_i x^{-1} = s_{\theta(i)}$ for all $i \in [r]$, we display one such element $x$ in the third column of the table.

\begin{table}[!h]
\caption{Dynkin diagram automorphisms.}\label{table_D_diagram_auto}
\begin{tabular}{@{}cll@{}}
\toprule
Type & $\Gamma$ & $x$ \\
\midrule
$\dyn{A}_{r}$ & $\Z_2$ ($r \geq 2$) & $w_0$\\
$\dyn{B}_r$, $\dyn{C}_r$ & $1$ \\
$\dyn{D}_r$ & $
\begin{cases}
	\mathfrak{S}_3
	& (r = 4)
\\	\Z_2
	& (r > 4)
\end{cases}$ & $w_0$ if $r$ odd\\[2ex]
$\dyn{E}_6$ & $\Z_2$ & $w_0$\\
$\dyn{E}_7$ & $1$\\
$\dyn{E}_8$ & $1$\\
$\dyn{F}_4$ & $1$\\
$\dyn{G}_2$ & $1$ \\
\bottomrule
\end{tabular}
\end{table}

An element of $W$ is called a \emph{Coxeter element} if it can be written as a product of all simple reflections $s_1,\dots,s_{r}$. Let $\Cox_W$ denote the set of all Coxeter elements in $W$. Using Theorem~\ref{theo:graph_determines_Xw}, we obtain the following result.

\begin{proposition}\label{thm:dynkin1}
Let $W$ be the Weyl group of a simple algebraic group $G$. Let $w, w' \in \Cox_W$. The following statements are equivalent:
\begin{enumerate}
\item \label{statement_1} $X_w \cong X_{w'}$ as varieties;
\item \label{statement_2} $\mathcal G_{w} \cong \mathcal G_{w'}$ as edge-labeled digraphs. 
\end{enumerate}
If\/ $G$ is of type $\dyn{A}$, $\dyn{D}_r$ ($r$ odd), or\/ $\dyn{E}_6$, then the above statements are equivalent to the following:
\begin{enumerate}[resume]
\item \label{statement_3} $w' = w$ or $w' = w_0 w w_0$.
\end{enumerate}
\end{proposition}

\begin{remark}
The statement~\eqref{statement_3} implies the statement~\eqref{statement_1} in Proposition~\ref{thm:dynkin1} for any (not necessarily toric) Schubert variety (see~\cite{richmond2021isomorphism}). When $G$ is of type $A$, we consider the composition $\sigma$ of the two isomorphisms~$\tau,\rho$ of $\GL_n(\C)$ defined by
\[
\tau(A)=\! ^tA^{-1},
\qquad \rho(A)=w_0Aw_0.
\]
Since both $\tau$ and $\rho$ send the upper triangular Borel subgroup $B$ to the lower triangular Borel subgroup, their composition $\sigma$ preserves $B$ and hence induces a variety automorphism of $\GL_n(\C)/B$. Since $\tau$ fixes permutation matrices, we have $\sigma(BwB)=B\sigma(w)B=Bw_0ww_0B$, which implies $\sigma(X_w)=X_{w_0ww_0}$.
\end{remark}

Recall that for a Coxeter element $w \in \Cox_W$, the underlying graph of $\G_w$ is the same as the Dynkin diagram of $W$. Accordingly, by Theorem~\ref{theo:graph_determines_Xw}, the number of isomorphism classes of toric Schubert varieties given by $\Cox_W$ is the same as that of orientations and the edge labels of the Dynkin diagram of~$W$.

\begin{proposition}\label{cor_enumeration_isom_Coxeter_Schubert}
For each simple algebraic group $G$, the number of isomorphism classes of Fano or weak Fano toric Schubert varieties in $G/B$ given by Coxeter elements is displayed in Table~\ref{table_cardinality_iso_classes_Coxeter}.
\end{proposition}

\begin{table}[!h]
\caption{The cardinalities of the isomorphism classes of toric Schubert varieties given by Coxeter elements.}\label{table_cardinality_iso_classes_Coxeter}
\begin{tabular}{@{}clll@{}}
\toprule
Type & $\#\braces[\big]{X_w \st[\big] w \in \Cox_W } \big/{\sim}$ & weak Fano & Fano\\
\midrule
$\dyn{A}_r$ & $
\begin{cases}
	2^{r-2}
	& \text{when $r$ is even and $r\geq 2$}
\\	\makebox[\widthof{$7\times 2^{r-4} + 2^{\frac{r-5}{2}}$}][l]{$2^{r-2} + 2^{\frac{r-3}{2}}$}
	& \text{when $r$ is odd and $r \geq 3$}
\end{cases} $ & $
\begin{system}
	2^{r-2}
\\	2^{r-2} + 2^{\frac{r-3}{2}}
\end{system} $ & $
\begin{system}
	\frac{r}{2}
\\	\frac{r+1}{2}
\end{system}$
\\[2.5ex]
$\dyn{B}_r$ & $
\begin{cases}
	2^{r-1}
	& \text{when $r=2$ or $3$}
\\	7 \times 2^{r-4}
	& \text{when $r$ is even and $r \geq 4$}
\\	7 \times 2^{r-4} + 2^{\frac{r-5}{2}}
	& \text{when $r$ is odd and $r \geq 5$}
\end{cases}$ & $
\begin{system}
	2^{r-1}
\\	{7} \times 2^{r-4}
\\	{7} \times 2^{r-4} + 2^{\frac{r-5}{2}}
\end{system}$ & $1$
\\[4.5ex]
$\dyn{C}_r$ & $
\begin{cases}
	4
	& \text{when $r=3$}
\\	7 \times 2^{r-4}
	& \text{when $r$ is even and $r \geq 4$}
\\	7 \times 2^{r-4} + 2^{\frac{r-5}{2}}
	& \text{when $r$ is odd and $r \geq 5$}
\end{cases}$ & $
\begin{system}
	3
\\	{5} \times 2^{r-4}
\\	{5} \times 2^{r-4} + 2^{\frac{r-5}{2}}
\end{system}$ & {$
\begin{system}
	2
\\	\frac{r}{2}
\\	\frac{r+1}{2}
\end{system}$}
\\[4.5ex]
$\dyn{D}_r$ & $
\begin{cases}
	4
	& \text{if $r = 4$}
\\	\makebox[\widthof{$7\times 2^{r-4} + 2^{\frac{r-5}{2}}$}][l]{$3\cdot 2^{r-3}$}
	& \text{otherwise}
\end{cases}$ & $
\begin{system}
	3
\\	5 \cdot 2^{r-4}
\end{system}$ & $
\begin{system}
	2
\\	r-1
\end{system}$
\\[2.5ex]
$\dyn{E}_6$ & $20$ & $17$ & $4$ \\
$\dyn{E}_7$ & $2^6$ & $56$ & $7$ \\
$\dyn{E}_8$ & $2^7$ & $112$ & $8$ \\
$\dyn{F}_4$ & $2^3$ & $6$ & $2$ \\
$\dyn{G}_2$ & $2$ & $1$ & $1$ \\
\bottomrule
\end{tabular}
\end{table}

Before providing a proof, we recall some terminology on (labeled) digraphs. We say that a vertex $k$ is a \emph{sink} if $\outdeg(k) = 0$, and a \emph{source} if $\indeg(k) = 0$. For instance, for the digraph $\mathcal{G}_w$ in Example~\ref{exam:digraph_Gi}\eqref{example_3.2_1}, the vertices $1$ and $4$ are sources, $2$ is a sink, and the sink set is $\{2\}$. Moreover, if a sink has indegree $1$, we call it a \emph{leaf}. For instance, $1$ is a leaf in the graph $\mathcal{G}_w$ in Example~\ref{exam:digraph_Gi}\eqref{example_3.2_2}.

\begin{proof}[Proof of Proposition~\ref{cor_enumeration_isom_Coxeter_Schubert}]
We provide proof by analyzing directed Dynkin diagrams case-by-case. We will use the numbering of the vertices in Table~\ref{table_finite}. We consider the simply laced cases first and then treat non-simply laced cases.

Before proceeding, we explain the strategy. By Proposition~\ref{thm:dynkin1}, the enumeration reduces to counting orientations up to Dynkin diagram automorphisms.

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{A}$}}
For type $\dyn{A}$, since there are at most two edges at each vertex, any toric Schubert variety is weak Fano by Theorem~\ref{thm_Fano_weak_Fano}.

We enumerate the isomorphism classes depending on the parity of the rank $r$. When $r$ is even, consider any orientation on the corresponding Dynkin diagram, which is a path graph having $r$ vertices. The involution on the Dynkin diagram does not preserve the orientation. Therefore, the involution on the Dynkin diagram provides an involution on the set $\braces[\big]{X_w \st[\big] w \in \Cox_W}$ of toric Schubert varieties, and this involution has no fixed element. Since there are $r-1$ edges, the number of isomorphism classes is $2^{r-1}/2 = 2^{r-2}$.

When $r$ is odd, there exist orientations which are fixed by the involution on the Dynkin diagram. In fact, a directed path graph $\G$ on $[r]$ is fixed by the involution if and only if
\[
(i,i+1) \in E(\G) \quad \iff \quad (r-i+1, r-i) \in E(\G)
\quad \text{for any $1 \leq i \leq \tfrac{r-1}{2}$}.
\]
Therefore, the number of orientations fixed by the involution is $2^{\frac{r-1}{2}}$. Hence, the number of orientations that are not fixed by the involution is $2^{r-1}-2^{\frac{r-1}{2}}$. Accordingly, the number of isomorphism classes is
\[
(2^{r-1}-2^{\frac{r-1}{2}})/2 + 2^{\frac{r-1}{2}} = 2^{r-2} + 2^{\frac{r-3}{2}}.
\]

Now we enumerate the isomorphism classes of Fano toric Schubert varieties in $\braces[\big]{X_w \st[\big] w \in \Cox_W}$. By Theorem~\ref{thm_Fano_weak_Fano}, we consider orientations, where each vertex has at most one incoming edge. Since the underlying graph is a path graph, any sink is a leaf of the graph. Indeed, an internal sink on a path would have two incident edges, both directed toward it, and hence indegree $2$, contradicting the Fano condition. Consider the following possibilities.
\begin{enumerate}[leftmargin=*]
\item The sink set is $\{1,r\}$: there is a unique source, and it can be any one of the vertices $2,\dots,r-1$; once the source is chosen, the orientation is forced. Considering the involution reversing the path, there are $\frac{r-1}{2}$ orientations when $r$ is odd; $\frac{r}{2} -1$ orientations when $r$ is even.
\item The sink set is $\{1\}$: the vertex $r$ must be a source, and there is a unique orientation having a source at $r$ and sink at $1$. Moreover, this is isomorphic to the orientation having a source at $1$ and a sink at $r$.
\end{enumerate}
The above list exhausts all Fano orientations up to the Dynkin diagram involution since the sink set $\{r\}$ is the involution image of sink set $\{1\}$, and no other sink configurations are possible under the indegree $\leq 1$ condition on a path. Therefore, we have $\frac{r-1}{2}+1 = \frac{r+1}{2}$ orientations satisfying the Fano condition (up to isomorphism) when $r$ is odd; $\frac{r}{2}-1+1 = \frac{r}{2}$ orientations satisfying the Fano condition (up to isomorphism) when $r$ is even. This proves the claim.
\end{enonce*}

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{D}$}}
For type $\dyn{D}_r$, when $r = 4$, we have the four directed Dynkin diagrams up to isomorphism:
\[
\begin{tabular}{ccccc}
$\G_w$ &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (180:1) {};
\node[Dnode] (4) at (300:1) {};

\draw[<-] (1) to (2) ;
\draw[<-] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (180:1) {};
\node[Dnode] (4) at (300:1) {};

\draw[<-] (2) to (1) ;
\draw[<-] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (180:1) {};
\node[Dnode] (4) at (300:1) {};

\draw[<-] (2) to (1) ;
\draw[<-] (3) to (1) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (180:1) {};
\node[Dnode] (4) at (300:1) {};

\draw[<-] (2) to (1) ;
\draw[<-] (3) to (1) ;
\draw[<-] (4) to (1) ;
\end{tikzpicture} \\[2ex]
maximum of $\indeg(k)$ & $3$ & $2$ & $1$ & $0$
\end{tabular}
\]
When $r > 4$, the Dynkin subgraph consisting of vertices $1,\dots,r-2$ contributes $2^{r-3}$ orientations, and the two leaf edges incident to $r-2$ contribute $3$ possibilities up to the diagram automorphism swapping $r-1$ and $r$ (both edges oriented into $r-2$, both oriented out of $r-2$, or one in/one out). Accordingly, we have $2^{r-3} \times 3$ directed Dynkin diagrams up to isomorphism.

To enumerate Fano or weak Fano toric Schubert varieties, we consider directions on the Dynkin diagram of type $\dyn{D}_r$. By Theorem~\ref{thm_Fano_weak_Fano}, to enumerate Fano toric Schubert varieties, it is enough to consider orientations in which each vertex has at most one incoming edge. Accordingly, any sink is a leaf of the graph and there should be at least two sinks. Indeed, an internal sink would force indegree $\geq 2$, and if there were only one sink then the trivalent vertex $r-2$ would necessarily have at least two incoming edges, violating the Fano indegree $\leq 1$ condition. Consider the following possibilities:
\begin{enumerate}
\item \label{sink_1} the sink set is $\{1, r\}$: there is only one orientation satisfying the Fano condition; in this case, the vertex $r-1$ is the unique source;
\item \label{sink_2} the sink set is $\{1, r-1\}$: there is only one orientation satisfying the Fano condition; in this case, the vertex $r$ is the unique source;
\item the sink set is $\{r-1, r\}$: there is only one orientation satisfying the Fano condition; in this case, the vertex $1$ is the unique source;
\item the sink set is $\{1, r-1, r\}$: there are ${r-3}$ orientations satisfying the Fano condition; indeed, each of the vertices $2,\dots,r-2$ can be a source.
\end{enumerate}
Since the orientations in~\eqref{sink_1} and~\eqref{sink_2} yield isomorphic digraphs (via the Dynkin diagram automorphism swapping the two leaf vertices $r$ and $r-1$), we have $1 + 1 + r-3 = r-1$ orientations satisfying the Fano condition (up to isomorphism).

Now we enumerate weak Fano toric Schubert varieties. There is only one trivalent vertex in the graph and because of Theorem~\ref{thm_Fano_weak_Fano}, it is enough to consider the orientations such that the indegree of the trivalent vertex $r-2$ is at most $2$. Indeed, since all edge labels are $1$ and every other vertex has degree $\leq 2$, those vertices automatically have indegree $\leq 2$, so the only potential obstruction is the condition $\indeg(r-2) \leq 2$. Considering the orientations on the emanating edges of the vertex $r-2$, there are $2^3 - 1 = 7$ allowed orientations of the three incident edges (excluding the case $\indeg(r-2)=3$), and modulo the symmetry swapping $r-1$ and $r$ these fall into five isomorphism types:
\[
\begin{tabular}{cccccc}
$\G_w$ &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (-60:1) {};
\node[Dnode] (4) at (180:1) {};

\draw[dashed] (4) to (-2,0) ;
\draw[<-] (1) to (2) ;
\draw[<-] (1) to (3) ;
\draw[->] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (-60:1) {};
\node[Dnode] (4) at (180:1) {};

\draw[dashed] (4) to (-2,0) ;
\draw[<-] (1) to (2) ;
\draw[->] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (-60:1) {};
\node[Dnode] (4) at (180:1) {};

\draw[dashed] (4) to (-2,0) ;
\draw[->] (1) to (2) ;
\draw[->] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (-60:1) {};
\node[Dnode] (4) at (180:1) {};

\draw[dashed] (4) to (-2,0) ;
\draw[->] (1) to (2) ;
\draw[<-] (1) to (3) ;
\draw[->] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (60:1) {};
\node[Dnode] (3) at (-60:1) {};
\node[Dnode] (4) at (180:1) {};

\draw[dashed] (4) to (-2,0) ;
\draw[->] (1) to (2) ;
\draw[->] (1) to (3) ;
\draw[->] (1) to (4) ;
\end{tikzpicture} \\[2ex]
$\indeg(r-2)$ & $2$ & $2$ & $1$ & $1$ & $0$
\end{tabular}
\]
Since there is no condition on the ${r-4}$ edges in the path subgraph on the vertices $1,\dots,r-3$, there are $5 \cdot 2^{r-4}$ orientations satisfying weak Fano condition.
\end{enonce*}

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{E}$}}
For type $\dyn{E}_6$, the longest element $w_0$ provides a Dynkin diagram involution. Since the Dynkin diagram of $\dyn{E}_6$ can be constructed by adding one more edge to the middle vertex in the Dynkin diagram of type $\dyn{A}_5$, the number of isomorphism classes is the same as twice that of~$\dyn{A}_5$. This is because once an orientation on the $\dyn{A}_5$-subdiagram is fixed up to the involution, there are two choices for the orientation of the additional edge attached to the branching vertex. Therefore, we obtain $2 \times 10 = 20$. For types $\dyn{E}_7$ and $\dyn{E}_8$, there does not exist a nontrivial Dynkin diagram automorphism. Accordingly, there are $2^6$ (respectively, $2^7$) isomorphism classes in type~$\dyn{E}_7$ (re\-spectively,~$\dyn{E}_8$).

To enumerate (weak) Fano toric Schubert varieties in type $\dyn{E}_r$, we consider (weak) Fano in type~$\dyn{A}_{r-1}$ first and then see how many choices of orientations there are for the extra edge attached to the vertex $4$. The extra edge contributes two choices of orientations exactly when the vertex~$4$ is the unique source in the~$\dyn{A}_{r-1}$ orientation; otherwise its direction must point outward the vertex~$4$. For instance, for type $\dyn{E}_6$, there are three Fano toric Schubert varieties in type $\dyn{A}_5$, where they are distinguished by the position of the source. Now we look at the orientation on the extra edge of $\dyn{E}_6$. If the source is the middle point, that is the vertex~$4$, then both orientations are fine but otherwise the orientation will be unique. Therefore we obtain $2+1+1$ orientations satisfying Fano condition for Type $\dyn{E}_6$. A similar observation will work for $\dyn{E}_7$ and $\dyn{E}_8$. Indeed, for $r = 7,8$, an orientation satisfying the Fano condition in type $\dyn{A}_{r-1}$ has only one source. In fact, there is a unique source and it can be any of the vertices $[r] \setminus \{4\}$ on the $\dyn{A}_{r-1}$-subgraph. Moreover, the Dynkin diagram automorphism on $\dyn{A}_{r-1}$ does not induce that on~$\dyn{E}_r$. Therefore, we have $r-2+2$ orientations satisfying the Fano condition for type $\dyn{E}_r$. Here, $r-2$ orientations are induced from that on~$\dyn{A}_{r-1}$ which do not have the source on the vertex $4$; and two orientations coming from that on $\dyn{A}_{r-1}$ which has the source on the vertex $4$.

Now we consider weak Fano toric Schubert varieties. For type $\dyn{E}_6$, we consider orientations in type $\dyn{A}_5$. There are ten different orientations in type $\dyn{A}_5$ (up to isomorphism). We have the following three possibilities by looking at the middle vertex (which corresponds to the vertex $4$ in~$\dyn{E}_6$):
\[
\begin{tabular}{cccc}
$\G_w$ &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (90:1) {};
\node[Dnode] (3) at (0:1) {};
\node[Dnode] (4) at (180:1) {};

\node[Dnode] (5) at (-2,0) {};
\node[Dnode] (6) at (2,0) {};

\draw[dashed] (4) to (5) (3) to (6) (1) to (2);
\draw[->] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (90:1) {};
\node[Dnode] (3) at (0:1) {};
\node[Dnode] (4) at (180:1) {};

\node[Dnode] (5) at (-2,0) {};
\node[Dnode] (6) at (2,0) {};

\draw[dashed] (4) to (5) (3) to (6) (1) to (2);

\draw[<-] (1) to (3) ;
\draw[<-] (1) to (4) ;
\end{tikzpicture} &
\begin{tikzpicture}[scale=0.7, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode] (1) at (0,0) {};
\node[Dnode] (2) at (90:1) {};
\node[Dnode] (3) at (0:1) {};
\node[Dnode] (4) at (180:1) {};

\node[Dnode] (5) at (-2,0) {};
\node[Dnode] (6) at (2,0) {};

\draw[dashed] (4) to (5) (3) to (6) ;

\draw[dashed] (4) to (-2,0) (1) to (2);
\draw[->] (1) to (3) ;
\draw[->] (1) to (4) ;
\end{tikzpicture} \\[1ex]
\renewcommand{\arraystretch}{1}
\begin{tabular}{c}
$\#$ of orientations \\
(up to isomorphism)
\end{tabular} & $4$ & $3$ & $3$
\end{tabular}
\]
If the middle vertex is a sink, then we have only one choice for the orientation of the additional edge to be weak Fano but otherwise, both orientations are fine. Indeed, if the vertex $4$ already has indegree $2$ in the $\dyn{A}_5$ orientation, then the extra edge must point outward to keep indegree $\leq 2$; otherwise either direction works. Accordingly, we have $4 \times 2 + 3 + 3 \times 2 = 17$ orientations satisfying the weak Fano condition on type $\dyn{E}_6$ (up to isomorphism).

Now we consider $\dyn{E}_r$ for $r = 7,8$. Considering three edges emanating from the vertex $4$, there are $2^3 -1 = 7$ orientations such that the vertex $4$ has at most $2$ indegree. Since there are no other conditions on the remaining $r-4$ edges, we have $7\times 2^{r-4}$ orientations satisfying the weak Fano condition on type $\dyn{E}_r$ for $r = 7,8$.
\end{enonce*}

For the other Lie types, we use the following observation. Suppose that the Dynkin diagram has a double or triple edge of the following form:
\[
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode, label=below:{$i$}] (2) {};
\node[Dnode, label=below:{$j$}] (3) [right=of 2] {};

\draw[double line] (2)--(3);
\end{tikzpicture}
\qquad
\begin{tikzpicture}[scale =.5, baseline=-.5ex]
\tikzset{every node/.style={scale=0.7}}

\node[Dnode, label=below:{$i$}] (2) {};
\node[Dnode, label=below:{$j$}] (3) [right=of 2] {};

\draw[triple line] (2)--(3);
\draw (2)--(3);
\end{tikzpicture}
\]
From the definition of the edge labeling on the graph $\G_{w}$ and the definition of the Cartan integers, $s_j$ appears ahead of $s_i$ in the reduced word of $w$ if and only if the corresponding edge in the graph~$\G_w$ has the label $-c_{i,j}$.

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{B}$}}
We have the label $2$ on the edge $(r-1,r)$ on the graph $\G_w$ if $s_{r}$ appears before $s_{r-1}$ in the reduced expression of $w$. In this case, we have $\indeg(r) = 2$:
\[
\begin{tikzpicture}
\node (1) at (1,0) {$r-1$};
\node[right = of 1] (2) {$r$};
\draw[->] (1) to node[above, midway] {$2$} (2);
\end{tikzpicture}
\]
Otherwise, we have label $1$ on the edge $(r,r-1)$, so $\indeg(r)=0$ and $\indeg(r-1)\geq 1$. When $r=2$, there are two toric Schubert varieties which are weak Fano and one of which is Fano. When $r=3$, there are four toric Schubert varieties which are weak Fano and one of which is Fano. Now we assume $r\geq 4$.

We first enumerate the isomorphism classes of toric Schubert varieties.
\begin{enumerate}[leftmargin=*]
\item If we have the edge $(r-1,r)$, then there are $2^{r-2}$ different digraphs up to isomorphism.
\item If we have the edge $(r,r-1)$, then the vertex $r$ should be a source. We consider the following two cases separately.
\begin{enumerate}
\item If the first vertex (with label 1 in the Dynkin diagram) is a sink, then $2^{r-3}$ digraphs are all different. Indeed, if the vertex $1$ is required to be a sink, then the edge $(2,1)$ is forced. Hence, there is no nontrivial diagram symmetry identifying distinct orientations.
\item If the first vertex (with label 1 in the Dynkin diagram) is a source, then the number of isomorphism classes of the directed graphs is the same as that of the directed graphs of Lie type $\dyn{A}_{r-2}$.
\end{enumerate}
\end{enumerate}
By adding all possible cases above, we obtain that if $r$ is even and $r \geq 4$, then the number of isomorphism classes is $7 \times 2^{r-4}$; if $r$ is odd and $r \geq 5$, then the number of isomorphism classes is $7\times 2^{r-4} + 2^{\frac{r-3}{2} - 1}$.

Moreover, we note that all orientations provide weak Fano toric Schubert varieties. To enumerate the isomorphism classes of Fano toric Schubert varieties, we consider the directed Dynkin diagrams which do not have label $2$, that is, we have the edge $(r,r-1)$ in $\G_w$. Since $r$ is a source and (under the indegree $\leq 1$ condition) any sink must be a leaf, the only possible sink is vertex~$1$, which forces the unique orientation $1 \leftarrow 2 \leftarrow \dots \leftarrow r-1 \leftarrow r$. Indeed, the Coxeter element $s_1s_2 \dotsm s_r$ provides a Fano toric Schubert variety.
\end{enonce*}

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{C}$}}
We have the label $2$ on the edge $(r,r-1)$ if $s_{r-1}$ appears before $s_r$ in the reduced expression of $w$. Otherwise, we have the edge $(r-1,r)$ with label $1$. Therefore, we obtain the same number of isomorphism classes in a similar way to the case of type $\dyn{B}_r$. When $r=3$, there are four toric Schubert varieties, three of which are weak Fano and two of which are Fano. Now we assume $r\geq 4$.

If the graph $\G_w$ has the edge $(r,r-1)$ with label $2$ and the edge $(r-2,r-1)$, then $r-1$ is a sink with $\indeg(r-1) = 1+2= 3$:
\[
\begin{tikzpicture}
\node (1) at (1,0) {$r-2$};
\node[right = of 1] (2) {$r-1$};
\node[right = of 2] (3) {$r$};
\draw[<-] (2) to (1);
\draw[<-] (2) to node[above, midway] {$2$} (3) ;
\end{tikzpicture}
\]
Therefore, any completion of the remaining edges necessarily violates the weak Fano condition $\indeg(k) \leq 2$ for any vertex $k \in [r]$. The number of such orientations is $2^{r-3}$. Since all the other orientations provide weak Fano, we obtain the number of isomorphism classes of weak Fano toric Schubert varieties as in Table~\ref{table_cardinality_iso_classes_Coxeter}.

Recall that $X_w$ is Fano if and only if $\indeg(k) \leq 1$ for any vertex $k \in [r]$. Therefore, for $X_w$ to be Fano, there is no edge with label $2$ in the graph $\G_w$ and no sink on the internal vertices $\{2,\dots,r-1\}$. Accordingly, the vertex $r$ should be a sink and there exists only one source. Then the problem reduces to the same path-orientation counting as in the type $\dyn{A}$ case. Enumerating such orientations, we obtain $\frac{r+1}{2}$ orientations when $r$ is odd; $\frac{r}{2}$ orientations when $r$ is even.
\end{enonce*}

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{F}_4$}}
There are eight directed Dynkin diagrams and we display the value $\max\braces[\big]{\indeg(k)}$ in Table~\ref{table_F4}. Accordingly, there are six isomorphism classes of weak Fano toric Schubert varieties, and two isomorphism classes of Fano toric Schubert varieties.
\end{enonce*}

\begin{table}[!h]
\caption{Coxeter elements and corresponding directed Dynkin diagrams in type $\dyn{F}_4$.}\label{table_F4}
\resizebox{\linewidth}{!}{
\begin{tabular}{@{}cccc@{}}
\midrule $w$ & $s_1s_2s_3s_4$ & $s_2s_1s_3s_4$ & $s_1s_2s_4s_3$ \\
$\G_w$ &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (1) to node[above, midway] {\vphantom{$2$}} (2);
\draw[<-] (2) to (3);
\draw[<-] (3) to (4);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (2) to (1);
\draw[<-] (2) to (3);
\draw[<-] (3) to (4);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (1) to (2);
\draw[<-] (2) to (3);
\draw[<-] (4) to (3);
\end{tikzpicture} \\[1ex]
maximum of $\indeg(k)$ & 1 & 2 & 1 \\
\midrule $w$ & $s_2s_1s_4s_3$ & $s_1s_3s_2s_4$ & $s_3s_2s_1s_4$ \\
$\G_w$ &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (2) to (1);
\draw[<-] (2) to (3);
\draw[<-] (4) to (3);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (1) to (2);
\draw[<-] (3) to node[above, midway] {$2$} (2);
\draw[<-] (3) to (4);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (2) to (1);
\draw[<-] (3) to node[above, midway] {$2$} (2);
\draw[<-] (3) to (4);
\end{tikzpicture} \\[1ex]
maximum of $\indeg(k)$ & 2 & 3 & 3 \\
\midrule $w$& $s_1 s_4 s_3 s_2$ & $s_4s_3s_2s_1$ \\
$\G_w$ &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (1) to (2);
\draw[<-] (3) to node[above, midway] {$2$} (2);
\draw[<-] (4) to (3);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,4}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (2) to (1);
\draw[<-] (3) to node[above, midway] {$2$} (2);
\draw[<-] (4) to (3);
\end{tikzpicture} \\[1ex]
maximum of $\indeg(k)$ & 2 & 2 & \\
\midrule
\end{tabular}}
\end{table}

\begin{enonce*}[remark]{Type~\boldmath{$\dyn{G}_2$}}
There are two directed Dynkin diagrams corresponding to two Coxeter elements:
\[
\begin{tabular}{cc}
$s_1s_2$ & $s_2s_1$
\\
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,2}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (1) to node[above, midway] {$3$} (2);
\end{tikzpicture} &
\begin{tikzpicture}[baseline = -0.5ex]
\foreach \x in {1,...,2}{
\node(\x) at (\x,0) {$\x$};}
\draw[<-] (2) to (1);
\end{tikzpicture}
\end{tabular}
\]
One provides a Fano toric Schubert variety and the other is not weak Fano. This completes the proof. \qedhere
\end{enonce*}
\let\qed\relax
\end{proof}

\section{The cohomology ring distinguishes toric Schubert varieties for simply-laced types}\label{section_cohomology_ring_ADE}

Throughout this section, we assume that $X_w$ is a toric Schubert variety. We consider those $X_w$ for which all edges in $\G_w$ are labeled by $1$. This family includes every toric Schubert variety in $G/B$ for a simple algebraic group $G$ of simply-laced type. We show that for each toric Schubert variety~$X_w$ in the family above, the edge-labeled digraph $\G_w$ is recovered (up to isomorphism) from the cohomology ring~$H^\ast(X_w;\Z)$.

For the sake of simplicity, we define the following two sets.
\begin{align*}
	\eset{w}{j}&=\braces[\big]{k \st[\big] (k,j)\in E(\mathcal{G}_w)},
\\	\peset{w}{j}&=\braces[\big]{k \st[\big] (j,k)\in E(\mathcal{G}_w)}.
\end{align*}
That is, $\eset{w}{j}$ corresponds to the set of inward edges to $j$, and $\peset{w}{j}$ corresponds to the set of outward edges from ${j}$ in the digraph $\G_w$.

\begin{lemma}\label{lem:cohomology}
Let $X_w$ be a toric Schubert variety in $G/B$. Assume that all edges in $\mathcal{G}_w$ are labeled by $1$. Then the cohomology ring $H^*(X_w)$ can be described in terms of the graph $\mathcal{G}_w$. If we further assume $w\in \Cox_W$, then
\[
{H^*(X_w)=\Z[x_{1},\dots,x_{r}]\left/\parens*{x_{j}^2-x_{j}\sum_{k\in \peset{w}{j}} x_{k}\ \middle\vert \ j=1,\dots,{r}}\right.},
\]
where $r$ is the rank of\/ $G$.
\end{lemma}

\begin{proof}
Let $w=s_{i_1} \dotsm s_{i_m}$. Applying Jurkiewicz's theorem to Theorem~\ref{thm_char_matrix_of_toric_Schubert} (see~\cite{Jurkiewicz}), we have
\[
{H^*(X_w)=\Z[y_1,\dots,y_{m}]\left/\parens*{y_j^2 + y_j \parens[\Bigg]{\sum_{k<j}c_{i_j,i_k}y_k } \ \middle\vert \ j=1,\dots,m}\right.},
\]
where $c_{i_j, i_k}$ is a Cartan integer. Since we assume that all the edges in $\G_w$ are labeled by $1$, the Cartan integer $c_{i_j,i_k}$ is either $0$ or $-1$. Moreover, $c_{i_j,i_k} = -1$ if and only if the vertices $i_j$ and $i_k$ are adjacent in the digraph $\G_w$. Hence, by changing $y_j \mapsto x_{i_j}$ for each $j=1,\dots,m$, we get
\begin{equation}\label{eq:graph_cohomology}
{H^*(X_w)=\Z[x_{i_1},\dots,x_{i_{m}}]\left/\parens*{x_{i_j}^2-x_{i_j}\sum_{(i_j,i_k) \in E(\mathcal{G}_w)}x_{i_k} \ \middle\vert \ j=1,\dots,m}\right.}.
\end{equation}
If $w\in\Cox_W$, then $m$ is the rank of $G$, and this proves the lemma.
\end{proof}

\begin{remark}\label{rmk:cohomology_decomp}
From the presentation of $H^\ast(X_w)$ in~\eqref{eq:graph_cohomology}, we obtain the following decomposition property. Since the defining ideal
\[
\parens*{x_{i_j}^2-x_{i_j}\sum_{(i_j,i_k)\in E(\mathcal{G}_w)}x_{i_k} \ \middle\vert \ j=1,\dots,m}
\]
depends only on the edge set $E(\mathcal{G}_w)$, if $\mathcal{G}_w$ is disconnected and admits a decomposition
\[
\mathcal{G}_w=\mathcal{G}_{w^{(1)}}\sqcup \dots \sqcup \mathcal{G}_{w^{(p)}},
\]
into connected components, then the defining ideal decomposes as the sum of the ideals corresponding to these components. It follows that
\[
H^\ast(X_w)\cong H^\ast(X_{w^{(1)}})\otimes \dots \otimes H^\ast(X_{w^{(p)}}).
\]
\end{remark}

We demonstrate Lemma~\ref{lem:cohomology} in the following example.

\begin{example}\label{example_cohomology_disconnected}
Suppose that $w= s_3s_1s_4s_5s_2 s_7s_8\in W_{\dyn{A}_8}$. Then the digraph $\G_w$ is given as follows:
\[
\begin{tikzpicture}[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (1) at (1,0) {$1$};
\node[node] (2) at (2.4,0) {$2$};
\node[node] (3) at (3.8,0) {$3$};
\node[node] (4) at (5.2,0) {$4$};
\node[node] (5) at (6.6,0) {$5$};
\node[node] (7) at (9.4,0) {$7$};
\node[node] (8) at (10.8,0) {$8$};
\draw[<-] (1) to (2);
\draw[<-] (3) to (2);
\draw[<-] (3) to (4);
\draw[<-] (4) to (5);
\draw[<-] (7) to (8);
\end{tikzpicture}
\]
By Lemma~\ref{lem:cohomology}, we obtain that the cohomology ring $H^*(X_w)$ is the truncated polynomial ring $\Z[x_1,\dots,x_5,x_7,x_8]/\mathcal{I}$, where the ideal $\mathcal I$ is the sum $\mathcal{I}'+\mathcal{I}''$, with $\mathcal{I}'$ generated by
\[
x_1^2,
\qquad x_2^2 - x_2(x_3+x_1),
\qquad x_3^2,
\qquad x_4^2 - x_4x_3,
\qquad x_5^2 - x_5x_4,
\]
and $\mathcal{I}''$ generated by
\[
x_7^2,
\qquad x_8^2-x_8x_7.
\]
Therefore,
\[
\begin{aligned}
H^\ast(X_w)&\cong \Z[x_1,\dots,x_5]/\mathcal{I}' \otimes \Z[x_7,x_8]/\mathcal{I}''\\
&\cong H^\ast(X_{w^{(1)}})\otimes H^\ast(X_{w^{(2)}}),
\end{aligned}
\]
where $w^{(1)}=s_3s_1s_4s_5s_2$, $w^{(2)}=s_7s_8$, and $w=w^{(1)} w^{(2)}$.
\end{example}

We notice that in Example~\ref{example_cohomology_disconnected}, $x_i^2 = 0$ in the cohomology ring $H^\ast(X_w)$ if and only if $i = 1$, $3$, or~$7$. On the other hand, the sinks of the digraph ${\G}_w$ are $1$, $3$, and $7$. We will see that this observation holds in general.

An element $z\in H^2(X_w)$ is \emph{primitive} if $z$ cannot be divisible by an integer greater than $1$ and is called \emph{square-zero} if $z^2=0$ in $H^\ast(X_w)$. From the ring presentation in Lemma~\ref{lem:cohomology}, we see that $x_i$ is a square-zero primitive element if the vertex $i$ is a sink of ${\G}_w$. There are more square-zero primitive elements in $H^2(X_w)$. For simplicity, we set
\begin{equation}\label{eq:alpha}
\alpha_j \coloneqq \sum_{k\in \peset{w}{j}}x_k
\quad \text{for $j=1,\dots,m$}.
\end{equation}
Hence if $j$ is a sink, then $\alpha_j=0$; otherwise it is the sum of at most three $x_k$'s.

\begin{lemma}\label{lemm:sve}
Let $X_w$ be a toric Schubert variety in $G/B$. Assume that all the edges in $\G_w$ are labeled by $1$. A square-zero primitive element in~$H^2(X_w)$ is of the form up to sign:
\[
x_j
\qquad \text{or} \qquad 
2x_k-x_j,
\]
where $j$ is a sink of ${\G}_w$. Moreover, the latter occurs only when $\peset{w}{k}=\{j\}$.
\end{lemma}

\begin{proof}
We first notice that $x_j^2=0$ in the cohomology ring $H^*(X_w)$ if and only if $j$ is a sink of the digraph~${\G}_w$. By~\cite[Corollary~2.1]{ch-ma12}, a square-zero primitive element is of the following form up to sign:
\[
x_k-\frac{1}{2}\alpha_k
\qquad \text{or} \qquad
2x_k-\alpha_k
\]
for some $k$ with $\alpha_k^2=0$, where the former case occurs when $\alpha_k$ is divisible by $2$ and the latter occurs otherwise. In our case, $\alpha_k$ is divisible by $2$ only when $\alpha_k=0$ (equivalently, $k$ is a sink). If $\alpha_k\neq 0$, then $\alpha_k^2=0$ only when $\alpha_{k}$ equals $x_j$ for a sink $j$ of $\G_w$, that is, $\peset{w}{k}=\{j\}$ for a sink $j$ of~$\G_w$. This completes the lemma.
\end{proof}

Since the mod~2 reduction of the elements in Lemma~\ref{lemm:sve} is $x_j$, we can find all sinks of ${\G}_w$ by looking at square-zero primitive elements in $H^2(X_w)$.

Following~\cite[Section~6]{ch-ma-ou17}, we call an element $\alpha\in H^2(X_w;\Z_2)$ (possibly $\alpha=0$) an \emph{eigenelement} if there exists $x\in H^2(X_w;\Z_2)$ such that
\[
x^2=\alpha x,
\quad x\neq0,
\quad \text{and}
\quad x\neq\alpha.
\]
Moreover, we call such $x$ an \emph{eigenvector} associated to $\alpha$. We define $E(\alpha)$ as the set of elements $x$ with $x^2=\alpha x$ ($x$ may be $0$ or $\alpha$) for an eigenelement $\alpha$, which is called the \emph{eigenspace} associated to $\alpha$. In fact, $E(\alpha)$ forms a vector space because we are working over $\Z_2$. We denote by $\bar{E}(\alpha)$ the quotient of $E(\alpha)$ by the one-dimensional subspace $\langle \alpha\rangle$ spanned by $\alpha$ and call $\bar{E}(\alpha)$ the \emph{reduced eigenspace} associated with $\alpha$. Note that eigenelements, eigenspaces, and reduced eigenspaces are preserved under a cohomology ring isomorphism. For an eigenelement $\alpha \in H^2(X_w;\Z_2)$, we define its \emph{multiplicity} to be the dimension of $\bar{E}(\alpha)$.

\begin{lemma}[{\cite[Lemma~6.2]{ch-ma-ou17}}] \label{lem:eigenelements}
Let $X_w$ be a toric Schubert variety in $G/B$. Assume that all the edges in $\G_w$ are labeled by $1$. The eigenelements are $\alpha_j$'s (regarded as elements over $\Z_2$) and the reduced eigenspace $\bar{E}(\alpha)$ is spanned by $x_j$'s for each eigenelement $\alpha_j=\alpha$.
\end{lemma}

We see an example for nonzero eigenelements and their eigenspaces.

\begin{example}\label{example1}
Let $G$ be of type $\dyn{D}_9$ and $w=s_8s_5 s_4 s_2s_1s_3 s_6s_7s_9$. Then we have:
\[
\G_w =
\begin{tikzpicture}
[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (7) at (0,0) {$7$};
\node[node] (8) at (60:1.4) {$8$};
\node[node] (9) at (-60:1.4) {$9$};
\node[node] (6) at (180:1.4) {$6$};
\node[node] (5) at (-2.8,0) {$5$};
\node[node] (4) at (-4.2,0) {$4$};
\node[node] (3) at (-5.6,0) {$3$};
\node[node] (2) at (-7,0) {$2$};
\node[node] (1) at (-8.4,0) {$1$};
\draw[<-] (5) to (6);
\draw[<-] (6) to (7);
\draw[<-] (8) to (7);
\draw[<-] (7) to (9);
\draw[<-] (4) to (3);
\draw[<-] (2) to (3);
\draw[<-] (2) to (1);
\draw[<-] (5) to (4);
\end{tikzpicture}
\]
Then $\alpha_2=\alpha_5=\alpha_8=0$, $\alpha_1=x_2$, $\alpha_3=x_2+x_4$, $\alpha_4=x_5$, $\alpha_6=x_5$, $\alpha_7=x_6+x_8$, and $\alpha_9=x_7$. Hence, for nonzero eigenelements of $H^2(X_w;\mathbb{Z}_2)$, their eigenspaces are given as follows:
\[
\begin{tabular}{@{}ccc@{}}
\toprule
$\alpha$ ($\neq 0$) & ${E}(\alpha)$ & $\dim\bar{E}(\alpha)$ \\
\midrule
$x_2$ & $\langle x_1, x_2\rangle$ & $1$ \\
$x_2+x_4$ & $\langle x_3,x_2+x_4\rangle$ & $1$ \\
$x_5$ & $\langle x_4,x_5,x_6\rangle$ & $2$ \\
$x_6+x_8$ & $\langle x_7,x_6+x_8\rangle$ & $1$ \\
$x_7$ & $\langle x_7,x_9\rangle$ & $1$ \\
\bottomrule
\end{tabular}
\]
\end{example}

By Lemma~\ref{lem:eigenelements} and the definition of $\alpha_j$, any eigenelement $\alpha$ is the element~$\alpha_j=\sum_{k\in \peset{w}{j}} x_k$ for some $j$. Hence, $\alpha_j$ is one of the following four forms:
\[
0,
\qquad x_k,
\qquad x_i+x_k,
\qquad x_i+x_k+x_\ell,
\]
where $i,j,k,\ell$ are pairwise distinct. In the following lemma, we describe the eigenspace $E(\alpha)$ in each case.

\begin{lemma}\label{lemm:eigenspace}
Let $X_w$ be a toric Schubert variety in $G/B$. Assume that all the edges in $\G_w$ are labeled by $1$. Then for each eigenelement $\alpha$, the following hold:
\begin{enumerate}
\item \label{lemm:eigenspace_1} if $\alpha=0$, then $\bar{E}(\alpha)=\langle x_k \mid \textnormal{$k$ is a sink}\rangle$;
\item \label{lemm:eigenspace_2} if $\alpha=x_k$, then
\[
\bar{E}(\alpha) = \angles[\big]{x_i \bigmid \peset{w}{i}=\{k\}};
\]
\item \label{lemm:eigenspace_3} if $\alpha=x_i+x_k$ with $i\neq k$, then $\bar{E}(\alpha)=\langle x_j \rangle$, where $\peset{w}{j}=\{i,k\}$;
\item \label{lemm:eigenspace_4} if $\alpha=x_i+x_k+x_\ell$ with pairwise distinct $i,k,\ell$, then $\bar{E}(\alpha)=\langle x_j\rangle$, where $\peset{w}{j}=\{i,k,\ell\}$.
\end{enumerate}
Therefore, if $\alpha$ is nonzero, then we get $1\leq \dim \bar{E}(\alpha)\leq 3$.
\end{lemma}

\begin{proof}
Since $\alpha$ is of the form $\alpha_j=\sum_{k\in \peset{w}{j}} x_k$, we prove the lemma according to the cardinality of $\peset{w}{j}$.
\begin{enumerate}[leftmargin=3.9em,labelwidth=\widthof{Case~3:},align=left]
\renewcommand{\labelenumi}{Case~\theenumi:}
\item Suppose that $\card[\big]{\peset{w}{j}}=0$. Then $j$ is a sink, and $\alpha_j=0$. Hence, we get
\[
\bar{E}(\alpha_j)=E(\alpha_j)=\langle x_k \mid \textnormal{$k$ is a sink}\rangle.
\]
Hence, $\dim \bar{E}(0)$ is the number of sinks in $\G_w$.
\item Suppose that $\card[\big]{\peset{w}{j}}=1$. Then $\alpha_j=x_k$, where $j\neq k$. If $\peset{w}{i}=\{k\}$, then $\alpha_i=x_k$; otherwise, $\alpha_i\neq x_k$. Therefore, we obtain
\[
\bar{E}(x_k) = \angles[\big]{x_i \bigmid \peset{w}{i}=\{k\}}.
\]
Therefore, $1\leq \dim\bar{E}(x_k)\leq \deg^{-}(k)\leq 3$.
\item Suppose that $\card[\big]{\peset{w}{j}}=2$. Then $\alpha_j=x_{i}+x_{k}$, where $i,j,k$ are pairwise distinct. If $\eset{w}{j}=\emptyset$, then $j$ is a source; otherwise, $j$ is a trivalent vertex of $\G_w$. In any case, {$E(\alpha_j) = \langle x_j, x_i+x_k\rangle$ and} $\bar{E}(\alpha_j)=\langle x_j\rangle$.
\item Suppose that $\card[\big]{\peset{w}{j}}=3$. Then $\alpha_j=x_i+x_k+x_\ell$, where $i,j,k,\ell$ are pairwise distinct. Then $j$ is a trivalent vertex, and it is a source. Furthermore, {$E(\alpha_j) = \langle x_j, x_i+x_k+x_{\ell} \rangle$ and} $\bar{E}(\alpha_j)=\langle x_j\rangle$. \qedhere
\end{enumerate}
\let\qed\relax
\end{proof}

Note that, as we saw in Lemma~\ref{lemm:eigenspace}\eqref{lemm:eigenspace_1} and~\eqref{lemm:eigenspace_2}, the same $\alpha_j$ may occur for different indices $j$; nevertheless, when counted with their indices, the total number of eigenelements is exactly the number of vertices of $\G_w$.

\begin{theorem}\label{theo:recover}
Let $X_w$ be a toric Schubert variety in $G/B$. The edge-labeled digraph $\mathcal{G}_w$ can be recovered (up to isomorphism) from the cohomology ring $H^*(X_w;\Z)$ if all the labels in\/ $\G_w$ are $1$.
\end{theorem}

Recall that for a toric Schubert variety $X_w$ in $G/B$ for a simple Lie group of type $\dyn{A}$, $\dyn{D}$, or $\dyn{E}$, every edge in $\G_w$ is labeled by~$1$. Hence we get the following result.

\begin{corollary}\label{thm:dynkin2}
Let $W$ and $W'$ be the Weyl groups of simple Lie groups $G$ and $G'$ of type $\dyn{A}$, $\dyn{D}$, or $\dyn{E}$. Let $w$ and $w'$ be elements in $W$ and $W'$, respectively. The following statements are equivalent:
\begin{enumerate}
\item $H^{\ast}(X_w;\Z) \cong H^{\ast}(X_{w'};\Z)$ as graded rings;
\item $\G_{w} \cong \G_{w'}$ as digraphs.
\end{enumerate}
\end{corollary}

\begin{remark}
Richmond and Slofstra~\cite{richmond2021isomorphism} studied a relation between isomorphism classes of (not necessarily toric) Schubert varieties and their cohomology rings. Indeed, they proved that two Schubert varieties are isomorphic (as algebraic varieties) if and only if there is a graded cohomology ring isomorphism preserving the Schubert bases. Here, for a toric Schubert variety of dimension~$m$, the cohomology classes $\braces[\big]{x_{i_1} \dotsm x_{i_k} \st[\big] 1 \leq i_1 < \dots < i_k \leq m}$ form the Schubert basis in terms of the cohomology ring presentation in Lemma~\ref{lem:cohomology}.
\end{remark}

\begin{remark}\label{rmk_F4}
As mentioned in Remark~\ref{rema:1}, the assumption in Theorem~\ref{theo:recover} cannot be weakened. In this remark, we consider the cohomology rings of two toric Schubert varieties $X_{s_3s_2s_1s_4}$ and $X_{s_2s_1s_4s_3}$ of type $\dyn{F}_4$. The cohomology rings are given as follows:
\begin{align*}
	H^{\ast}(X_{s_3s_2s_1s_4}) & \cong \Z[x_1,x_2,x_3,x_4] \big/ \parens[\big]{x_1^2 - x_1x_2, \: x_2^2 - 2 x_2x_3, \: x_3^2, \: x_4^2 - x_3x_4},
\\	H^{\ast}(X_{s_2x_1s_4s_3}) & \cong \Z[y_1,y_2,y_3,y_4] \big/ \parens[\big]{y_1^2 - y_1 y_2, \: y_2^2, \: y_3^2 - y_2 y_3 - y_3 y_4, \: y_4^2}.
\end{align*}
The graded ring homomorphism given by $\varphi(x_1) = y_3 - y_2 - y_4$, $\varphi(x_2) = - y_2 - y_4$, $\varphi(x_3) = - y_2$, $\varphi(x_4) = - y_2 + y_1$ provides the graded ring isomorphism between two cohomology rings because
\begin{alignat*}{2}
	\varphi(x_1^2 - x_1x_2) & = y_3^2 - y_2 y_3 - y_3 y_4,
	& \qquad
	\varphi(x_2^2 - 2 x_2x_3) & = - y_2^2 + y_4^2,
\\	\varphi(x_3^2) & = y_2^2,
	& \qquad
	\varphi(x_4^2 - x_3x_4) & = y_1^2 - y_1 y_2.
\end{alignat*}
However, two toric varieties are not isomorphic as proved in Proposition~\ref{thm:dynkin1} (see Table~\ref{table_F4} for the corresponding directed Dynkin diagrams).
\end{remark}

Since a directed graph $\mathcal{G}_w$ has at most one trivalent vertex, we divide the proof of Theorem~\ref{theo:recover} into two cases according to whether $\mathcal{G}_w$ has a trivalent vertex. We first treat the case with no trivalent vertex and then the case with exactly one trivalent vertex. Because the essential idea already appears in type $\dyn{A}$, we begin with an example of type $\dyn{A}$ to illustrate the argument. We then prove the case without a trivalent vertex. After that, we give an example of type $\dyn{D}$ to indicate the additional issues caused by the presence of a trivalent vertex, and finally complete the proof in the general case.

We begin with an example of type $\dyn{A}$, in which the directed graph has no trivalent vertex. This simple case already exhibits the essential mechanism of the proof.

\begin{example}
Let $G$ be of type $\dyn{A}$ and $w=s_9s_{10}s_5s_4s_6s_7s_8s_1s_2s_3$. Then we have:
\[
\G_w =
\begin{tikzpicture}
[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (7) at (0,0) {$7$};
\node[node] (8) at (1.4,0) {$8$};
\node[node] (9) at (2.8,0) {$9$};
\node[style={circle,draw, fill=white!20, inner sep = 0.25mm}] (10) at (4.2,0) {$10$};
\node[node] (6) at (180:1.4) {$6$};
\node[node] (5) at (-2.8,0) {$5$};
\node[node] (4) at (-4.2,0) {$4$};
\node[node] (3) at (-5.6,0) {$3$};
\node[node] (2) at (-7,0) {$2$};
\node[node] (1) at (-8.4,0) {$1$};
\draw[<-] (1) to (2);
\draw[<-] (2) to (3);
\draw[->] (3) to (4);
\draw[->] (4) to (5);
\draw[->] (6) to (5);
\draw[->] (7) to (6);
\draw[->] (8) to (7);
\draw[->] (8) to (9);
\draw[->] (10) to (9);
\end{tikzpicture}
\]
Note that $\alpha_1=\alpha_5=\alpha_9=0$ and
\[
\alpha_2=x_1,
\qquad \alpha_3=x_2+x_4,
\qquad \alpha_4=x_5,
\qquad \alpha_6=x_5,
\qquad \alpha_7=x_6,
\qquad \alpha_8=x_7+x_9,
\qquad \alpha_{10}=x_9.
\]
Now we recover $\G_w$ from $H^\ast(X_w)$ by the following three steps.

\newstepset
\begin{step}
We define the $\Z_2$-vector spaces $V_1,\dots,V_{10}$ using eigenelements $\alpha_1,\dots,\alpha_{10}$:
\[
V_j=
\begin{cases}
	\langle x_j\rangle
	& \text{if $\alpha_j=0$},
\\	E(\alpha_j)
	& \text{otherwise}.
\end{cases}
\]
Then the spaces $V_j$'s are given as follows:
%\begin{alignat*}{2}
%	& V_1 = \langle x_1\rangle,
%	&& V_6 = \langle x_4,x_5,x_6\rangle,
%\\	& V_2 = \langle x_1,x_2\rangle,
%	&& V_7 = \langle x_6, x_7\rangle,
%\\	& V_3 = \langle x_3,x_2+x_4\rangle, \qquad
%	&& V_8 = \langle x_8,x_7+x_9\rangle,
%\\	& V_4 = \langle x_4,x_5,x_6\rangle,
%	&& V_9 = \langle x_9\rangle,
%\\	& V_5 = \langle x_5\rangle,
%	&& V_{10} = \langle x_9,x_{10}\rangle.
%\end{alignat*}
\begin{alignat*}{5}
	& V_1 = \langle x_1\rangle, \qquad
	&& V_2 = \langle x_1,x_2\rangle, \qquad
	&& V_3 = \langle x_3,x_2+x_4\rangle, \qquad
	&& V_4 = \langle x_4,x_5,x_6\rangle, \qquad
	&& V_5 = \langle x_5\rangle,
\\	& V_6 = \langle x_4,x_5,x_6\rangle, \qquad
	&& V_7 = \langle x_6, x_7\rangle, \qquad
	&& V_8 = \langle x_8,x_7+x_9\rangle, \qquad
	&& V_9 = \langle x_9\rangle, \qquad
	&& V_{10} = \langle x_9,x_{10}\rangle.
\end{alignat*}
\end{step}

We draw a graph whose vertices are labeled by $V_1,\dots,V_{10}$ in the following Steps~\ref{example_36_step_2} and~\ref{example_36_step_3}.

\begin{step}\label{example_36_step_2}
We find a connected component starting from $V_1$, $V_5$, and $V_9$, which are one-dimensional. First, for each $j=1,5,9$, we find a space $V$ with $\dim(V\cap V_j)=1$, and then draw a directed edge $V \rightarrow V_j$. Then we get:
\[
V_1 \leftarrow V_2,
\qquad V_4 \rightarrow V_5 \leftarrow V_6,
\qquad V_9 \leftarrow V_{10}.
\]
Then $V_2$, $V_4$, $V_6$, and $V_{10}$ are the sources in the current graph. Next, for each source $V_j$, we find a space $V$ not yet contained in the current graph satisfying $\dim(V\cap V_j)=1$. Then $V_4$ and $V_6$ are the same, and $V_4\cap V_7=V_6\cap V_7=\langle x_6\rangle$. We choose one of $V_4$ and $V_6$, and the draw a directed edge from~$V$ to the chosen source. Note that any choice produces isomorphic graphs, so the construction is well defined up to isomorphism. In the following we choose $V_4$. Then we get:
\[
V_1 \leftarrow V_2,
\qquad V_7 \rightarrow V_4 \rightarrow V_5 \leftarrow V_6,
\qquad V_9 \leftarrow V_{10}.
\]
Let $\mathcal R$ be the set of spaces $V_j$ that are not contained in any connected component in the current graph, and let $\mathcal S$ be the set of sources in the current graph. Then $\mathcal R=\{V_3,V_8\}$ and $\mathcal S=\{V_2,V_6,V_7,V_{10}\}$. In this case, there is no more pair of $V\in \mathcal R$ and $V'\in\mathcal S$ such that $\dim(V\cap V')=1$.
\end{step}

Now we use a new rule to connect the components constructed in Step~\ref{example_36_step_2} by means of the remaining spaces.

\begin{step}\label{example_36_step_3}
Let $\mathcal L$ be the set of leaves of the connected components in the graph constructed in Step~\ref{example_36_step_2}, viewed as an undirected graph. We have $\mathcal{L}=\{V_1,V_2,V_6,V_7,V_9,V_{10}\}$. Then for each $V_j=E(\alpha_j)\in \mathcal{R}$, we can choose $V_p$ and $V_q$ in $\mathcal L$ minimal with respect to inclusion such that $\alpha_j\in V_p+V_q$. Indeed, since $\alpha_3=x_2+x_4\in V_2+V_6$ and $\alpha_8=x_7+x_9\in V_7+V_9\subset V_7+V_{10},$ we choose $V_2$ and $V_6$ for $V_3$, and we choose $V_7$ and $V_9$ for $V_8$. Then we draw the directed edges $V_j \rightarrow V_p$ and $V_j \rightarrow V_q$. The resulting graph is:
\[
V_1 \leftarrow V_2\leftarrow V_3 \rightarrow V_6\rightarrow V_5 \leftarrow V_4 \leftarrow V_7 \leftarrow V_8 \rightarrow V_9 \leftarrow V_{10},
\]
%\begin{center}
%\begin{tikzpicture}[baseline=(current bounding box.east),
%scale=.4, terminal/.style={ellipse, minimum width=.8cm, minimum height=.5cm, font=\itshape,},]
%\matrix[column sep=.4cm] {
%	\node [terminal](p1) {$V_1$};
%&&	\node [terminal](p2) {$V_2$};
%&&	\node [terminal](p3) {$V_3$};
%&&	\node [terminal](p4) {$V_4$};
%&&	\node [terminal](p5) {$V_5$};
%&&	\node [terminal](p6) {$V_6$};
%&&	\node [terminal](p7) {$V_7$};
%&\\
%};
%\draw (p2) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p1) edge [<-, shorten <=2pt, shorten >=2pt] (p3) (p3) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p4) edge [<-, shorten <=2pt, shorten >=2pt] (p5) (p5) edge [<-, shorten <=2pt, shorten >=2pt] (p6) (p7) edge [<-, shorten <=2pt, shorten >=2pt] (p6);
%\end{tikzpicture}
%\end{center}
which is isomorphic to the graph $\G_w$ as a directed graph.
\end{step}
\end{example}

Now we prove Theorem~\ref{theo:recover} when there is no trivalent vertex.

\begin{proof}[Proof of Theorem~\ref{theo:recover} in the absence of a trivalent vertex]
Assume $\mathcal{G}_w$ has no trivalent vertex. We recover the graph $\mathcal{G}_w$ from the cohomology ring $H^\ast(X_w)$ in three steps. In Step~\ref{theo:recover_step_1}, we determine the vertex set. In Step~\ref{theo:recover_step_2}, we first recover the sinks, and then iteratively recover all vertices except for the sources that are not leaves. In Step~\ref{theo:recover_step_3}, we recover the source vertices that are not leaves, which completes the reconstruction of $\mathcal{G}_w$.

For simplicity, we first prove the result for $w\in \Cox_W$, and then treat the general case.

\newstepset
\begin{step}\label{theo:recover_step_1}
For each~$j=1,\dots,r$, we define a $\Z_2$-vector space $V_j$ by
\[
V_j=
\begin{cases}
	\langle x_j\rangle
	& \text{if $\alpha_j=0$},
\\	E(\alpha_j)
	& \text{otherwise}.
\end{cases}
\]
\end{step}

In Steps~\ref{theo:recover_step_2} and~\ref{theo:recover_step_3}, we construct a directed graph with vertices $V_1,\dots, V_r$ using the following statements, which are obtained from Lemma~\ref{lemm:eigenspace}.
\begin{enumerate}
\item $V_i \subset V_j$, which occurs only when $i$ is a sink and $\peset{w}{j}=\{i\}$;
\item $\dim(V_i\cap V_j)=1$, which occurs only when $\peset{w}{j}=\{i\}$;
\item $\alpha_i \in V_a+V_b$, which occurs when $\peset{w}{i}=\{a,b\}$.
\end{enumerate}

\begin{step}\label{theo:recover_step_2}
For each index $j$ with $\alpha_j=0$, we construct a connected component as follows. Fix such a $j$ and initialize the directed graph with the vertex $V_j$. First, we find a space satisfying the following:
\begin{enumerate}
\renewcommand*{\theenumi}{J\textsubscript{\arabic{enumi}}}\relabel
\item \label{J1} a space $V$ satisfies $\dim(V\cap V_j)=1$.
\end{enumerate}

For such a space $V$, we draw a directed edge $V \rightarrow V_j$. Let $\mathcal S$ be the set of sources of the directed graph constructed so far.

Next, we incorporate the remaining spaces one by one. Given a space $V$ not yet contained in the current graph, if there exists a source $V_a\in\mathcal S$ such that $\dim(V\cap V_a)=1$, we choose one such~$V_a$ and draw a directed edge $V \rightarrow V_a$. We then update the set $\mathcal S$ of sources and continue. We repeat this process until no unused space $V$ has a one-dimensional intersection with any source in~$\mathcal S$.

If the choice of $V_a$ is not unique (which may occur when $V_a=V_{a'}$ for distinct indices), any choice produces isomorphic graphs, so the construction is well defined up to isomorphism.

At this stage, the resulting connected component is a path, and there are three possible forms~by
\begin{gather*}
\langle x_j\rangle
\leftarrow \underbrace{E(\alpha_{j+1})}_{= \langle x_j, \: x_{j+1}\rangle}
\leftarrow \cdots
\leftarrow \underbrace{E(\alpha_{j+p}),}_{= \langle x_{j+p-1}, \: x_{j+p}\rangle}
\\
\underbrace{E(\alpha_{j-q})}_{=\langle x_{j-q}, \: x_{j-q+1}\rangle}
\rightarrow \cdots
\rightarrow \underbrace{E(\alpha_{j-1})}_{=\langle x_{j-1}, \: x_{j}\rangle}
\rightarrow \langle x_j\rangle,
\\
E(\alpha_{j-q}) \rightarrow E(\alpha_{j-q+1})\rightarrow \dots\rightarrow E(\alpha_{j-1})\rightarrow \langle x_j\rangle\leftarrow E(\alpha_{j+1})\leftarrow \dots\leftarrow E(\alpha_{j+p-1})\leftarrow E(\alpha_{j+p}),
\end{gather*}
where
\begin{align*}
	& E(\alpha_{j-1})=E(\alpha_{j+1})={E}(x_j)=\langle x_{j-1},x_j,x_{j+1}\rangle,
\\	& {E}(\alpha_{j+t-1})=\langle x_{j+t-1},x_{j+t}\rangle
	\quad \text{for $2\le t\le p$},
\\	& {E}(\alpha_{j-s+1})=\langle x_{j-s},x_{j-s+1} \rangle
	\quad\text{for $2\le s\le q$}.
\end{align*}
(Note that the multiplicity of the eigenelement $x_j$ is two and the last two identities above make sense when $p\ge 2$, $q\ge 2$.)

If the graph obtained in this step is connected, then it is the desired graph and the construction is complete. Otherwise, some spaces have not yet been incorporated, and we proceed to the next step, where we use a new rule to connect the components constructed in Step~\ref{theo:recover_step_2} by means of the remaining spaces.
\end{step}

\begin{step}\label{theo:recover_step_3}
Let $\mathcal R$ be the set of subspaces $V_j$ that are not contained in any connected component constructed in Step~\ref{theo:recover_step_2}. Let $\mathcal L$ be the set of leaves of those connected components, viewed as an undirected graph. Then every space $V_j\in \mathcal{R}$ satisfies the following statement:
\begin{enumerate}
\renewcommand*{\theenumi}{J\textsubscript{\arabic{enumi}}}\relabel
\stepcounter{enumi}
\item \label{J2} for $V_j=E(\alpha_j)\in \mathcal{R}$, there exist $V_p,V_q\in \mathcal{L}$ such that $\alpha_j\in V_p+ V_q$.
\end{enumerate}
We choose $V_p,V_q\in \mathcal{L}$ minimal with respect to inclusion, and then we draw directed edges $V_j \rightarrow V_p$ and $V_j \rightarrow V_q$. In some cases, the choice of $V_p$ and $V_q$ is not uniquely determined. This occurs when $V_p=V_{p'}$ for distinct indices $p$ and $p'$. In this situation, we have $\peset{w}{p}=\peset{w}{p'}=\{s\}$ for some $s$, and hence $p$ and $p'$ are symmetric neighbors of $s$. Consequently, either choice leads to the same resulting graph, and the construction is completed.
\end{step}

It remains to explain why it suffices to treat the case where $w$ is a Coxeter element. For a general element $w=s_{i_1} \dotsm s_{i_m}$, the associated graph $\mathcal{G}_w$ may be disconnected, and the indices are given by $i_1,\dots,i_m$ rather than $1,\dots,m$. However, by Remark~\ref{rmk:cohomology_decomp}, the cohomology ring $H^\ast(X_w)$ decomposes as the tensor product of the cohomology rings associated with the connected components of $\mathcal{G}_w$. Therefore, the reconstruction problem reduces to the connected case, namely the case where $w$ is a Coxeter element. The same argument as above then applies on each connected component, with $j$ replaced by $i_j$.
\end{proof}

We now discuss the modifications needed when $\mathcal{G}_w$ has a trivalent vertex. If the outdegree of the trivalent vertex is at most one, then the reconstruction proceeds by repeating Step~\ref{theo:recover_step_2} exactly as in the case with no trivalent vertex. If the outdegree is two, then after Step~~\ref{theo:recover_step_3} has been carried out, one can return to Step~~\ref{theo:recover_step_2} and continue the reconstruction. If the trivalent vertex is a source, then the argument is analogous to Step~~\ref{theo:recover_step_3}, although a new rule is needed to handle the trivalent vertex. The following example of type $\dyn{E}$ illustrates the case where Step~~\ref{theo:recover_step_2} must be applied again after Step~~\ref{theo:recover_step_3}.

\begin{example}
Let $G$ be of type $\dyn{E}_7$ and $w=s_7s_2s_1s_3s_4s_5s_6$. Then we have:
\[
\G_w =
\begin{tikzpicture}
[node/.style={circle,draw, fill=white!20, inner sep = 0.7mm}, baseline = -0.5ex]
\node[node] (7) at (0,0) {$7$};
\node[node] (6) at (180:1.4) {$6$};
\node[node] (5) at (-2.8,0) {$5$};
\node[node] (4) at (-4.2,0) {$4$};
\node[node] (3) at (-5.6,0) {$3$};
\node[node] (2) at (-4.2,1) {$2$};
\node[node] (1) at (-7,0) {$1$};
\draw[<-] (5) to (6);
\draw[->] (6) to (7);
\draw[<-] (4) to (5);
\draw[<-] (3) to (4);
\draw[<-] (2) to (4);
\draw[<-] (1) to (3);
\end{tikzpicture}
\]
Then $\alpha_1=\alpha_2=\alpha_7=0$ and $\alpha_3=x_1$, $\alpha_4=x_2+x_3$, $\alpha_5=x_4$, $\alpha_6=x_5+x_7$.

\newstepset
\begin{step}
The spaces $V_j$'s are given as follows:
\begin{alignat*}{4}
	& V_1 = \langle x_1\rangle, \qquad
	&& V_2 = \langle x_2\rangle, \qquad
	&& V_3 = \langle x_1,x_3\rangle, \qquad
	&& V_4 = \langle x_2+x_3,x_4\rangle,
\\	& V_5 = \langle x_4,x_5\rangle, \qquad
	&& V_6 = \langle x_5+x_7,x_6\rangle, \qquad
	&& V_7 = \langle x_7\rangle.
\end{alignat*}
\end{step}

\begin{step}\label{example_step_2}
We obtain the following connected components:
\[
V_1 \leftarrow V_3,
%\begin{tikzpicture}[baseline=-.5ex,
%scale=.4, terminal/.style={ellipse, minimum width=.8cm, minimum height=.5cm, font=\itshape}]
%\matrix[column sep=.4cm] {\node [terminal](p1) {$V_1$};&&\node [terminal](p3) {$V_3$,};\\};
%\draw (p1) edge [<-, shorten <=2pt, shorten >=2pt] (p3);
%\end{tikzpicture}
\qquad V_2,
\qquad V_7.
\]
\end{step}

\begin{step}\label{example_step_3}
We have $\mathcal{R}=\{V_4,V_5,V_6\}$ and $\mathcal{L}=\{V_1,V_2,V_3,V_7\}$. Since $\alpha_4\in V_2+ V_3$, we combine the connected components obtained from Step~\ref{example_step_2}, we get the following graph:
%\begin{center}
%\begin{tikzpicture}[baseline=(current bounding box.east),
%scale=.4, terminal/.style={ellipse, minimum width=.8cm, minimum height=.5cm, font=\itshape,},]
%\matrix[row sep=.7cm,column sep=.4cm] {&&& & \node[terminal](p2) {$V_2$}; & & & & & &&\\
%\node [terminal](p1) {$V_1$};&&\node [terminal](p3) {$V_3$};& & \node [terminal](p4) {$V_4$}; & &\node [terminal](p5) {$~$}; &&\node [terminal](p6) {$~$};& & \node [terminal](p7) {$V_7$};&\\
%};
%\draw (p2) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p1) edge [<-, shorten <=2pt, shorten >=2pt] (p3) (p3) edge [<-, shorten <=2pt, shorten >=2pt] (p4);
%\end{tikzpicture}
%\end{center}
\[
\begin{tabular}{@{}*{13}{W{c}{1.5em}@{}}}
	& & & & & $V_{2}$
\\	& & & & & $\uparrow$
\\	& $V_{1}$ & $\leftarrow$ & $V_{3}$ & $\leftarrow$ & $V_{4}$ & \phantom{$\leftarrow$} & \phantom{$V_{5}$} & \phantom{$\leftarrow$} & \phantom{$V_{6}$} & \phantom{$\leftarrow$} & $V_{7}$
\end{tabular}
\]
Unlike in the case with no trivalent vertex, for the spaces $V_5$ and $V_9$ in $\mathcal{R}$, there is no pair $V_p, V_q\in \mathcal{L}$ satisfying~\eqref{J2}. Nevertheless, $V_4$ satisfies $\dim(V_4\cap V_5)=1$, so the criterion from Step~\ref{example_step_2} still applies.
\end{step}

\begin{step}
From the condition $V_4\cap V_5=\langle x_4\rangle$, we draw a directed edge from $V_5$ to $V_4$. Then, we get the following graph:
%\begin{center}
%\begin{tikzpicture}[baseline=(current bounding box.east),
%scale=.4, terminal/.style={ellipse, minimum width=.8cm, minimum height=.5cm, font=\itshape,},]
%\matrix[row sep=.7cm,column sep=.4cm] {&&& & \node[terminal](p2) {$V_2$}; & & & & & &&\\
%\node [terminal](p1) {$V_1$};&&\node [terminal](p3) {$V_3$};& & \node [terminal](p4) {$V_4$}; & &\node [terminal](p5) {$V_5$}; &&\node [terminal](p6) {$~$};& & \node [terminal](p7) {$V_7$};&\\
%};
%\draw (p2) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p1) edge [<-, shorten <=2pt, shorten >=2pt] (p3) (p3) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p4) edge [<-, shorten <=2pt, shorten >=2pt] (p5);
%\end{tikzpicture}
%\end{center}
\[
\begin{tabular}{@{}*{13}{W{c}{1.5em}@{}}}
	& & & & & $V_{2}$
\\	& & & & & $\uparrow$
\\	& $V_{1}$ & $\leftarrow$ & $V_{3}$ & $\leftarrow$ & $V_{4}$ & $\leftarrow$ & $V_{5}$ & \phantom{$\leftarrow$} & \phantom{$V_{6}$} & \phantom{$\leftarrow$} & $V_{7}$
\end{tabular}
\]
Finally, $V_6$ satisfies that $\alpha_6\in V_5+ V_7$, so we can apply the criterion from Step~\ref{example_step_3} again. We draw the directed edges $V_6 \rightarrow V_5$ and $V_6 \rightarrow V_7$, which recover the graph $\G_w$:
%\begin{center}
%\begin{tikzpicture}[baseline=(current bounding box.east),
%scale=.4, terminal/.style={ellipse, minimum width=.8cm, minimum height=.5cm, font=\itshape,},]
%\matrix[row sep=.7cm,column sep=.4cm] {&&& & \node[terminal](p2) {$V_2$}; & & & & & &&\\
%\node [terminal](p1) {$V_1$};&&\node [terminal](p3) {$V_3$};& & \node [terminal](p4) {$V_4$}; & &\node [terminal](p5) {$V_5$}; &&\node [terminal](p6) {$V_6$};& & \node [terminal](p7) {$V_7$};&\\
%};
%\draw (p2) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p1) edge [<-, shorten <=2pt, shorten >=2pt] (p3) (p3) edge [<-, shorten <=2pt, shorten >=2pt] (p4) (p4) edge [<-, shorten <=2pt, shorten >=2pt] (p5) (p5) edge [<-, shorten <=2pt, shorten >=2pt] (p6) (p7) edge [<-, shorten <=2pt, shorten >=2pt] (p6);
%\end{tikzpicture}
%\end{center}
\[
\begin{tabular}{@{}*{13}{W{c}{1.5em}@{}}}
	& & & & & $V_{2}$
\\	& & & & & $\uparrow$
\\	& $V_{1}$ & $\leftarrow$ & $V_{3}$ & $\leftarrow$ & $V_{4}$ & $\leftarrow$ & $V_{5}$ & $\leftarrow$ & $V_{6}$ & $\rightarrow$ & $V_{7}$
\end{tabular}
\]
\end{step}
\end{example}

Let us prove Theorem~\ref{theo:recover} when there is a trivalent vertex.

\begin{proof}[Proof of Theorem~\ref{theo:recover} in the presence of a trivalent vertex]
Now we assume that $\mathcal{G}_w$ has a trivalent vertex. As in the case without a trivalent vertex, it is enough to prove when $w$ is a Coxeter element. Then the arguments in Steps~\ref{theo:recover_step_1} and~\ref{theo:recover_step_2} remain unchanged. We will modify Step~\ref{theo:recover_step_3} by adding one more criterion and add Step~\ref{theo:recover_step_4}, which repeats Steps~\ref{theo:recover_step_2} and~\ref{theo:recover_step_3}.

We remark that if the trivalent vertex is already recovered after Step~\ref{theo:recover_step_2}, then its outdegree is necessarily at most~$1$. Indeed, this happens only when the trivalent vertex itself is a sink, or when there is a sink in the branch determined by the outward edge from the trivalent vertex.

\begingroup
\newstepset
\setcounter{step}{2}
\renewcommand{\thestep}{\arabic{step}\textsuperscript{$\boldsymbol{\prime}$}}
\begin{step}
As before, we begin with the sets $\mathcal{R}$ and $\mathcal{L}$, where $\mathcal{R}$ is the set of subspaces $V_j$ that are not contained in any connected component constructed in Step~\ref{theo:recover_step_2}, and $\mathcal{L}$ is the set of leaves of those connected components viewed as an undirected graph. Now we add another criterion as follows:\hypertarget{theo:recover_step_3prime}{}
\begin{enumerate}
\renewcommand*{\theenumi}{J\textsubscript{\arabic{enumi}}}\relabel
\setcounter{enumi}{2}
\item \label{J3} for $V_j=E(\alpha_j)\in\mathcal{R}$, there exist $V_p, V_q, V_r \in \mathcal{L}$ such that $\alpha_j\in V_p+V_q+V_r$.
\end{enumerate}

Note that there exists a space $V_j\in \mathcal{R}$ satisfying~\eqref{J3} only when the trivalent vertex of $\mathcal{G}_w$ is a source. Then each $V_j\in\mathcal{R}$ satisfies~\eqref{J2},~\eqref{J3}, or neither. Hence, we divide $\mathcal{R}$ into three subsets:
\begin{align*}
	\mathcal{R}_1 & = \braces[\big]{V_j\in\mathcal{R} \st[\big] \text{$V_j$ satisfies neither~\eqref{J2} nor~\eqref{J3}}},
\\	\mathcal{R}_2 & = \braces[\big]{V_j\in\mathcal{R} \st[\big] \text{$V_j$ satisfies~\eqref{J2}}},
\\	\mathcal{R}_3 & = \braces[\big]{V_j\in\mathcal{R} \st[\big] \text{$V_j$ satisfies~\eqref{J3}}}.
\end{align*}
\begin{enumerate}[leftmargin=3.9em,labelwidth=\widthof{Case~3:},align=left]
\renewcommand{\labelenumi}{Case~\theenumi:}
\item Assume that the trivalent vertex has outdegree at most $1$. Then both $\mathcal{R}_1=\mathcal{R}_3=\emptyset$, and the graph $\mathcal{G}_w$ can be recovered under the same procedures as in Step~\ref{theo:recover_step_3} for the case with no trivalent vertex.
\item Assume that the trivalent vertex is a source. In this situation, $\mathcal{R}_1=\emptyset$ and $\mathcal{R}_3$ is a singleton, say $\mathcal{R}_3=\{V\}$. Then we can choose $V_p$, $V_q$, $V_r$ minimal with respect to inclusion. If such a choice is not unique, we choose one of them and draw the directed edges $V \rightarrow V_p$, $V \rightarrow V_q$, and $V \rightarrow V_r$. As in the case with~\eqref{J2}, either choice leads to the same resulting graph. For each $V\in \mathcal{R}_2$, we draw the two directed edges as in the case with no trivalent vertex. Then we can recover the graph $\mathcal{G}_w$.
\item \label{case_3} Assume that the trivalent vertex has outdegree $2$. In this case, $\mathcal{R}_1\neq \emptyset$ and $\mathcal{R}_3=\emptyset$. If $\mathcal{R}_2\neq \emptyset$, then we proceed with Step~\ref{theo:recover_step_3} as in the case with no trivalent vertex, and then go to Step~\ref{theo:recover_step_4}.
\end{enumerate}

Note that each space in $\mathcal{R}_1$ satisfies~\eqref{J1} or~\eqref{J2}.
\end{step}
\endgroup

\begin{step}\label{theo:recover_step_4}
Let $\mathcal{V}$ be the set of $V_j$'s contained in some connected component of the graph obtained from Step~\hyperlink{theo:recover_step_3prime}{3\textsuperscript{${\prime}$}}. Since one of the connected components contains the vertex $V_b$ corresponding to the trivalent vertex recovered in Step~\hyperlink{theo:recover_step_3prime}{3\textsuperscript{${\prime}$}}, Case~\ref{case_3}, there exists a unique subspace $V_a=E(\alpha_a)\in\mathcal R_1$ such that $\dim(V_a\cap V_b)=1$ or $\alpha_a\in V_b+V_c$ for some $V_c \in \mathcal{V}$. We distinguish two cases according to the structure of $\peset{w}{a}$.
\begin{enumerate}[leftmargin=3.9em,labelwidth=\widthof{Case~3:},align=left]
\renewcommand{\labelenumi}{Case~\theenumi:}
\item Assume $\alpha_a\in V_b+V_c$ for some $V_c \in \mathcal{V}$. We choose such spaces $V_b$ and $V_c$ minimal with respect to inclusion. If such a choice is not unique, we choose one of them and draw $V_a \rightarrow V_b$ and $V_a \rightarrow V_c$. As before, either choice leads to the same resulting graph. This situation corresponds to $\mathcal{R}_1=\{V_a\}$, and the resulting graph is the desired one.
\item Assume $\dim(V_a\cap V_b)=1$. In this case, we draw a directed edge $V_a \rightarrow V_b$ as in Step~\ref{theo:recover_step_2}. We then continue this procedure as in Step~\ref{theo:recover_step_2}, successively attaching spaces whose intersections are one-dimensional. This process terminates either when all remaining spaces have been incorporated, or when a configuration of the form satisfying~\eqref{J2} appears. In the latter case, we apply Step~\ref{theo:recover_step_3} once, and the construction is complete.
\end{enumerate}
\end{step}
This completes the proof.
\end{proof}

\begin{remark}
We cannot extend Theorem~\ref{theo:recover} to other Lie types by simply following the steps considered in its proof. For a simple Lie group $G$ of type~$\dyn{B}_2$, the cohomology ring $H^\ast(X_{s_2s_1})$ is isomorphic to $\Z[x,y] \big/ \langle x^2,y^2\rangle$. Hence, there are two primitive square-zero elements $x$ and $y$. This produces a graph consisting of two vertices with no edges by following the steps in the proof of Theorem~\ref{theo:recover}. However, the digraph $\G_{s_2s_1}$ consists of two vertices and a directed edge with label~$2$. This failure occurs because the procedure relies solely on the cohomology ring data and cannot detect nontrivial edge labels such as the label~$2$.
\end{remark}

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