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\newcommand{\NN}{\mathbb{N}} % natural numbers
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%% define your title in the usual way
\title[The Isomorphism Problem for (Co)adjoint Schubert Varieties]{The Isomorphism Problem for (Co)adjoint Schubert Varieties}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Linji Chen, Yunhao Lai, and Zhehan Xu]{\texorpdfstring{Linji Chen\addressmark{1}\thanks{\href{mailto:cmuuuu7@gmail.com}{cmuuuu7@gmail.com}}
, Yunhao Lai\addressmark{2}\thanks{\href{mailto:laiyh36@mail2.sysu.edu.cn}{laiyh36@mail2.sysu.edu.cn}},
\and Zhehan Xu\addressmark{3}\thanks{\href{mailto:kxu3nive@gmail.com}{kxu3nive@gmail.com}}}{Linji Chen, Yunhao Lai, and Zhehan Xu}
}
%\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}

%% then use \addressmark to match authors to institutions here
\address{
\addressmark{1}School of Mathematical Science, Peking University, Beijing, China  \\ 
%\addressmark{2}School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, China \\
\addressmark{2}School of Mathematics, Sun Yat-sen University, Guangzhou, China \\
\addressmark{3}School of Mathematical Science, Peking University, Beijing, China 
}

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\received{November 14, 2024}

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\abstract{We classify isomorphism classes of Schubert varieties coming from adjoint and coadjoint partial flag varieties across all Dynkin types via Hasse diagrams given by the Chevalley formula.}

%% put your French abstract here, or comment this out if you don't have one
%\resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Schubert variety, flag variety, adjoint, coadjoint}

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\newcommand{\XXX}{Chevalley--Hasse}



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\begin{document}

\maketitle 
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above




\section{Introduction}


Being relatively simple families of homogeneous spaces, adjoint and coadjoint flag varieties have been extensively  studied \cites{BP,CP,KS,PS,SY}.  There is one of each for every complex simple reductive group \(G\).  The adjoint flag variety is given by the unique closed \(G\)-orbit in \(\PP(\mathfrak g)\), where \(G\) acts on its Lie algebra \(\mathfrak g\) via the adjoint representation.  It is of the form \(G/P_\text{adjoint}\), where \(P_\text{adjoint}\subset G\) is the parabolic subgroup whose corresponding weight is the highest (long) root. The coajoint flag variety is \(G/P_\text{coadjoint}\), where \(P_\text{coadjoint}\subset G\) is the parabolic subgroup whose corresponding weight is the highest short root. In simply-laced Lie types, adjoint and codjoint flag varieties coincide. See \Cref{ClaOfAdjAndCoaFlaVar} for a list of adjoint and coadjoint flag varieties and their corresponding Dynkin diagrams. In each case, the subgroup \(P_\text{adjoint}\) (resp. \(P_\text{coadjoint}\)) is determined by the subset of empty nodes, which are simple roots perpendicular to the highest root (resp. highest short root). Except in type $A$, these parabolic subgroups are maximal. 

% The adjoint and coadjoint partial flag varieties, being generalizations of grassmannians and bearing nice properties (for example, their cohomology ring can be presented, their Schubert calculus take a simple form, and they are sometimes hyperplane sections of other varieties), have been extensively studied (\cite{BP}, \cite{CP}, \cite{KS}, \cite{PS},\cite{SY}).  They are homogeneous varieties in the form of $G/P$, where $G$ is a complex reductive algebraic group and $P$ is a parabolic subgroup, and such a quotient $G/P$ is called adjoint (resp.\ coadjoint) if the weight corresponding to $P$ is the highest long root (resp.\ highest short root) in the root system. \xwh{Added 14 here. Add the orbit description.}

% See \Cref{ClaOfAdjAndCoaFlaVar} for a list of adjoint and coadjoint flag varieties. In each Dynkin diagram, the (co)adjoint roots (i.e., the root whose corresponding weight is the (co)adjoint weight) are filled.  There is one adjoint partial flag variety and one coadjoint partial flag variety in each Lie type, and except for type $A$, their parabolic subgroups are maximal. In simply-laced Lie types, adjoint and codjoint flag varieties coincide.  \xwh{you didn't define (co)adjoint weight.}


The adjoint (resp.\ coadjoint) Schubert varieties, being the closure of Borel orbits in an adjoint (resp.\ coadjoint) flag variety, are naturally indexed by long roots (resp.\ short roots) in the root system. We give the following combinatorial criterion for distinguishing their isomorphism classes. 

% We give the following combinatorial criterion for distinguishing isomorphism classes of Schubert varieties coming from (co)adjoint flag varieties. 
%In \Cref{SecIntPro}, we define the \XXX\ diagram associated to such a Schubert variety, which can be determined from combinatorics of the cooresponding root system, but also represents the structure of the Chow group with action of a chosen divisor in the Schubert variety. Then, we managed to show \Cref{Theorem 1}, which solves the isomorphism problem for adjoint and coadjoint Schubert varieties except for two cases.

\begin{thm}\label{Theorem 1}
    Let $X_\alpha\subseteq X$ and $Y_\beta\subseteq Y$ be adjoint or coadjoint Schubert varieties. Then the following are equivalent:
    \begin{enumerate}
    \item[(i)] $X_\alpha$ and $Y_\beta$ are isomorphic as abstract algebraic varieties;
    \item[(ii)] Their corresponding \XXX\ diagrams, $P_\alpha$ and $P_\beta$, are isomorphic.
\end{enumerate}
\end{thm}



The \XXX\ diagram of a (co)adjoint Schubert variety $X_\alpha$ is a weighted, directed graph whose vertices index a well-chosen basis (the Schubert basis) of $\CH(X_\alpha)$ and whose edges represent how $\Pic(X_\alpha)$ acts on the basis elements via the intersection product. For a precise definition, as well as an equivalent combinatorial definition based on the root system, see \Cref{SecIntPro}.

\begin{table}[h]\centering 
    \begin{adjustbox}{center}
    \begin{tabularx}{15cm}{ |m{4.14cm}|m{4.14cm}|m{5.4cm}| }\hline
    \smallbox{Adjoint}&\smallbox{Coadjoint}&\smallbox{Description}\\\hline
    \multicolumn{2}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [dot=below:$\alpha_1$];
    \node (2) at (2,0) [circ=below:$\alpha_2$];
    \node (3) at (3,0){$\ldots$};
    \node (4) at (4,0) [circ=below:$\alpha_{n-1}$];
    \node (5) at (5,0) [dot=below:$\alpha_{n}$];
    \draw (1)--(2)--(3)--(4)--(5);
    \end{tikzpicture}
    }
    &
    \smallbox{Type $A_n$ ($n\ge 1$):\\$G/P_\text{(co)adjoint}\simeq \Fl(1,n;n+1)$}
    \\\hline
    \multicolumn{1}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [circ=below:$\alpha_1$];
    \node (2) at (2,0) [dot=below:$\alpha_2$];
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    \draw[double] (4)--(5);
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    &
    \multicolumn{1}{c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [dot=below:$\alpha_1$];
    \node (2) at (2,0) [circ=below:$\alpha_2$];
    \node (3) at (3,0){$\ldots$};
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    \draw (4.572,0)--(4.4,0.1) (4.572,0)--(4.4,-0.1) ;
    \end{tikzpicture}
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    &
    \smallbox{Type $B_n$ ($n\ge 2$):\\$G/P_\txt{adjoint}\simeq \OGr(2,2n+1)$\\$G/P_\txt{coadjoint}\simeq Q^{2n-1}$}
    \\\hline
    \multicolumn{1}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [dot=below:$\alpha_1$];
    \node (2) at (2,0) [circ=below:$\alpha_2$];
    \node (3) at (3,0){$\ldots$};
    \node (4) at (4,0) [circ=below:$\alpha_{n-1}$];
    \node (5) at (5,0) [circ=below:$\alpha_{n}$];
    \draw (1)--(2)--(3)--(4);
    \draw[double] (4)--(5);
    \draw (4.428,0)--(4.6,0.1) (4.428,0)--(4.6,-0.1);
    \end{tikzpicture}
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    &
    \multicolumn{1}{c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [circ=below:$\alpha_1$];
    \node (2) at (2,0) [dot=below:$\alpha_2$];
    \node (3) at (3,0){$\ldots$};
    \node (4) at (4,0) [circ=below:$\alpha_{n-1}$];
    \node (5) at (5,0) [circ=below:$\alpha_{n}$];
    \draw (1)--(2)--(3)--(4);
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    \draw (4.428,0)--(4.6,0.1) (4.428,0)--(4.6,-0.1);
    \end{tikzpicture}
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    &
    \smallbox{Type $C_n$ ($n\ge 3$):\\$G/P_\txt{adjoint}\simeq \PP^{2n-1}$\\$G/P_\text{coadjoint}\simeq \IG(2,2n)$}
    \\\hline
    
    \multicolumn{2}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [circ=below:$\alpha_1$];
    \node (2) at (2,0) [dot=below:$\alpha_2$];
    \node (3) at (3,0) [circ=below:$\alpha_3$];
    \node (4) at (4,0){$\ldots$};
    \node (5) at (5,0) [circ=below:$\alpha_{n-2}$];
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    &
    \parbox[c]{\linewidth}{$\phantom{x}$\\[-11pt]Type $D_n$ ($n\ge 4$):\\$G/P_\text{(co)adjoint}\simeq \OGr(2,2n)$}
    \\\hline
    \multicolumn{2}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [circ=below:$\alpha_1$];
    \node (3) at (2,0) [circ=below:$\alpha_3$];
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    \draw (1)--(3)--(4)--(5)--(6) (2)--(4);
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    &
    \parbox[c]{\linewidth}{$\phantom{x}$\\[-5pt]Type $E_6$}
    \\\hline
    
    \multicolumn{2}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [dot=below:$\alpha_1$];
    \node (3) at (2,0) [circ=below:$\alpha_3$];
    \node (4) at (3,0) [circ=below:$\alpha_4$];
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    &
    \parbox[c]{\linewidth}{$\phantom{x}$\\[-5pt]Type $E_7$}
    \\\hline
    \multicolumn{2}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (1,0) [circ=below:$\alpha_1$];
    \node (3) at (2,0) [circ=below:$\alpha_3$];
    \node (4) at (3,0) [circ=below:$\alpha_4$];
    \node (5) at (4,0) [circ=below:$\alpha_5$];
    \node (6) at (5,0) [circ=below:$\alpha_6$];
    \node (7) at (6,0) [circ=below:$\alpha_7$];
    \node (8) at (7,0) [dot=below:$\alpha_8$];
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    \draw (1)--(3)--(4)--(5)--(6)--(7)--(8) (2)--(4);
    \end{tikzpicture}
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    &
    \parbox[c]{\linewidth}{$\phantom{x}$\\[-5pt]Type $E_8$}
    \\\hline
    \multicolumn{1}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (0,0) [dot=left:$\alpha_1$];
    \node (2) at (1,0) [circ=below:$\alpha_2$];
    \node (3) at (2,0) [circ=below:$\alpha_3$];
    \node (4) at (3,0) [circ=right:$\alpha_4$];
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    &
    \multicolumn{1}{c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (0,0) [circ=left:$\alpha_1$];
    \node (2) at (1,0) [circ=below:$\alpha_2$];
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    \draw (1.572,0)--(1.4,0.1);
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    &
    \parbox[c]{\linewidth}{$\phantom{x}$\\[-5pt]Type $F_4$}
    
    \\\hline
    \multicolumn{1}{|c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (0,0) [dot=below:$\alpha_1$];
    \node (2) at (1,0) [circ=below:$\alpha_2$];
    \draw[scaling nfold=3] (1)--(2);
    \draw (0.572,0)--(0.4,0.1);
    \draw (0.572,0)--(0.4,-0.1);
    \end{tikzpicture}
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    &
    \multicolumn{1}{c|}{
    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.75]
    \node (1) at (0,0) [circ=below:$\alpha_1$];
    \node (2) at (1,0) [dot=below:$\alpha_2$];
    \draw[scaling nfold=3] (1)--(2);
    \draw (0.572,0)--(0.4,0.1);
    \draw (0.572,0)--(0.4,-0.1);
    \end{tikzpicture}
    }
    &
    \smallbox{Type $G_2$: $G/P_\text{coadjoint}\simeq Q^5$}
    \\\hline  
    
    \end{tabularx}
    \end{adjustbox}
    \caption{(Co)adjoint flag varieties. 
    % The filled-in circles denote the (co)adjoint roots.
    }\label{ClaOfAdjAndCoaFlaVar}
    \end{table}

    



% \xzh{I prefer first giving this algebraic definition and simply mentioning that there will be a combinatorial definition. This is because the order $\le$ on roots is pretty complicated; so is the combinatorial description of the coefficients in the Chevalley formula. Giving a simpler definition first gives a reader a better impression.} %\textcolor{red}{\bf To sound more combinatorial, I would say: vertices are (rep. short)long roots \(\beta\) such that \(\beta\leq \alpha\) (and ... when \(\alpha\) is simple) and edges are certain reflections. I also recommend defining them in this way in Sec 4. You can explain the connection to Chevalley formula in a remark.}

Our work is analogous to the work of Richmond, Tarigradschi and Xu in \cite{RTX}, where they proved that two cominuscule Schubert varieties are isomorphic if and only if their corresonding labeled posets of roots are isomorphic. Other related works include the classification of a class of smooth Schubert varieties in type A partial flag varieties by Develin, Martin, Reiner in \cite{DMR} and Richmond and Slofstra's work on the isomorphism problem of Schubert varieties in complete flag varieties in \cite{RS21}. The cases of Grassmannian Schubert varieties are handled using Young diagrams, by Tarigradschi and Xu in \cite{TX}.

We illustrate examples in \Cref{SecExa} and discuss preliminaries in \Cref{SecPre}. Then in \Cref{SecIntPro}, we will construct the \XXX\ diagram associated with a (co)adjoint Schubert variety. Our proof of ~\Cref{Theorem 1} will be sketched very briefly in \Cref{CalDisofMAITHE}. 
The direction from isomorphic (co)adjoint Schubert varieties to isomorphic \XXX\ diagram is easy by definition, see \Cref{CHDisoinv}.
However, the converse part of \Cref{Theorem 1} is more complicated: two main techniques, namely minimal embeddings and foldings, are applied to embed (co)adjoint Schubert varieties into other classes of Schubert varieties.

    
\section{Examples of \texorpdfstring{\Cref{Theorem 1}}{Theorem 1}}\label{SecExa}
Let us first apply \Cref{Theorem 1} in various examples.
\begin{convention}
\textbf{Throughout the remaining sections}: for convenience, for a fixed root system $\Phi$ of some Lie type, we index the simple roots in $\Phi$ following \Cref{ClaOfAdjAndCoaFlaVar}. A general root $\alpha$ can be expressed uniquely as a linear combination of simple roots, and we will denote $\alpha$ simply by the coefficients of this expression. For example, the root $\alpha_2+\alpha_4+\alpha_5$ in $E_7$ will be denoted $0101100$, and the root $-\alpha_{3}-2\alpha_4$ in $B_4$ will be denoted $-(0012)$.

We will omit the directions of edges when we present the \XXX\ diagram of a (co)adjoint Schubert variety $X_\alpha$; instead, we will give the set of vertices a partial order such that the ``heights'' of vertices (defined as the minimal length of paths to the minimal vertex) represent the dimension $d$ of the corresponding class in $\CH_*(X_\alpha)$. And every edge will automatically be directed from the lower-dimensional class to the higher-dimensional one.  See \Cref{SecIntPro} for the full definition. %\xwh{Here the words "order" and "height" can have mathematical meanings that may or may not be compatible with your description.}
%\lyh{I don't exactly know what do mentor mean. I slightly modify.}
%xzh: I think the word order is fine, but height is indeed only an intuitive description. To be strict it should probably be the rank of vertices in a finite poset. But I think using ``heights'' is just okay.

We will denote in this article the (co)adjoint Schubert variety and its \XXX\ diagram respectively by $X(T,\alpha)$ and $P(T,\alpha)$, where $T$ is the Cartan type of the root system $\Phi$ and $\alpha$ is a short/long root in $\Phi$ labeling the Schubert variety. We will sometimes use $X_\alpha$ and $P_\alpha$ if the partial flag variety is clear in the context.
\end{convention}



\begin{eg}\label{egofthm1}
Consider the root system $\Phi$ of Cartan type $C_2$; we know that the adjoint partial flag variety corresponding to this type is $\PP^3$. Letting $\alpha=(01)=\alpha_2$, we get an adjoint Schubert variety $X(C_2,\alpha)$.

On the other hand, consider the root system $\Phi'$ of Cartan type $B_3$; the corresponding adjoint flag variety is $\OGr(2,2n+1)$. Letting $\beta=-(012)$, we get another Schubert variety $X(B_3,\beta)$.

The combinatorial view of \XXX\ diagrams will be introduced later. Let us admit the following results: the Chevalley--Hasse diagrams $P(C_2,\alpha)$ and $P(B_3,\beta)$ are both isomorphic to the diagram in \Cref{AdjC201}. Therefore, \Cref{Theorem 1} says $X(C_2,\alpha)$ and $X(B_3,\beta)$ are isomorphic as algebraic varieties.

Note that there is exactly one vertex of each height in \Cref{AdjC201}, which means the Chow group of this algebraic variety is free of rank $1$ in dimensions $0,1,2$. This is verified by the fact that $X(C_2,\alpha)$ and $X(B_3,\beta)$ are both isomorphic to $\PP^2$.
\end{eg}

 \begin{figure}[ht]
    \begin{tabular}{p{4.5cm}p{4.6cm}p{4.5cm}}
               \begin{minipage}[t]{0.4\textwidth}
               %\centering
               \captionsetup{skip=-8pt,position=bottom}
               \begin{ctikzpicture}[scale=0.5]
\node (-2x1) at (0,-1) [circ];
\node (-3x1) at (0,-2) [circ];
\node (-4x1) at (0,-3) [circ];
\draw (-4x1)--(-3x1)--(-2x1);
       \end{ctikzpicture}
       \caption{$\begin{matrix}\mathop{\phantom{\simeq}}P(C_2,\alpha)\\ \simeq P(B_3,\beta)\end{matrix}$}\label{AdjC201}
           \end{minipage}
       &
       \begin{minipage}[t]{0.45\textwidth}
           %\centering
       \captionsetup{skip=-8pt,position=bottom}
       \begin{ctikzpicture}[scale=0.5]

\node (-4x2) at (5,-3) [circ];
\node (-5x2) at (6,-4) [circ];

\node (-3x2) at (4,-2) [circ];
\node (-3x3) at (6,-2) [circ];

\node (-2x3) at (5,-1) [circ];

\draw (-3x2)--(-2x3)--(-3x3)--(-4x2)--(-3x2) (-4x2)--(-5x2);

                   \end{ctikzpicture}
       \caption{$\begin{matrix}\mathop{\phantom{\simeq}}P(B_4,-(0112))\\ \simeq P(C_4,-(0111))\\ \simeq P(D_4,-(0101))\end{matrix}$}\label{AdjB4-0112}
       \end{minipage}
&
        \begin{minipage}[t]{0.45\textwidth}
            %\centering
        \captionsetup{skip=-8pt,position=bottom}
        \begin{ctikzpicture}[scale=0.5]
                      
 
 
 \node (-6x2) at (7,-5) [circ];
 
 \node (-5x2) at (6,-4) [circ];
 
 \draw (-5x2)--(-6x2);
 
 \node (-4x2) at (5,-3) [circ];
 \node (-4x3) at (7,-3) [circ];
 
 \draw (-4x2)--(-5x2)--(-4x3);
 
 \node (-3x2) at (4,-2) [circ];
 \node (-3x3) at (6,-2) [circ];
 
 \draw (-4x2)--(-3x3)--(-4x3);
 \draw (-3x2)--(-4x2);
 
 
 \node (-2x2) at (3,-1) [circ];
 \node (-2x3) at (5,-1) [circ];
 
 \draw (-3x2)--(-2x2) (-3x3)--(-2x3);
 \draw (-2x3)--(-3x2);
 
 
 \node (-1x3) at (4,0) [circ];
 
 \draw (-2x2)--(-1x3);
 \draw (-1x3)--(-2x3);
        \end{ctikzpicture}
        \caption{$P(C_4,-(0100))$}\label{CoadjC4-0100}
        \end{minipage}
        
       \\[2cm]
       
       \begin{minipage}[t]{0.45\textwidth}
           %\centering
       \captionsetup{skip=-8pt,position=bottom}
       \begin{ctikzpicture}[scale=0.5]

 \node (-6x2) at (7,-5) [circ];
 
 \node (-5x2) at (6,-4) [circ];
 
 \draw (-5x2)--(-6x2);
 
 \node (-4x2) at (5,-3) [circ];
 \node (-4x3) at (7,-3) [circ];
 
 \draw (-4x2)--(-5x2)--(-4x3);
 
 \node (-3x2) at (4,-2) [circ];
 \node (-3x3) at (6,-2) [circ];
 
 \draw (-4x2)--(-3x3)--(-4x3);
 \draw (-3x2)--(-4x2);
 
 
 \node (-2x3) at (5,-1) [circ];
 \node (-2x4) at (7,-1) [circ];
 
 \draw (-3x2)--(-2x3) (-3x3)--(-2x4);
 \draw (-2x3)--(-3x3);
 
 
 \node (-1x4) at (6,0) [circ];
 
 \draw (-2x3)--(-1x4);
 \draw (-1x4)--(-2x4);
                   \end{ctikzpicture}
                   \caption{$P(C_4,-(0010))$}\label{CoadjC4-0010}
       \end{minipage}
&
        \begin{minipage}[t]{0.45\textwidth}
            %\centering
        \captionsetup{skip=-8pt,position=bottom}
        \begin{ctikzpicture}[scale=0.5]
                      
 
 
 \node (-6x2) at (7,-5) [circ];
 
 \node (-5x2) at (6,-4) [circ];
 
 \draw (-5x2)--(-6x2);
 
 \node (-4x2) at (5,-3) [circ];
 \node (-4x3) at (7,-3) [circ];
 
 \draw (-4x2)--(-5x2)--(-4x3);
 
 \node (-3x2) at (4,-2) [circ];
 \node (-3x3) at (6,-2) [circ];
 
 \draw (-4x2)--(-3x3)--(-4x3);
 \draw[double] (-3x2)--(-4x2);
 
 
 \node (-2x2) at (3,-1) [circ];
 \node (-2x3) at (5,-1) [circ];
 
 \draw (-3x2)--(-2x2);
 \draw[double] (-2x3)--(-3x3);
 \draw (-2x3)--(-3x2);
 
 
 \node (-1x3) at (4,0) [circ];
 
 \draw (-2x2)--(-1x3);
 \draw (-1x3)--(-2x3);
        \end{ctikzpicture}
        \caption{$P(B_4,-(0100))$}\label{AdjB4-0100}
        \end{minipage}
    &
       \begin{minipage}[t]{0.45\textwidth}
        %\centering
    \captionsetup{skip=-8pt,position=bottom}
    \begin{ctikzpicture}[scale=0.5]
                  
 \node (-6x2) at (7,-5) [circ];
 
 \node (-5x2) at (6,-4) [circ];
 
 \draw (-5x2)--(-6x2);
 
 \node (-4x2) at (5,-3) [circ];
 \node (-4x3) at (7,-3) [circ];
 
 \draw (-4x2)--(-5x2)--(-4x3);
 
 \node (-3x2) at (4,-2) [circ];
 \node (-3x3) at (6,-2) [circ];
 
 \draw (-4x2)--(-3x3)--(-4x3);
 \draw[double] (-3x2)--(-4x2);
 
 
 \node (-2x3) at (5,-1) [circ];
 \node (-2x4) at (7,-1) [circ];
 
 \draw (-3x2)--(-2x3) (-3x3)--(-2x4);
 \draw[double] (-2x3)--(-3x3);
 
 
 \node (-1x4) at (6,0) [circ];
 
 \draw (-2x3)--(-1x4);
 \draw[double] (-1x4)--(-2x4);
    \end{ctikzpicture}
    \caption{$P(B_4,-(0010))$}\label{AdjB4-0010}
    \end{minipage}
\end{tabular}
\end{figure}



   % \begin{figure}
   %  \begin{tabular}{p{6cm}p{8cm}}

   %     \end{tabular}
   % \end{figure}

\begin{eg}\label{egofthm2}
    Now, consider the root system of type $D_4$ with the root $-(0101)$, the root system of type $C_4$ with the \textit{short} root $-(0111)$, and the root system $B_4$ with the \textit{long} root $-(0112)$. So we get a coadjoint Schubert variety $X(C_4,-(0111))$, an adjoint Schubert variety $X(B_4,-(0112))$, and a Schubert variety $X(D_4,-(0101))$ which is both coadjoint and adjoint (as $D_4$ is simply laced).

    The \XXX\ diagrams of all the three varieties are isomorphic to the one in \autoref{AdjB4-0112}. Since \Cref{Theorem 1} is applicable to one adjoint Schubert variety and one coadjoint Schubert variety, we deduce that the three Schubert varieties are all isomorphic. They are all isomorphic to the following Schubert variety in the Grassmannian $\Gr(2,8)$:
    \begin{align*}
\Gr_{2,4}(2,8)\defeq\setC{\Sigma\in\Gr(2,8)}{\dim(\Sigma\cap E_2)\ge 1,\ \Sigma\subseteq E_4},
    \end{align*}
    where $0=E_0\subset E_1\subset \ldots \subset E_8=\CC^8$ is a given flag in $\CC^8$.
\end{eg}

\begin{eg}\label{egofthm3}
Consider the coadjoint Schubert varieties $X(C_4,-(0100))$, $X(C_4,-(0010))$ and the adjoint Schubert varieties $X(B_4,-(0100))$, $X(B_4,-(0010))$. As is illustrated in \Cref{CoadjC4-0100}, \Cref{CoadjC4-0010}, \Cref{AdjB4-0100}, and \Cref{AdjB4-0010}, their \XXX\ diagrams are pairwise non-isomorphic. Therefore, although the Chow groups these varieties are free of the same rank in each dimension, \Cref{Theorem 1} says that these varieties are pairwise non-isomorphic. The point is that an isomorphism between the Chow groups of two distinct varieties in this example cannot preserve the action of Picard groups.
\end{eg}

\section{Preliminaries}\label{SecPre}
\subsection{Adjoint Schubert varieties and Coadjoint Schubert varieties}

First, let us recall some definitions. 
Let $G$ be a complex reductive algebraic group with a fixed Borel subgroup $B$ and a maximal torus $T$.
The triple $(G,B,T)$ determines a root system $\Phi$ in the weight lattice $M=\Hom(T,\CC^*)$, with a root basis $\Delta$ and Weyl group $W=N_G(T)/T=\spnC{s_\alpha}{\alpha\in \Delta}$, where $s_\alpha$ is the reflection with respect to the simple root $\alpha$.
We consider the cases when $W$ is finite.
For a subset $I\subseteq \Delta$,  $W_I$ denotes $\spnC{s_\alpha}{\alpha\in I}\subset W$. There is a minimal parabolic subgroup $P\leqslant  G$ containing $B$ and $\spnC{s_\alpha}{-\alpha\in I}$.
The quotient space $G/P$ is called a partial flag variety, and the closure of orbits $X_w=\overline{BwP/P} $ are called Schubert varieties.
All Schubert varieties are indexed by $W/W_I$, or equivalently by $W^I$, the set of all minimal length coset representatives of $W/W_I$. If $w\in W^I$ ,  $\dim X_w=\ell(w)$.
 
Let $\Theta$ be the highest (long) root in the root system. The weight $\varpi=\Theta$ is called \textit{adjoint weight}. The partial flag variety $G/P$ is called \textit{adjoint} if $P$ is the parabolic subgroup associated to $\varpi$\footnote{That is, $P$ corresponds to the simple roots $\alpha_i$ such that $\left<\varpi,\alpha_i\right>=0$.}. In \Cref{ClaOfAdjAndCoaFlaVar},  set $I$ consists of the unfilled nodes in the Dynkin diagram. In this case, we have $W_I=\Stab_W(\Theta)$, and $W$ acts transitively on $\Phi_\text{long}$; thus there is a bijection
\begin{align*}
\Map{}{W^I}{\Phi_\text{long}}{w}{w(-\Theta),}
\end{align*}
so we may just define $X_{w(-\Theta)}\coloneqq X_w$.\footnote{In fact, our indexing of roots differs from the definition in literature, such as in \cite{PS}, by an action of $-w_0$ on the root space. Here $w_0\in W$ is the element with maximal Coxeter length. But note that $-w_0$ permutates the simple roots which induces an automorphism of the Dynkin diagram, so this difference does not affect anything in our consideration.} Therefore, the Schubert varieties in an adjoint partial flag variety can be indexed by long roots in the root system.

The coadjoint case is pretty much the same, except that `long' shall be replaced with `short': 
% \lyh{these can be organized in the manner of \cite{SY}.!!}
% \xzh{To lyh: Do you mean the description by cominuscule/minuscule roots? By know I have not yet understood them \ldots I think the current version is just fine.}
 We define $\varpi=\theta$, the highest short root, as the \textit{coadjoint} weight\footnote{Note that our definition of coadjoint weight agrees with \cite{CP} but differs from \cite{BP} by a multiplicity of $2$ in type $B$. The resulting parabolic subgroups and varieties are the same, but we will need $\varpi=\theta$ for convenience in \Cref{SecIntPro}.}; there is a bijection between coadjoint Schubert varieties and the set of short roots. Also, we define $X_{w(-\theta)}\coloneqq X_w$.




\subsection{Chow groups, Picard groups and the Chevalley formula}
Here, we list several well-known results for the Chow groups of the Schubert varieties, which we will refer to in the rest of the paper.

\begin{lem}[{\cite[Corollary of Theorem 1]{FMSS}; also see \cite[Example 1.9.1]{F}}]\label{ChoGroStrOfSchVar}
The Schubert classes $[X_u]$ such that $X_u \subseteq X_w$ are exactly the minimal elements in the extremal rays of the effective cone in $\CH_*(X_w)$. Moreover, these classes form an integral basis of $\CH_*(X_w)$.
% The Schubert classes $[X_u]$ such that $X_u \subseteq X_w$ form an integral basis for $\CH_*(X_w)$. Moreover, these classes are exactly the minimal elements in the extremal rays of the effective cone in $\CH_*(X_w)$.
\end{lem}

\begin{lem}[{\cite[Proposition 2.8, part (ii)]{CP}}]\label{EquConOnConOfSchVar}
\begin{align*}
X_\alpha\subseteq X_\beta\Leftrightarrow\cas{&\beta-\alpha=\sum_{\alpha_i\in \Delta} c_i\alpha_i\ \text{with $c_i\ge 0$},&\quad &\text{if $\alpha,\beta$ are of the same sign;}\\& \Supp(\alpha)\cup \Supp(\beta)\ \text{is connected,}&\quad&\text{if $\alpha\in\Phi^-$ and $\beta\in\Phi^+$.}}
\end{align*}
Here, the connectedness is counted as in the Dynkin diagram.
\end{lem}
We mainly focus on the multiplication of $\Pic(X_w)$ on $\CH_*(X_w)$, however, the following lemma allows us to reduce to the multiplication of $\Pic(X)$ on $\CH_*(X)$ by projection formula.

\begin{lem}[{\cite[Proposition 6]{M}; also see \cite[Proposition 2.2.8, part (ii)]{B}} ]\label{SurOfPulOfDiv}
    The pullback map $\Pic(X) \to \Pic(X_w)$ is surjective.
    \end{lem}
By Poincar\'e duality we may identify the cup product with the cap product $H^2(X)\times H_{2N-2q}(X)\to H_{2N-2q-2}(X)$. By \cite[Example 19.1.11]{F} and \cite[Proposition 19.2]{F}, we may moreover identify homology groups and Chow groups of $X$, while the cup product is identified with the intersection product $\Pic(X)\times \CH_{N-q}(X)\to \CH_{N-q-1}(X)$. Using these identifications, we can give an explicit Chevalley formula for coadjoint and adjoint partial flag variety. We first define a suitable divisor such that the formula for its action on the Chow group has a unified form.

\begin{definition}
    Given an adjoint partial flag variety $X$, we define $D_X'$ to be the divisor corresponding to the adjoint weight $\varpi=\Theta$. Let $D_X\defeq \frac{1}{2} D_{X}^{\prime}$ in adjoint type $C$, and let $D_X\defeq D_X^{\prime}$ in other cases. The definition of $D_X'$ in coadjoint cases is similar; moreover, in coadjoint cases, we always let $D_X\defeq D_X'$.
    
    Finally, if $X_w\subseteq X$ is a (co)adjoint Schubert variety, we define $D_{X_w}\defeq i^*D_X$, where $i:X_w\hookrightarrow  X$ is the embedding.
    \end{definition}

\begin{rem}\label{Cherem}
    In adjoint cases, it turns out that $D_X'\defeq D_{\alpha_1}+D_{\alpha_n}$ in type $A$,  $D_X'\defeq 2D_{\alpha_1}$ in type $C$, and  $D_X'\defeq D_{\alpha_i}$ ($\alpha_i\in \Delta\setminus I$ is unique) in other cases. 
    In coadjoint cases, it turns out that $D_X'\defeq D_{\alpha_1}+D_{\alpha_n}$ in type $A$, and  $D_X'\defeq D_{\alpha_i}$ ($\alpha_i\in\Delta\setminus I$ is unique) in other cases.
    
    The reason for the case-by-case definition of $D_X$ can be seen in Table (1) in \cite{CP}: in adjoint type C, the weight $\Theta$ and the simple root $\varpi_1$ differ by a scalar $2$, so in order to make $D_{X_w}$ an invariant on the algebraic structure of $X_w$, we need to offset this scalar. 
\end{rem}
Now, we restate the Chevalley formula by Fulton and Woodward for $D_X'$:
 
\begin{lem}[{\cite[Lemma 8.1]{FW}}]\label{CheForAdj}
    Let $X$ be an adjoint (resp. coadjoint) partial flag variety. Then for $\alpha\in\Phi_\text{long (resp. short)}$ we have
    \begin{align*}
    D_X' \cdot [X_{\alpha}]&=\cas{
           \sum_{\gamma\in\Delta,\ (\gamma^\dual,\alpha)>0
           % \substack{\gamma\in\Delta\\(\gamma^\dual,\alpha)>0}
           } (\gamma^\dual,\alpha)[X_{s_\gamma\alpha}]&\quad\text{if $\alpha\notin\Delta_\text{long (resp. short)}$;}\\
           \sum_{\gamma\in\Delta_\text{long (resp. short)}} (\gamma^\dual,\alpha)[X_{-\gamma}]&\quad\text{if $\alpha\in\Delta_\text{long (resp. short)}$.}
    }
    \end{align*}
    \end{lem}


\section{Chevalley--Hasse Diagrams}\label{SecIntPro}
\subsection{Intersection Product and the Definition of \XXX\ Diagram}

    \begin{definition}[Chevalley--Hasse diagram for (co)adjoint Schubert varieties]\label{DefChe-HasDia}
    % \lyh{mentor suggests to give a more combinatorial definition. But I still adopt this definition in two reasons. First, I don't know how to describe the order of root in more easier way when they have different signs. Then, the definition of wedges of edges still need the Chevalley formula. So i think we will get a more complicated definition if we only take a root-theoretic perspective.}
% \xzh{I agree. I have also slightly modified the definitions.}
        Let $X_\beta$ be a Schubert variety in the adjoint partial flag variety $X=X_\Theta$. We then construct the \XXX\ diagram $P_\beta$ as follows:
        
        \textbf{Vertices: } all Schubert classes $[X_\alpha]$ such that $X_\alpha\subseteq X_\beta$. By \Cref{EquConOnConOfSchVar}, these classes coincide with those long roots $\alpha$ such that $\alpha\le\beta$ when they are of the same sign, or such that $\Supp(\alpha)\cup \Supp(\beta)$ is connected when they are of different signs. The vertices have a natural partial order of inclusion relationship .
        
        \textbf{Edges: } for two long roots $\alpha,\alpha'$ in the diagram with\footnote{The depth of a root is a standard notation in Coxeter systems. For $\alpha\in\Phi^+$, $\depth(\alpha)$ is the minimal number of consecutive simple reflections needed to send $\alpha$ to $\Phi^-$, and for $\alpha\in\Phi^-$, $\depth(\alpha)$ is just $1-\depth(-\alpha)$.} $\depth(\alpha)=\depth(\alpha')-1$, we draw $n$ oriented edge(s) from $\alpha$ to $\alpha'$ if the coefficient of $[X_\alpha]$ in $D_{X_\beta}\cdot [X_{\alpha'}]$ under the selected basis is $n$. Now \Cref{CheForAdj}, together with the relations between $D_{X_\beta}$, $D_X$ , and $D_X'$, allows us to write the weight as an inner product, which relies only on combinatorial data of the root system.
        
        
        We do a similar construction for coadjoint Schubert varieties, with `long' replaced by `short'.
        \end{definition}

\begin{rem}
    \begin{enumerate}
    \item[(1)] Let $p$ be a vertex in the \XXX\ diagram. Then the dimension of the associated Schubert class is the minimal length of the path from the (co)adjoint root to $p$. 
    \item[(2)] If $X_\beta\subseteq X$ is a Schubert variety in the (co)adjoint partial flag variety $X$. By definition,  $D_{X_\beta}=i^*D_X$.  Let $P_\beta$ and $P_X$ be the corresponding \XXX\ diagrams. Then it turns out that $P_\beta$ is a lower ideal of $P_X$ with the unique maximal vertex $\beta$. %For example, take $\beta=-(00122)$ in $B_5$, and $P(B_5,\beta)$ is exactly the lower ideal of $P(B_5,\Theta)$ which is marked in red in \Cref{B5AdjFig2}. 
    \item[(3)] The \XXX\ diagrams of partial flag varieties are all vertically symmetrical, with the upper half corresponding to the positive roots and the lower half corresponding to the negative roots. For simplicity, most of the diagrams shown in this article are only the lower half and the negative roots of the whole diagram. % \xwh{negative?} \clj{ This is because we start with the lowest short/long root} xzh: corrected.
    \item[(4)] One may now use \Cref{DefChe-HasDia}, together with the formula given in \Cref{CheForAdj}, to revisit \Cref{SecExa} and work out the diagrams explicitly!
\end{enumerate}
\end{rem}
% D6 & E7

\begin{eg}\label{eg:F4E7}
Consider the root system of type $F_4$; the corresponding coadjoint partial flag variety, which is itself the maximal coadjoint variety, is denoted $X(F_4,\theta)$. Its \XXX\ diagram is illustrated in \Cref{F4Fig}. Similarly, the \XXX\ diagram of $X(E_7,\theta)$ is illustrated in \Cref{E7Fig}.

Note that the colored parts in \Cref{F4Fig} and \Cref{E7Fig} are isomorphic. In fact, as an example of the techniques we applied to prove \Cref{Theorem 1}, we will show the isomorphism between those Schubert varieties indexed by the roots marked in blue in $F_4$ and $E_7$ later in \Cref{eg:E7F42}. We will deal with the red parts similarly in \Cref{eg:E7F43}.
\begin{figure}[ht]
    \begin{center}
    \begin{tabular}{m{6.5cm} m{7cm}}
        \begin{minipage}[t]{0.45\textwidth}
            \centering
            \begin{ctikzpicture}[scale=0.5]
                \node (0x1) at (0,0) [bdot];
                \node (1x1) at (0,1) [bdot];
                \node (2x1) at (0,2) [bdot];
                \node (3x1) at (0,3) [bdot];
                \node (4x1) at (-1,4) [rdot];
                \node (4x2) at (1,4) [bdot];
                \node (5x1) at (0,5) [rdot];
                \node (5x2) at (2,5) [bdot];
                \node (6x1) at (-1,6) [rdot];
                \node (6x2) at (1,6) [rdot];
                \node (7x1) at (-2,7) [rdot];
                \node (7x2) at (0,7) [rdot];
                \draw[red]  (4x1)--(5x1)--(6x2); 
                \draw [red] (7x1)--(6x1)--(7x2) (5x1)--(6x1) (6x2)--(7x2);
                \draw[red] (3x1)--(4x1) (4x2)--(5x1) (5x2)--(6x2);
                \draw [blue] (0x1)--(1x1)--(2x1)--(3x1)--(4x2)--(5x2);
                \node (-7x1) at (-2,8) [circ];
                \node (-7x2) at (0,8) [circ];
                \draw[double] (-7x1)--(7x1) (-7x2)--(7x2);
                \draw (-7x1)--(7x2) (7x1)--(-7x2);

            % \node[anchor=west] at (0.35,3.35){\textcolor{blue}{$s_3$}};
            % \node[anchor=west] at (1.35,4.35){\textcolor{blue}{$s_4$}};
            % \node[anchor=west] at (0,2.5){\textcolor{blue}{$s_2$}};
            % \node[anchor=west] at (0,1.5){\textcolor{blue}{$s_3$}};
            % \node[anchor=west] at (0,0.5){\textcolor{blue}{$s_4$}};
                \end{ctikzpicture}
                \caption{Lower half of $P(F_4,\theta)$ }\label{F4Fig}
        \end{minipage}
    &
    \begin{minipage}[t]{0.5\textwidth}
		\centering
        \captionsetup{skip=-6pt}
        \begin{ctikzpicture}[scale=0.3]
        \node (0x1) at (0,0) [bdot];
\node (1x1) at (0,1) [bdot];
\node (2x1) at (0,2) [bdot];
\node (3x1) at (0,3) [bdot];
\node (4x1) at (-2,4) [rdot];
\node (4x2) at (1,4) [bdot];
\node (5x1) at (-1,5) [rdot];
\node (5x2) at (2,5) [bdot];

\node (6x1) at (-3,6) [rdot];
\node (6x2) at (0,6) [rdot];
\node (6x3) at (3,6) [circ];
\node (7x1) at (-5,7) [rdot];
\node (7x2) at (-2,7) [rdot];
\node (7x3) at (1,7) [circ];
\node (8x1) at (-7,8) [circ];
\node (8x2) at (-4,8) [circ];
\node (8x3) at (-2,8) [circ];
\node (8x4) at (-1,8) [circ];
\node (9x1) at (-6,9) [circ];
\node (9x2) at (-4,9) [circ];
\node (9x3) at (-3,9) [circ];
\node (9x4) at (-1,9) [circ];
\node (10x1) at (-6,10) [circ];
\node (10x2) at (-5,10) [circ];
\node (10x3) at (-4,10) [circ];
\node (10x4) at (-3,10) [circ];
\node (10x5) at (0,10) [circ];
\node (11x1) at (-6,11) [circ];
\node (11x2) at (-5,11) [circ];
\node (11x3) at (-4,11) [circ];
\node (11x4) at (-3,11) [circ];
\node (11x5) at (-2,11) [circ];
\node (12x1) at (-8,12) [circ];
\node (12x2) at (-6,12) [circ];
\node (12x3) at (-5,12) [circ];
\node (12x4) at (-4,12) [circ];
\node (12x5) at (-3,12) [circ];
\node (12x6) at (-2,12) [circ];
\node (13x1) at (-8,13) [circ];
\node (13x2) at (-7,13) [circ];
\node (13x3) at (-5,13) [circ];
\node (13x4) at (-4,13) [circ];
\node (13x5) at (-2,13) [circ];
\node (13x6) at (-1,13) [circ];
\node (14x1) at (-8,14) [circ];
\node (14x2) at (-7,14) [circ];
\node (14x3) at (-6,14) [circ];
\node (14x4) at (-4,14) [circ];
\node (14x5) at (-3,14) [circ];
\node (14x6) at (-1,14) [circ];
\node (15x1) at (-8,15) [circ];
\node (15x2) at (-7,15) [circ];
\node (15x3) at (-6,15) [circ];
\node (15x4) at (-5,15) [circ];
\node (15x5) at (-3,15) [circ];
\node (15x6) at (-1,15) [circ];
\node (16x1) at (-8,16) [circ];
\node (16x2) at (-7,16) [circ];
\node (16x3) at (-6,16) [circ];
\node (16x4) at (-5,16) [circ];
\node (16x5) at (-4,16) [circ];
\node (16x6) at (-3,16) [circ];
\node (16x7) at (-1,16) [circ];

\draw[blue] (5x2)--(4x2)--(3x1)--(2x1)--(1x1)--(0x1);
\draw [red] (4x1)--(3x1) (4x1)--(5x1)--(4x2);
\draw[red] (5x1)--(6x2)--(5x2);
\draw (5x2)--(6x3);
\draw[red] (7x1)--(6x1)--(5x1) (6x1)--(7x2)--(6x2);
\draw (6x2)--(7x3)--(6x3);
\draw (8x1)--(7x1)--(8x2)--(7x2)--(8x3) (7x2)--(8x4)--(7x3);
\draw (8x1)--(9x1)--(8x2)--(9x2)--(8x3)--(9x4)--(8x4) (8x2)--(9x3)--(8x4);
\draw (9x1)--(10x1)--(9x2)--(10x4)--(9x4)--(10x5) (9x1)--(10x2)--(9x3)--(10x4) (9x2)--(10x3);
\draw (10x1)--(11x1)--(10x3)--(11x4)--(10x4)--(11x5)--(10x5) (10x1)--(11x2)--(10x4) (10x2)--(11x2) (10x3)--(11x3);
\draw (12x1)--(11x1)--(12x2)--(11x3) (12x3)--(11x1) (12x3)--(11x2) (12x3)--(11x4) (11x2)--(12x4)--(11x5) (11x3)--(12x5)--(11x4)--(12x6)--(11x5);
\draw (12x1)--(13x1)--(12x2)--(13x3)--(12x3)--(13x4)--(12x4) (12x1)--(13x2)--(12x3) (13x3)--(12x5)--(13x5)--(12x6)--(13x6) (13x4)--(12x6);
\draw (14x1)--(13x1)--(14x2)--(13x2)--(14x3)--(13x4)--(14x4)--(13x5)--(14x6)--(13x6) (14x2)--(13x3)--(14x4)--(13x5) (14x4)--(13x4)--(14x5)--(13x6);
\draw (15x1)--(14x1)--(15x2)--(14x2)--(15x3)--(14x3)--(15x4)--(14x5)--(15x5)--(14x6)--(15x6) (15x3)--(14x4)--(15x5)--(14x5);
\draw (16x1)--(15x1)--(16x2)--(15x2)--(16x3)--(15x3)--(16x4)--(15x4)--(16x5) (16x4)--(15x5)--(16x6)--(15x6)--(16x7);

\node (17x1) at (-8,17) [circ];
\node (17x2) at (-7,17) [circ];
\node (17x3) at (-6,17) [circ];
\node (17x4) at (-5,17) [circ];
\node (17x5) at (-4,17) [circ];
\node (17x6) at (-3,17) [circ];
\node (17x7) at (-1,17) [circ];

\draw[double] (16x1)--(17x1) (16x2)--(17x2) (16x3)--(17x3) (16x4)--(17x4) (16x5)--(17x5) (16x6)--(17x6) (16x7)--(17x7);
\draw (16x1)--(17x2) (16x2)--(17x1);
\draw (16x3)--(17x2) (16x2)--(17x3);
\draw (16x3)--(17x4) (16x4)--(17x3);
\draw (16x5)--(17x4) (16x4)--(17x5);
\draw (16x4)--(17x6) (16x6)--(17x4);
\draw (16x7)--(17x6) (16x6)--(17x7);
        \end{ctikzpicture}
        \caption{Lower half of $P(E_7,\theta)$}\label{E7Fig}
    \end{minipage}
    \end{tabular}
    \end{center} 
    \end{figure}


\end{eg}
% \begin{nota} From the examples of \XXX diagrams we can see that there are many subdiagrams of the following shape in \Cref{DiaPabFig}, and we denote it as $P_{a,b}$. For example, $P(C_4,-(0111)) \cong P(D_4,-(0101))\cong P(B_4,-(0112)) \cong P_{2,1}$ as directed diagram.\end{nota}

  % \begin{figure}
  %   \begin{tabular}{m{4cm}m{9.5cm}}
  %          \begin{minipage}[t]{0.4\textwidth}
  %              %\centering
  %              \captionsetup{skip=-8pt,position=bottom}
  %              \begin{ctikzpicture}[scale=0.5]
  %  \node (0y0) at (0,0) [circ];
  %  \node (0y1) at (-1,1) [circ];
  %  \node (0y2) at (-2,2) {$\ldots$};
  %  \node (1y2) at (0,2) [circ];
  %  \node (0y3) at (-3,3) [circ];
  %  \node (1y3) at (-1,3) [circ];
   
  %  \node (0y4) at (-4,4) [circ];
  %  \node (1y4) at (-2,4) [circ];
  %  \node (2y4) at (0,4) [circ];
   
  %  \node (1y5) at (-3,5) {$\ldots$};
  %  \node (2y5) at (-1,5) [circ];
   
  %  \node (0y5) at (-2,6) [circ];
   
  %  \draw (0y1)--(0y0) (0y2)--(0y1)--(1y2) (0y3)--(0y2)--(1y3)--(1y2) (0y4)--(0y3)--(1y4)--(1y3)--(2y4) (0y4)--(1y5)--(1y4)--(2y5)--(2y4) (0y5)--(1y5) (0y5)--(2y5);
   
  %  \node[rotate = 315] at (-2.4, 1.6) {$\underbrace{\hspace{3cm}}$};
  %  \node[rotate = 225] at (-3.4, 5.4) {$\underbrace{\hspace{1.5cm}}$};
   
  %  \node[red, anchor=south east] at (-3.5, 5.35){$b$};
  %  \node[red, anchor=north east] at (-2.4, 1.6){$a$};
  %  \end{ctikzpicture}
  %  \caption{diagram $P_{a,b}$}\label{DiaPabFig}
  %          \end{minipage}
  %      &
  %      \begin{minipage}[t]{0.85\textwidth}
  %          %\centering
  %      \captionsetup{skip=-3pt,position=bottom}
  %      \begin{tabular}{m{3cm}m{0.1cm}m{3cm}m{0.1cm}m{3cm}}
  %       \begin{ctikzpicture}[scale=0.5]
  %  \node (-6x2) at (7,-5) [rdot];
  %  \node (-5x2) at (6,-4) [rdot];
   
  %  \draw[red] (-5x2)--(-6x2);
   
  %  \node (-4x2) at (5,-3) [rdot];
  %  \node (-4x3) at (7,-3) [rdot];
   
  %  \draw[red] (-4x2)--(-5x2)--(-4x3);
   
  %  \node (-3x3) at (6,-2) [rdot=left:\textcolor{red}{-(0111)} ];
   
  %  \draw[red]  (-4x2)--(-3x3)--(-4x3);
  %              \node[anchor=east] at (6.6,-4.7){\textcolor{red}{$s_2$}};
  %              \node[anchor=east] at (5.6,-3.7){\textcolor{red}{$s_{3}$}};
  %              \node[anchor=east] at (6.5,-2.8){\textcolor{red}{$s_{1}$}};
  %  \end{ctikzpicture}
  %      & 
  %       $\cong $
  %       & 
  %       \begin{ctikzpicture}[scale=0.5]
  %           \node (-6x2) at (7,-5) [rdot];
  %           \node (-5x2) at (6,-4) [rdot];
            
  %           \draw[red] (-5x2)--(-6x2);
            
  %           \node (-4x2) at (5,-3) [rdot];
  %           \node (-4x3) at (7,-3) [rdot];
            
  %           \draw[red] (-4x2)--(-5x2)--(-4x3);
            
  %           \node (-3x3) at (6,-2) [rdot=left:\textcolor{red}{-(0101)} ];
            
  %           \draw[red]  (-4x2)--(-3x3)--(-4x3);
  %                       \node[anchor=east] at (6.6,-4.7){\textcolor{red}{$s_2$}};
  %                       \node[anchor=east] at (5.6,-3.7){\textcolor{red}{$s_3$}};
  %                       \node[anchor=east] at (6.5,-2.8){\textcolor{red}{$s_1$}};
  %           \end{ctikzpicture}
  %       & 
  %       $\cong $
  %       & 
  %       \begin{ctikzpicture}[scale=0.5]
  %           \node (-6x2) at (7,-5) [rdot];
  %           \node (-5x2) at (6,-4) [rdot];
            
  %           \draw[red] (-5x2)--(-6x2);
            
  %           \node (-4x2) at (5,-3) [rdot];
  %           \node (-4x3) at (7,-3) [rdot];
            
  %           \draw[red] (-4x2)--(-5x2)--(-4x3);
            
  %           \node (-3x3) at (6,-2) [rdot=left:\textcolor{red}{-(0112)} ];
            
  %           \draw[red]  (-4x2)--(-3x3)--(-4x3);
  %                       \node[anchor=east] at (6.6,-4.7){\textcolor{red}{$s_2$}};
  %                       \node[anchor=east] at (5.6,-3.7){\textcolor{red}{$s_3$}};
  %                       \node[anchor=east] at (6.5,-2.8){\textcolor{red}{$s_1$}};
  %           \end{ctikzpicture}
  %  \end{tabular}
  %       \caption{$P(C_4,-(0111)) \cong P(D_4,-(0101))\cong P(B_4,-(0112))$}\label{AdjB5isoD5}
  %      \end{minipage}
      
  %      \end{tabular}
  %  \end{figure}

    \subsection{Chevalley--Hasse diagrams serve as Isomorphic Invariants}

\begin{prop}\label{CHDisoinv}
    If two (co)adjoint Schubert varieties are isomorphic, then they have the same \XXX\  diagrams. In other words, we have derived one direction of \Cref{Theorem 1}.
    %In other words, the \XXX\ diagram of a (co)adjoint Schubert variety $X_\beta=X(T,\beta)$ can be characterized by the structure of $X_\beta$ as an algebraic variety.
    \end{prop}
    \begin{proof}[Sketch of proof]
    The Chow groups of $X$, together with the chosen basis $\BC{[X_\alpha]}$, can be recovered from purely algebraic structures of $X_\beta$ by \Cref{ChoGroStrOfSchVar}. It suffices to give a characterization of $D_{X_\beta}$ independent of the root system. It can be easily verified that
    \begin{enumerate}[topsep=.5em]
    \item[($\dagger$)] $D=D_{X_\beta}$ is the only divisor in $\Pic(X_\beta)$ such that $D\cdot [X_\alpha]=[\pt]$ for every Schubert class $[X_\alpha]\in \CH_1(X_\beta)$.
    \end{enumerate}
    Now ($\dagger$) determines $D_{X_\beta}$ by the algebraic structure of $X_\beta$ only.
    \end{proof}



\section{Two Techniques for Embedding}\label{Techofemb}
Now, we introduce some techniques that allow us to embed some of the (co)adjoint Schubert varieties into a smaller partial flag variety $X'=G'/P'$ (minimal embedding) or larger partial flag variety $\widehat{G}/ \widehat{P} $ (folding). Note that these embeddings are isomorphisms when we restrict them to Schubert varieties.

\subsection{Minimal embedding}

To define the minimal embedding of Schubert varieties, let us first recall the support of a Weyl group element.

\begin{definition}
Let $W$ be a Weyl group generated by the simple reflections in $S$. For $w\in W$, the support $S(w)$ of $w$ is defined as the set of all simple reflections that have appeared in reduced expressions of $w$.
\end{definition}

\begin{lem}[{\cite[Lemma 4.8]{RS}}]\label{TheEmb}
Let $G'$ be the reductive subgroup of $P_{S(w)}$ with Weyl group $W'= W_{S(w)}$ and $P'_{I'}= G' \cap P_I$ as the parabolic subgroup of $G'$, corresponding to $I' :=I\cap S(w)$. Set $X' :=G'/P'$. Note that $w \in W'^{I'}$.

Now $X'$  is again a partial flag variety, whose Dynkin diagram is the full subdiagram $D'$ of $D$ with underlying set $D\cap S(w)$. Let $X'_w$ be the Schubert variety in $X'$ induced by $w\in W'^{I'}$. Then the natural embedding $X'\hookrightarrow  X$ induces an isomorphism $X'_w\cong X_w$.
\end{lem}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
 
\begin{eg}\label{C5R-10000Sol}\label{eg:E7F42}
    In the coadjoint partial flag variety of type $F_4$, take $\beta=-(1110)$. Then $P_\beta$ is a path of length $5$ (see \Cref{F4FigNew}; it is exactly the subdiagram marked in blue in \Cref{F4Fig}). We have
   % \begin{align*}
%    \centerline{k
   \[ \beta=s_4s_3s_2s_3s_4(-\theta), \]
   % \end{align*}
    with the simple reflections marked in \Cref{F4FigNew}. By \Cref{TheEmb} we can embed $X_\beta$ as $X_w$ in $X'$, a partial flag variety with $D'=\BC{\alpha_2,\alpha_3,\alpha_4}$ of type $C_3$ and $I'=D'\setminus\BC{\alpha_4}$. This procedure is illustrated in \Cref{C5R-10000SolFig}. As $X_\beta\subseteq X'=C_3/P_1\simeq\PP^5$, it follows that $X_\beta\simeq \PP^5$ as $\dim X_\beta=5$. Moreover, $[X_{-(1111)}]$ is an effective Cartier divisor in $\PP^5$ which is a generator of $\CH_4(\PP^5)\cong \CaDiv(\PP^5)$, so we must have $X_{-(1111)}\simeq \PP^4$. Similarly, $X_{-(1121)}\simeq \PP^3$ and so on.

    Similarly, we can show that the blue part in \Cref{E7Fig} also corresponds to some Schubert varieties in $E_7$ isomorphic to projective spaces. This fact presents isomorphisms between Schubert varieties from coadjoint types $F_4$ and $E_7$, as promised in \Cref{eg:F4E7}.
    \end{eg}
    
\begin{figure}[ht]
 \begin{tabular}{p{5cm}p{5cm}}
        \begin{minipage}[t]{0.5\textwidth}
            %\centering
            \captionsetup{skip=-8pt,position=bottom}
            \begin{ctikzpicture}
    \node (1) at (0,0) [circ=left:$\alpha_1$];
    \node (2) at (1,0) [rcirc=below:$\textcolor{red}{\alpha_2}$];
    \node (3) at (2,0) [rcirc=below:$\textcolor{red}{\alpha_3}$];
    \node (4) at (3,0) [rdot=right:$\textcolor{red}{\alpha_4}$];
    \draw (1)--(2);
    \draw[red] (3)--(4);
    \draw[double, red] (2)--(3);
    \draw[red] (1.572,0)--(1.4,0.1);
    \draw[red] (1.572,0)--(1.4,-0.1);
    \end{ctikzpicture}
    \caption{taking support of $w$ in $F_4$}\label{C5R-10000SolFig}
        \end{minipage}
    &
    \begin{minipage}[t]{0.7\textwidth}
		%\centering
    \captionsetup{skip=-8pt,position=bottom}
    \begin{ctikzpicture}[scale=0.75]
                \node (0x1) at (5,0) [bdot=above:\tiny\textcolor{blue}{-(1232)}];
                \node (1x1) at (4,0) [bdot=above:\tiny\textcolor{blue}{-(1231)}];
                \node (2x1) at (3,0) [bdot=above:\tiny\textcolor{blue}{-(1221)}];
                \node (3x1) at (2,0) [bdot=above:\tiny\textcolor{blue}{-(1121)}];
                \node (4x2) at (1,1) [bdot=left: $ \mathbb{P}^4\cong X_{-(1111)}$ ];
                \node (5x2) at (0,2) [bdot=left: $ \mathbb{P}^5\cong X_{-(1110)}$];
                \draw [blue] (0x1)--(1x1)--(2x1)--(3x1)--(4x2)--(5x2);
                \node[anchor=east] at (2,0){$ \mathbb{P}^3\cong X_{-(1121)}$ };
            \node[anchor=east] at (0.65,1.35){\textcolor{blue}{$s_4$}};
            \node[anchor=east] at (1.65,0.35){\textcolor{blue}{$s_3$}};
            \node[anchor=north] at (2.5,0){\textcolor{blue}{$s_2$}};
            \node[anchor=north] at (3.5,0){\textcolor{blue}{$s_3$}};
            \node[anchor=north] at (4.5,0){\textcolor{blue}{$s_4$}};
                \end{ctikzpicture}
                \caption{diagram $P(F_4,-(1110))$}\label{F4FigNew}
    \end{minipage}
   
    \end{tabular}
\end{figure}
 %     \Cref{C5R-10000Sol} can be generalized to any \XXX diagrams, we conclude that:
 %    \begin{prop}\label{AppOfTheEmb1}
 %    Let $X$ be a (co)adjoint partial flag variety and $X_\beta$ is the Schubert variety in $X$ indexed by $\beta$. If $P_\beta\simeq P_{a,0}$ is a path of length $a$, then $X_\beta\simeq \PP^a$.
 %    \end{prop}
 %    Next, we can apply this method into further cases. For example, in \Cref{AdjB5isoD5}, we discover that $P(C_4,-(0111)) \cong P(D_4,-(0101))\cong P(B_4,-(0112))$. Now we apply \Cref{TheEmb} to explicitly show that they are isomorphic. 
 % \begin{eg}\label{egofminbed}
 % From the root associated to the Schubert varieties, we can determine their corresponding Weyl group element directly as in \Cref{AdjB5isoD5}. That is $-(0111)=s_1 s_3 s_2(-\theta_{C})$ in $C_4$, $-(0101)=s_1 s_3 s_2(-\theta_{D})$ in $D_4$, $-(0112)=s_1 s_3 s_2(-\theta_{B})$ in $B_4$. By taking the support of these Weyl group elements, as is visualized in the Dynkin diagram (\Cref{C4-0111supp}, \Cref{D4-0101supp}, \Cref{B4-0112supp}) and applying the minimal embedding, we can identify these (co)adjoint Schubert varieties as the same cominuscule Schubert variety in the partial flag variety $X'=A_3/P_2\simeq \Gr(2,4)$, which corresponds to the Weyl group element $s_1 s_3 s_2$ in type A. Hence, we show that $X(C_4,-(0111)) \cong X(D_4,-(0101))\cong X(B_4,-(0112))$ isomorphic as Schubert varietries.
 %    \end{eg}
 %     \begin{figure}
 %    \begin{center}
 %    \begin{tabular}{p{5cm}p{5cm}p{5cm}}
 %        \begin{minipage}[t]{0.38\textwidth}
% 		\centering
 %        \captionsetup{skip=-6pt,position=bottom}
 %        \begin{ctikzpicture}
 %        \node (2) at (2,0) [rcirc=below:\textcolor{red}{$\alpha_1$}];
 %        \node (3) at (3,0) [rdot=below:\textcolor{red}{$\alpha_2$}];
 %        \node (6) at (4,0) [rcirc=below:\textcolor{red}{$\alpha_3$}];
 %        \node (7) at (5,0) [circ=below:$\alpha_4$];
 %        \draw[red] (2)--(3)--(6);
 %        \draw[double] (7)--(6);
 %        \draw (4.428,0)--(4.6,0.1);
 %        \draw (4.428,0)--(4.6,-0.1);
 %        \end{ctikzpicture}
 %    \caption{taking support of $w_1=s_1 s_3 s_2$ in $C_4$}\label{C4-0111supp}
% 	\end{minipage}
 %    &
 %        \begin{minipage}[t]{0.38\textwidth}
 %            \centering
 %            \captionsetup{skip=-6pt,position=bottom}
 %             \begin{ctikzpicture}
 %    \node (2) at (2,0) [rcirc=below:$\textcolor{red}{\alpha_1}$];
 %    \node (4) at (3,0) [rdot=below:$\textcolor{red}{\alpha_2}$];
 %    \node (5) at (4,0.5) [circ=right:$\alpha_3$];
 %    \node (6) at (4,-0.5) [rcirc=right:$\textcolor{red}{\alpha_4}$];
 %    \draw [red](2)--(4);
 %    \draw [red](4)--(6);
 %    \draw (4)--(5);
 %    \end{ctikzpicture}
 %    \caption{taking support of $w_2=s_1 s_3 s_2$ in $D_4$}\label{D4-0101supp}
% 	\end{minipage}
 %    &
 %        \begin{minipage}[t]{0.38\textwidth}
% 		\centering
 %        \captionsetup{skip=-6pt,position=bottom}
 %        \begin{ctikzpicture}
 %        \node (2) at (2,0) [rcirc=below:\textcolor{red}{$\alpha_1$}];
 %        \node (3) at (3,0) [rdot=below:\textcolor{red}{$\alpha_2$}];
 %        \node (6) at (4,0) [rcirc=below:\textcolor{red}{$\alpha_3$}];
 %        \node (7) at (5,0) [circ=below:$\alpha_4$];
 %        \draw[red] (2)--(3)--(6);
 %        \draw[double] (7)--(6);
 %        \draw (4.572,0)--(4.4,0.1) (4.572,0)--(4.4,-0.1) ;
 %        \end{ctikzpicture}
 %    \caption{taking support of $w_3=s_1 s_3 s_2$ in $B_4$}\label{B4-0112supp}
% 	\end{minipage}
 %    \end{tabular}
 %    \end{center} 
 %    \end{figure}
    
\subsection{Folding}

\begin{table}[h]
    \centering
    \begin{tabular}{cc}
      \hline
      $G/P$ &  $\widehat{G}/\widehat{P}$ \\ \hline
      $B_n/P_1=Q_{2n-1}$ & $D_{n+1}/P_1=Q_{2n}$ \\
      $C_n/P_2=IG(2,2n)$ & $A_{2n-1}/P_2=G(2,2n)$ \\
      $F_4/P_4$ & $E_6/P_1$ \\
      $G_2/P_1=Q_5$ & $D_4/P_1=Q_6$ \\
      \hline
      
    \end{tabular}\caption{Hyperplane sections}
      \label{foldhyp}
\end{table}
The main reference of this type of embedding is \cite{PS}. Firstly, we introduce the folding operation between linear algebraic groups. Let $\widehat{G} $ be a simply connected simple algebraic group whose Dynkin diagram is as in the second column of \Cref{foldhyp}. 
Let $\sigma \in \Aut (\widehat{G} )$ be the automorphism induced by the symmetry of the Dynkin diagram of $\widehat{G}$. Define a closed subgroup $G\coloneqq {\widehat{G} }^{\sigma}$ as the fixed locus of $\sigma$; other standard subgroups of $G$ are defined as $T:=G\cap \widehat{T} $, $B:=G\cap \widehat{B} $,  $P:=G\cap \widehat{P} $. The embedding of maximal tori gives rise to a surjective map of root systems $\pi:\Phi_{\widehat{G} }\twoheadrightarrow \Phi_{G}$.
There is a closed embedding $j$ from $G/P$ to $\widehat{G}/\widehat{P}$.
Moreover, $G/P$ can be identified as hyperplane sections of $\widehat{G}/\widehat{P}$. For more details, see \cite[Lemma 4.1]{PS}.

The map of root systems $\pi$ induces an embedding $\iota:W_G\to W_{\widehat{G}}$ of Weyl groups, by sending each $s_\alpha$ to $\prod_{\widehat{\alpha}\in\pi^{-1}(\alpha)}s_{\widehat{\alpha}}$. The product is well-defined since for every $\alpha\in\Phi$ and every $\widehat\alpha, \widehat\alpha'\in\pi^{-1}(\alpha)$, the reflections $s_{\widehat\alpha}$ and $s_{\widehat\alpha'}$ commute.

The lemma below serves as a useful tool for verifying isomorphisms between Schubert varieties when the minimal embedding method fails.

\begin{lem}[{\cite[Lemma 4.5]{PS}}]\label{foldiso}
    For $w\in W^P_G$ with $2 \ell (w)\leq \dim G/P$, the closed embedding gives rise to a isomorphism between Schubert varieties $j((G/P)_{w})=(\widehat{G}/\widehat{P}  )_{w_{\ast }}$.
\end{lem}

\noindent
Now let us witness the power of \Cref{foldiso} by revisiting examples in \Cref{F4Fig,E7Fig}.



\begin{eg}\label{foldmapF4toE6}\label{eg:E7F43}
The folding of Dynkin diagrams from $E_6$ to $F_4$ is illustrated in \Cref{foldF4toE6}.
        \begin{figure}[htbp]
            \centering
            \captionsetup{skip=-3pt,position=bottom}
            \begin{minipage}[t]{0.6\textwidth}
                \centering
                \begin{minipage}[t]{0.45\textwidth}
                    \centering
                    \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.9]
                \node (1) at (1,0) [circ=below:$2^{\prime }$];
                \node (2) at (2,0) [circ=below:$4^{\prime }$];
                \node (3) at (3,0.5) [circ=above:$3^{\prime}$];
                \node (6) at (3,-0.5) [circ=below:$5^{\prime}$];
                \node (4) at (4,1) [circ=above:$1^{\prime}$];
                \node (7) at (4,-1) [circ=below:$6^{\prime }$];
                \draw (1)--(2)--(3)--(4);
                \draw (2)--(6)--(7);
                \end{tikzpicture}
                \end{minipage}
                $\longrightarrow $
                \begin{minipage}[t]{0.45\textwidth}
                  \centering
                  \begin{tikzpicture}[baseline=(current bounding box.center), scale=0.9]
                  \node (1) at (0,0) [circ=below:$1 $];
                  \node (2) at (1,0) [circ=below:$2 $];
                  \node (3) at (2,0) [circ=below:$3 $];
                  \node (4) at (3,0) [dot=below:$4 $];
                  \draw (1)--(2) (3)--(4);
                  \draw[double] (2)--(3);
                  \end{tikzpicture}
              
              \end{minipage}
            \end{minipage}\raisebox{2em}{
            \begin{minipage}[t]{0.3\textwidth}
                \centering
                $\iota : W_{F_4}\hookrightarrow  W_{E_6} $
        
                $s_1\mapsto s_2^{\prime}$
        
                $s_2\mapsto s_4^{\prime}$
        
                $s_3\mapsto s_3^{\prime}s_5^{\prime}$
        
                $s_4\mapsto s_1^{\prime}s_6^{\prime} $
            \end{minipage}}
            \caption{Folding from $E_6$ to $F_4$}
            \label{foldF4toE6}
        \end{figure}
By applying \Cref{foldiso}, the Schubert varieties of coadjoint type $F_4$ indexed by the red vertices in \Cref{F4Fig} are isomorphic to some Schubert varieties in the \textit{cominuscule} partial flag variety $E_6/P_1$. On the other hand, by applying \Cref{TheEmb}, the Schubert varieties of type $E_7$ indexed by the red vertices in \Cref{E7Fig} are isomorphic to the same set of Schubert varieties in $E_6/P_1$. Therefore we have shown the result promised in \Cref{eg:F4E7}. 
\end{eg}

% \begin{eg}\label{caloftypeC}
%     We claim that  all coadjoint Schubert varieties from type $C_n$ (note that n can vary here) with a fixed Chevalley--Hasse diagram $P_{a,b}$ $(a,b\in \mathbb{N} , b\leq a)$ are isomorphic.

%     Denote that for isotropic grassmannian the Schubert Varieties are specified as $S_{p_1,p_2}(2,2n)\defeq \brak{\Sigma\in \IG(2,2n)\mid \dim \Sigma \cap \spn(e_1,\ldots, e_{p_1})\geq 1,\ \Sigma\subset \spn(e_1,\ldots,e_{p_2})}$, 
% where $1\leq p_1< p_2\leq 2n,\ p_1+p_2\neq 2n+1$. Similarly for grassmannian we also have $G_{p_1,p_2}(2,2n)\defeq \brak{\Sigma\in \Gr(2,2n)|\dim \Sigma \cap \spn(e_1,\ldots, e_{p_1})\geq 1,\ \Sigma\subset \spn(e_1,\ldots,e_{p_2})}$. If $X_\beta\subseteq X(C_n,\theta)$ corresponds to the diagram $P_\beta\simeq P_{a,b}$ with $a+b\le 2n-3$, applying the embedding of folding, we have $X_\beta = S_{b+1,a+2}(2,2n)\simeq G_{b+1,a+2}(2,2n)$.
% Note that $G_{b+1,a+2}=G_{b+1,a+2}(2,2n)$ is actually independent of $n$ by definition, which prove our claim.

% \end{eg}


% \begin{eg}\label{caloftypeEandF}
%     In \Cref{F4Fig} and \Cref{E7Fig}, if $\alpha$ is among the short roots which is marked in red in \Cref{F4Fig}, then the diagram $P(F_4,\alpha)$ coincides with some subdiagram of type $E_7$ (see \Cref{E7Fig}). Applying the method of folding, it turns out that the corresponding Schubert varieties are isomorphic.

%     For example, if we take $\alpha _1=-(0110)=s_4^{\prime\prime}s_3^{\prime\prime}s_1^{\prime\prime}s_2^{\prime\prime}s_3^{\prime\prime}s_4^{\prime\prime}(-\theta )\in \phi _{F_4}$ and $\alpha _2=-(1123211)=s_6^{\prime}s_5^{\prime}s_2^{\prime}s_4^{\prime}s_3^{\prime}s_1^{\prime}(-\theta )\in \phi _{E_7}$. Here $\{ s_i^{\prime\prime}\}, \{s^{\prime}_{j}\} , \{s_k\} $ denote the simple reflections in the root systems of type $F_4, E_7$ and $E_6$. For $X(F_4, \alpha _1)$, we can embed it into flag variety of $E_6/P_1$ (See \Cref{foldhyp}) through folding. Under the map of \Cref{foldmapF4toE6}, we can directly compute that $\iota (s_{\alpha _1})=(s_1s_6)(s_3s_5)(s_2)(s_4)(s_3s_5)(s_1s_6)$. And $X_{\alpha_1}\cong X^{\prime}_{\iota (s_{\alpha _1})}$ by \Cref{foldiso}. For  $X(E_7, \alpha _2)$, we can also embed it into flag variety of $E_6/P_1$ through minimal embedding. In this case, we also use $\iota(s_{\alpha _2})$ to denote the corresponding Weyl element in $E_6$, which is simply the restriction $ \iota (s_{\alpha _2})=(s_6)(s_5)(s_2)(s_4)(s_3)(s_1)$. We have $X_{\alpha_2}\cong X^{\prime}_{\iota (s_{\alpha _2})}$.

%     Then, if we consider ${\iota (s_{\alpha _1})}^{-1}\iota (s_{\alpha _2}) $. It turn out to be $s_6s_5s_4s_3\in P_1$. That is,  ${\iota (s_{\alpha _1})}$ and $\iota (s_{\alpha _2}) $ belong to the same class of $W^{P_1}_{E_6}$, which mean they correspond to same Schubert variety in partial flag Variety of $E_6/P_1$. As a result, we verify that $X(F_4, \alpha _1)\simeq X(E_7, \alpha _2) $.
% \end{eg}

\section{Proof Sketch of \texorpdfstring{\Cref{Theorem 1}}{Theorem 1}}\label{CalDisofMAITHE}
We give a brief sketch of our strategy proving the other direction of \Cref{Theorem 1}:
\begin{enumerate}[itemsep=0pt,topsep=.1pt]
\item[(1)] We classify the Schubert varieties $X(T,\beta)$ whose \XXX\ diagrams are unique, up to at most an automorphism of the Dynkin diagram. For example, the  (co)adjoint Schubert varieties corresponding to positive roots in most Cartan types are unique in their isomorphic types.
\item[(2)] We use the two techniques for embedding to give the isomorphism between two Schubert varieties whose \XXX\ diagrams fall into the same isomorphic class. \Cref{eg:E7F42} and \Cref{eg:E7F43} provide a good demonstration.
\end{enumerate}


\acknowledgements{
The authors express their sincere gratitude to Weihong Xu for introducing them to partial flag varieties, Schubert varieties and intersection theory. The authors thank Vladimiro Benedetti for pointing out the reference on the embeddings induced by foldings. 

The authors also thank Yibo Gao for organizing the PACE (PKU Algebraic Combinatorics Experience) program, which allowed for active learning and heated discussions. Without this REU experience, this project would not have been possible.

This project was partially supported by NSFC grant 12426507. }




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