%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{147}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{mtheorem}[theorem]{Main Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}


%% define your title in the usual way
\title[On the Classification of Schubert Varieties in Partial Flag Varieties]{On the Classification of Schubert Varieties in Partial Flag 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[Yanjun Chen]{\texorpdfstring{Yanjun Chen\thanks{\href{mailto:121090063@link.cuhk.edu.cn}{121090063@link.cuhk.edu.cn}. Partially supported by NSFC grant 12426507.}\addressmark{1}}{Yanjun Chen}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
    We generalize the classification of isomorphism classes of Schubert varieties in complete flag varieties $G/B$ (E. Richmond and W. Slofstra, 2021) to a class of partial flag varieties $G/P$. 
    In particular, we classify all Schubert varieties in $G/P$ where $P$ is a minimal parabolic subgroup, and all Schubert surfaces which are two-dimensional Schubert varieties.
    }

%% 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 varieties, partial flag varieties, Schubert surfaces, cohomology}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{sample.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\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}
Schubert varieties form an extensively studied class of algebraic varieties whose properties are often characterized by combinatorics. The isomorphism problem for Schubert varieties, first raised by Develin, Martin, and Reiner in \cite{DMR07}, asks for a classification of all Schubert varieties up to algebraic isomorphism. In the same paper, they classified a class of smooth Schubert varieties in type $A$ partial flag varieties. Using Cartan equivalence, Richmond and Slofstra solved this problem for Schubert varieties in complete flag varieties in \cite{RS21}. On the other hand, Richmond, Țarigradschi, and Xu solved this problem for cominuscule Schubert varieties in \cite{RTX24} using labeled posets.

To describe our results, we set the following notations. We only consider Schubert varieties of finite types over $\mathbb{C}$ for simplicity. Let $G$ be a complex reductive Lie group, $T$ a maximal torus, and $B$ a Borel subgroup containing $T$. Then a root system is defined, with the correspondent set of simple reflections $S$, Cartan matrix $A = (a_{st})_{(s, t) \in S^2}$, and the Weyl group $W$. The pair $(W, S)$ forms a Coxeter system. A standard parabolic subgroup $P$ contains $B$ corresponding to a subset $I$ of $S$. We denote the flag variety $G/P$ by $X(A, I)$. The subset $I$ generates a subgroup $W_I \subset W$. In every coset $W/W_I$, we take the elements with minimal length and form a set $W^I$. Then a Schubert variety is the closure of the $B$ orbit $BwP/P$ for some $w \in W^I$, and we write this variety as $X(w, A, I)$ since it is uniquely determined by the triple $(w, A, I)$. We denote the \emph{support} of $w$ to be
\begin{equation*}
S(w) = \{ s \in S|s \leq w \}.
\end{equation*}

In our notations, Richmond and Slofstra classified Schubert varieties of the form $X(w, A, \varnothing)$. Motivated by their methods, we extend their work.

\begin{mtheorem}[See \Cref{not}, \Cref{neces}, and \Cref{suff}]\label{MT}
Let
\begin{align*}
A = (a_{st})_{(s, t) \in S^2} && \mbox{and} && A' = (a'_{s't'})_{(s', t') \in S'^2}
\end{align*}
be two Cartan matrices with associated Weyl groups $W$ and $W'$, and sets of simple reflections $S$ and $S'$, respectively. Let $I \subset S$ and $I' \subset S'$. Take $w \in W^I$ and $w' \in W'^{I'}$. Assume that $|S(w) \cap I| \leq 1$ and $|S(w') \cap I'| \leq 1$. Then the following are equivalent:
\begin{enumerate}
\item the Schubert varieties $X(w, A, I)$ and $X(w', A', I')$ are algebraically isomorphic;
\item there exists a bijection $\tau: S(w) \to S(w')$ sending $S(w) \cap I$ to $S(w') \cap I'$, such that:
\begin{enumerate}
\item[(a)] for some reduced word $w = s_1 \cdots s_k$, $w' = \tau(s_1) \cdots \tau(s_k)$ is also a reduced word;
\item[(b)] for any $t_1, t_2 \in S(w)$, $a_{t_1, t_2} = a'_{\tau(t_1), \tau(t_2)}$ whenever $t_1 t_2 \leq w$.
\end{enumerate}
\end{enumerate}
In particular, if $X(w, A, I)$ and $X(w', A', I')$ are isomorphic, then $|S(w) \cap I| = |S(w') \cap I'|$.
\end{mtheorem}

Since we allow the subset $I$ to satisfy the condition $|S(w) \cap I| \leq 1$, a larger class of Schubert varieties is classified. It also completely solves the isomorphism problem of Schubert varieties in flag varieties corresponding to minimal parabolic subgroups, in which case, $|I| = 1$.

Schubert varieties of dimension zero and one are isomorphic to a point and the projective line $\mathbb{P}^1$, respectively. Applying the main theorem, we can easily classify the Schubert varieties of dimension two without explicit computation.

The last statement of our main theorem gives a necessary condition for the isomorphism of Schubert varieties, which can be slightly generalized; see \Cref{not}.

For two Schubert varieties $X(w, A, I)$ and $X(w', A', I')$, if $|S(w)| = |S(w')|$, then we say these two Schubert varieties are \emph{equally supported}. The way we prove our criterion enables us to attain a sufficient condition (\Cref{suff}) to determine whether the given equally supported pair is an isomorphism.

It is also interesting to consider whether our sufficient condition is necessary. It is still unknown, and we will leave it for future study.





\subsection{Outline of the Abstract}
In \Cref{EC}, we show some illustrative examples of \Cref{MT}. At the end of this section, we show the classification of Schubert surfaces; see \Cref{cla}. Then we discuss preliminaries in \Cref{pre}. The following two sections give a type-independent proof for our main theorem. In \Cref{co}, we derive some necessary conditions of the isomorphism of Schubert varieties, and we prove that condition 1 implies condition 2 of the main theorem (\Cref{neces}). Our primary tool is the cohomology ring whose cup product can be computed by Chevalley’s formula in \cite[Lemma 8.1]{FW01}. We need more explicit computation and discussion than what was done in \cite{RS21}. Next, in \Cref{con iso}, we obtain a practical sufficient condition for equally supported Schubert varieties (\Cref{suff}), and, as a corollary, we prove that condition 2 implies condition 1 of the main theorem, finishing the proof of the main theorem.







\section{Classification of Schubert Surfaces}\label{EC}
To illustrate our context, we provide several examples of Schubert varieties. In the following examples, we will use $s_i$ to denote the simple reflection corresponding to the $i$-th row or column of the Cartan matrix.

\begin{example}
Considering the three Schubert varieties
\begin{align*}
X \left( s_4 s_3 s_2 s_1, \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{pmatrix}, \{ s_4 \} \right), 
X \left( s_1 s_2 s_3 s_4, \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -2 \\ 0 & 0 & -1 & 2 \end{pmatrix}, \{ s_4 \} \right), \end{align*}\begin{align*}
X \left( s_1 s_2 s_3 s_4, \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -2 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{pmatrix}, \{ s_4 \} \right),
\end{align*}
of types $A_4$, $B_4$, and $F_4$, respectively. The lower triangular parts of three Cartan matrices are the same, so by the main theorem, they are isomorphic.
\end{example}

We call a two-dimensional Schubert variety a \emph{Schubert surface}. For $n = 1, 2, 3$, we let $M_n$ denote the Cartan matrix
\begin{equation*}
M_n = \begin{pmatrix} 2 & -1 \\ -n & 2 \end{pmatrix}.
\end{equation*}
In other words, $M_1$, $M_2$, and $M_3$ are the Cartan matrices of type $A_2$, $B_2$, and $G_2$, respectively. 

\begin{example}\label{12}
Consider the Schubert surfaces $X(s_1 s_2, M_n, \{ s_1 \})$. Since the entries $a_{12} = -1$ of Cartan matrices are equal, by the main theorem, these three surfaces are isomorphic. Taking $n = 1$, $X(s_1 s_2, M_1, \{ s_1 \})$ is a Schubert surface in the flag variety $Gr(1, 3) = \mathbb{P}^2$, which is also two dimensional. It follows that we have $
X(s_1 s_2, M_n, \{ s_1 \})$ is isomorphic to $\mathbb{P}^2$. Similarly, the Schubert surfaces $X(s_1 s_2, M_n, \varnothing)$ are isomorphic. By \cite[Example 1.1]{RS21}, they are isomorphic to the \emph{first Hirzebruch surface} $\Sigma_1$.
\end{example}

\begin{example}\label{21,1}
Consider the Schubert surfaces $X(s_2 s_1, M_n, \{ s_2 \})$. The corresponding entries $a_{21} = -n$, so these three varieties are \emph{not} isomorphic to each other. Note that geometrically, $X(s_2 s_1, M_1, \{ s_2 \})$ is the same as $X(s_1 s_2, M_1, \{ s_1 \})$, so it is isomorphic to $\mathbb{P}^2$. On the other hand, it can be shown that $X(s_2 s_1, M_2, \{ s_2 \})$ is the cone over a smooth conic.
\end{example}

\begin{example}\label{21,0}
Consider the Schubert surfaces $X(s_2 s_1, M_n, \varnothing)$. The corresponding entries $a_{21} = -n$, so these four surfaces are \emph{not} isomorphic to each other. As mentioned in \cite[Example 1.1]{RS21}, $X(s_2 s_1, M_n, \varnothing)$ is isomorphic to the \emph{$n$-th Hirzebruch surface} $\Sigma_n$.
\end{example}

\begin{example}\label{*}
Consider the Schubert surface $X(s_1 s_2, 2I_2, \varnothing)$, where $I_2$ is the $2 \times 2$ identity matrix. The corresponding entry $a_{12} = 0$, so it differs from any Schubert surfaces mentioned above. The Cartan matrix is of type $A_1 \times A_1$, so the flag variety can be identified with $\mathbb{P}^1 \times \mathbb{P}^1$. Comparing the dimensions, we conclude that $X(s_1 s_2, 2I_2, \varnothing)$ is $\mathbb{P}^1 \times \mathbb{P}^1$.
\end{example}

In fact, the Schubert surfaces in the above four examples are all possible Schubert surfaces. To be more precise, we have the following theorem.

\begin{theorem}[Classification of Schubert Surfaces]\label{cla}
There are seven isomorphism classes of Schubert surfaces. Precisely, any Schubert surface is isomorphic to one and exactly one of the following varieties:
\begin{enumerate}
\item product of projective lines $\mathbb{P}^1 \times \mathbb{P}^1$,
\item projective plane $\mathbb{P}^2$,
\item the $n$-th Hirzebruch surface $\Sigma_n$ for $n = 1, 2, 3$,
\item the cone over a smooth conic, and
\item the variety $X(s_2 s_1, M_3, \{ s_2 \})$.
\end{enumerate}
\end{theorem}

\Cref{min emb} (or \cite[Lemma 4.8]{RS16}) implies that every Schubert surface is isomorphic to a Schubert surface lying in a partial flag variety of rank two. Since a root system of rank two have type $A_1 \times A_1$, $A_2$, $B_2$, or $G_2$, Examples \ref{12}, \ref{21,1}, \ref{21,0}, and \ref{*} exhaust all possibilities of isomorphism classes of Schubert classes. It follows that any Schubert surface is isomorphic to one of the seven surfaces listed above. Applying \Cref{MT}, these seven surfaces are not isomorphic to each other.







\section{Preliminaries}\label{pre}
Given a complex reductive group $G$ with a maximal torus $T$ and a Borel subgroup $B$ containing $T$, a root system $R$ is determined. Let $R = R^+ \cup R^-$ be the choice of positive/negative roots determined by $B$, and $\Delta$ the set of simple roots. With a choice of ordered for $\Delta$, the root system $R$ defines a Cartan matrix
\begin{align*}
A = (a_{st})_{(s,t) \in S^2}, && \mbox{where } a_{st} = \langle \alpha_s^\vee, \alpha_t \rangle = \frac{2 \langle \alpha_s, \alpha_t \rangle}{\langle \alpha_s, \alpha_s \rangle}.
\end{align*}
Any positive root $\alpha$ defines a reflection $s_\alpha$; conversely, a reflection $s$ defines a positive root $\alpha_s$. The set of simple reflections $S$ generates the Weyl group $W$, which can be identified with $W = N(T)/T$. The pair $(W, S)$ is a Coxeter system. For any Weyl group element $w \in W$, we denote
\begin{equation*}
S(w) = \{ s \in S|s \leq w \}
\end{equation*}
to be the \emph{support} of $w$. If $S(w) = S$, then we say that $w$ is \emph{fully supported}. $w$ can be written as a product of simple reflections, and if $w = s_1 \cdots s_k$ with $k$ minimal, then we say the sequence $(s_1, \cdots, s_k)$ (or the product $s_1 \cdots s_k$) to be a \emph{reduced word} and denote the set of reduced words to be $RW(w)$. To obtain a reduced word from an arbitrary word, we only need to delete an even number of simple reflections. The \emph{Coxeter length function} $\ell: W \to \mathbb{Z}_{\geq 0}$ is defined to be the length of any reduced word. There is a unique longest element, denoted by $w_0$. We define the \emph{right descent set} of $w \in W$ to be 
\begin{equation*}
D_R(w) = \{ s \in S|\ell(us) = \ell(u) - 1 \}.
\end{equation*}
In other words, $D_R(w)$ is the set of simple reflections $s$ that are the rightmost elements in some reduced words of $w$.



\subsection{The Weyl Group and the Root System}
A standard parabolic subgroup $P$ corresponds to a subset $I \subset S$. We let the subgroup $W_I \subset W$ be generated by $I$, and the set $W^I$ to be the set of representatives of $W/W_I$ with minimal length. It is easy to see that a simple reflection $s \in D_R(w)$ if and only if $w \notin W^{\{ s \}}$. For $w \in W$, We let $w^{\min} \in W^I$ be the minimal length representative in the coset $wW_I$. On the other hand, $I$ defines a subset of simple roots $\Delta_I$. We write the root subsystem spanned by $\Delta_I$ as $R_I$.

On the Weyl group $W$ (or, more generally, a Coxeter group), there is a partial order called the \emph{Bruhat order}. For simplicity, we write $[u, v]^I = [u, v] \cap W^I$. \cite[Proposition 2.5.1]{BB05} shows that the map $w \mapsto w^{\min}$ is order-preserving.

We define the \emph{inversion set} of $w \in W$ to be
\begin{equation*}
I(w) = R^+ \cap w(R^-).
\end{equation*}
The following statement is part of \cite[Lemma 1.6]{Hum92}.

\begin{lemma}\label{root}
Let $w \in W$ and $s \in S$. Then $\ell(sw) > \ell(w)$ if and only if $w^{-1}(\alpha_s) \in R^+$. Similarly, we have $\ell(ws) > \ell(w)$ if and only if $w(\alpha_s) \in R^+$. In particular, if $w \in W^I$, then $I(w^{-1}) \subset R^+ \setminus R_I^+$.
\end{lemma}



\subsection{Lie Algebras}
Denote the Lie algebra of $G$ and $T$ as $\mathfrak{g}$ and its Cartan subalgebra as $\mathfrak{h}$, respectively. Each coroot $\alpha^\vee = \frac{2 \alpha}{\langle \alpha, \alpha \rangle}$ can be identify as an element $h_\alpha$ in $\mathfrak{h}$, and $\{ h_\alpha \}_{\alpha \in \Delta}$ is a basis of $\mathfrak{h}$. We denote the root space corresponding to $\alpha$ as $\mathfrak{g}_\alpha$. Then we have the root space decomposition $\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in R} \mathfrak{g}_\alpha$. We also denote the nilpotent Lie algebras $\mathfrak{n}^\pm = \bigoplus_{\alpha \in R^\pm} \mathfrak{g}_\alpha$, and the Borel subalgebra $\mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}^+$. Write $e_\alpha$ and $f_\alpha$ to be nonzero elements in $\mathfrak{g}_\alpha$ and $\mathfrak{g}_{-\alpha}$ for any positive root $\alpha$, respectively. Then $\{ e_\alpha \}_{\alpha \in \Delta}$ and $\{ f_\alpha \}_{\alpha \in \Delta}$ generate $\mathfrak{n}^+$ and $\mathfrak{n}^-$, respectively. For $w \in W$, we denote $\mathfrak{n}^+_w = \bigoplus_{\alpha \in I(w)} \mathfrak{g}_\alpha$, which is a nilpotent Lie algebra.

Let $\lambda$ be a dominant integral weight. Then there is a unique irreducible $\mathfrak{g}$-module $L_\lambda$ with the highest weight $\lambda$. It can be constructed in the following way. Let $\mathbb{C}_\lambda$ be a one-dimensional complex vector space spanned by $\omega = \omega_\lambda$, whose $\mathfrak{b}$-module structure is given by
\begin{align*}
h \omega & = \lambda(h) \omega, && h \in \mathfrak{h}, \\
e \omega & = 0, && e \in \mathfrak{n}^+.
\end{align*}
Then the \emph{Verma module} with highest weight $\lambda$ is
\begin{equation*}
M_\lambda = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda,
\end{equation*}
where $U(\mathfrak{g}) $ and $U(\mathfrak{b})$ are the universal enveloping algebras of $\mathfrak{g}$ and $\mathfrak{b}$, respectively. It has a submodule, denoted by $M^1_\lambda$, which is generated by $f_\alpha^{\lambda(h_\alpha) + 1} \otimes 1$. Then by \cite[Theorem 8.28]{Kir08},
\begin{equation*}
L_\lambda = M_\lambda/M^1_\lambda.
\end{equation*}



\subsection{Schubert Varieties}
The variety $G/P$, called the \emph{(partial) flag variety}, is determined by the Cartan matrix $A$ and the subset $I$, so we denote it as $X(A, I)$. We say a weight $\lambda$ is an \emph{$I$-regular weight} if it is a dominant integral weight and satisfies, for a simple reflection $s \in S$, $\lambda(\alpha_s^\vee) = 0$ if and only $s \in I$. The following result is in \cite[Lemma 7.1.2]{Kum12}.

\begin{lemma}\label{clo imm}
Suppose that $\lambda$ is an $I$-regular weight, and $V$ is a finite-dimensional highest weight module with the highest weight vector $\omega$. Then there is a closed immersion
\begin{align*}
G/P \hookrightarrow \mathbb{P}(V), && gP \mapsto [g \omega],
\end{align*}
where $[g \omega]$ denote the line through $g \omega$.
\end{lemma}

A \emph{Schubert cell} in $G/P$ is a subvariety $BwP/P$. Here, to be more precise, we need to replace $w$ with a representative in $N(T)$, but any such choice gives the same variety. Different choices of $w$ may give the same Schubert variety. In fact, $BwP/P$ only depends on the coset $wW_I$, so we usually choose the minimal representative. If we take $w \in W^I$, then as varieties, $BwP/P \cong \mathbb{A}^{\ell(w)}$.

A \emph{Schubert variety} in $G/P$, denoted by $X(w, A, I)$, is the Zariski closure of the Schubert cell $BwP/P$. The triple $(w, A, I)$ determines the Schubert variety. $X(w, A, I)$ has an affine stratification. If we take $w \in W^I$, then we have a disjoint union
\begin{equation*}
X(w, A, I) = \bigcup_{v \in [1, w]^I} BuP/P.
\end{equation*}
\begin{definition}\label{sch basis}
There is a $\mathbb{Z}$-basis $\sigma_v$, indexed by $v \in [1, w]^I$, for the integral cohomology group $H^*(X(w, A, I))$. We call this basis to be \emph{Schubert basis}, denoted by $\Sigma(w, A, I)$. The collection of Schubert basis elements of degree two is denoted by $\Sigma_1(w, A, I)$.
\end{definition}

The following lemma is essentially the \cite[Proposition 4.1]{RS21}, whose proof can be extended into our case with slight changes.

\begin{lemma}\label{basis}
Suppose that $f: X(w, A, I) \to X(w', A', I')$ is an isomorphism of algebraic varieties. Then the induced map
\begin{equation*}
\varphi^*: H^*(X(w, A, I)) \to H^*(X(w', A', I'))
\end{equation*}
is a graded ring isomorphism that identifies Schubert bases $\Sigma(w, A, I) \to \Sigma(w', A', I')$ and $\Sigma_1(w, A, I) \to \Sigma_1(w', A', I')$.
\end{lemma}

In other words, both $\Sigma(w, A, I)$ and $\Sigma_1(w, A, I)$ are determined by the variety structure of the Schubert variety $X(w, A, I)$. From this definition, we have an identification
\begin{align*}
i: [1, w]^I \to \Sigma(w, A, I), && v \mapsto \sigma_v.
\end{align*}
Each basis element $\sigma_u$ has degree $2 \ell(u)$, written by $\deg \sigma_u = 2 \ell(u)$, i.e.,
\begin{equation*}
\sigma_u \in H^{2 \ell(u)}(X(w, A, I)).
\end{equation*}
Recall that the set of simple reflections in $[1, w]^I$ is just $S(w) \setminus I$. Hence the restriction of $i$ gives a one-one correspondence
\begin{align*}
i: S(w) \setminus I \to \Sigma_1(w, A, I), && s \mapsto \sigma_s.
\end{align*}

The cup product of $H^*(X(w, A, I))$ is given by the \emph{Chevalley's formula} in \cite[Lemma 8.1]{FW01}.

\begin{lemma}[Chevalley's formula]
Suppose that $\alpha \in \Delta \setminus \Delta_I$ and $w \in W^I$. Then
\begin{equation*}
\sigma_{s_\alpha} \sigma_w = \sum \omega_\alpha(\beta^\vee) \sigma_{(w s_\beta)^{\min}},
\end{equation*}
the sum over all positive roots $\beta$ such that $\ell((w s_\beta)^{\min}) = \ell(w) + 1$. Here, $\omega_\alpha$ is the fundamental weight corresponding to $\alpha$.
\end{lemma}

\begin{remark}
If $\ell((w s_\beta)^{\min}) = \ell(w) + 1$, then either $w s_\beta \leq w$ or $w s_\beta > w$. If the former one holds, then we have $(w s_\beta)^{\min} \leq w^{\min} = w$, which is impossible. Hence $w s_\beta > w$.
\end{remark}

When considering the variety structure of a Schubert variety, it is usually convenient to embed it into a smaller flag variety. This is given by \cite[Lemma 4.8]{RS16}.

\begin{lemma}\label{min emb}
We write $A_w = (a_{st})_{(s,t) \in S(w)^2}$ to be the Cartan submatrix of $A$. Then the inclusion $X(A_w, S(w) \cap I) \hookrightarrow X(A, I)$ restricts to an isomorphism
\begin{equation*}
X(w, A_w, S(w) \cap I) \cong X(w, A, I).
\end{equation*}
\end{lemma}

\begin{remark}
The Weyl group determined by $A_w$ is $W_{S(w)}$, and $w \in (W_{S(w)})^{S(w) \cap I}$, so the Schubert variety $X(w, A_w, S(w) \cap I)$ is well-defined.
\end{remark}







\section{The Cohomology Rings of Schubert Varieties}\label{co}
In this section, we will use the cohomology ring to study the intrinsic properties of a Schubert variety. In other words, we will obtain combinatorial information from an isomorphism class of Schubert variety. Some results have been proven in \cite[Section 4]{RS21} for the case $I$ being an empty set, and the proofs are generalized. In the following, we fix a Schubert variety $X(w, A, I)$ with $w \in W^I$. Due to \Cref{min emb}, we assume that $S = S(w)$.



\subsection{General Situation}\label{ge ca}
First, we show that we can reconstruct the poset $[1, w]^I$. For any $\sigma = \sum_{v \in W^I} a_v \sigma_v \in H^*(X(w, A, I))$, we define its \emph{support}, denoted by $\Sigma(\sigma)$, to be the collection of $\sigma_v$ with $a_v \neq 0$. Let $\prec$ be the partial order on $\Sigma(w, A, I)$, the Schubert basis, generated by the relation $\sigma_u \prec \sigma_v$ if $\sigma_v \in S(\sigma_s \sigma_u)$ for some $\sigma_s \in \Sigma_1(w, A, I)$. By \Cref{basis}, the poset $\Sigma(w, A, I)$ is determined by the variety structure of $X(w, A, I)$. 

\begin{lemma}
The identification $i: [1, w]^I \to \Sigma(w, A, I)$ is a poset isomorphism.
\end{lemma}

Given a subset $J'$ of $\Sigma_1(w, A, I)$, let $H^{J'}$ be the subring of $H^*(X(w, A, I))$ generated by
\begin{equation*}
\sigma_s \in \Sigma_1(w, A, I) \setminus J',
\end{equation*}
and let
\begin{equation*}
\Sigma(w, A, I)^{J'} = \bigcup_{\sigma \in H^{J'}} \Sigma(\sigma).
\end{equation*}

\begin{lemma}\label{^J}
Suppose that there is a subset $J \subset S \setminus I$. Then poset isomorphism $i: [1, w]^I \to \Sigma(w, A, I)$ restricts to another poset isomorphism
\begin{equation*}
i^J: [1, w]^{I \cup J} \to \Sigma(w, A, I)^{i(J)}.
\end{equation*}
\end{lemma}

Now we show how to reconstruct the right descent set. We define the \emph{right descent set} of $\sigma_v$ to be
\begin{equation*}
D_R(\sigma_v) = \left\{ \sigma_s \in \Sigma_1(w, A, I)|\sigma_v \notin \Sigma(w, A, I)^{\{ \sigma_s \}} \right\}.
\end{equation*}

\begin{lemma}
Suppose that $v \in [1, w]^I$. Then the poset isomorphism $i: [1, w]^I \to \Sigma(w, A, I)$ restricts to a bijection $i_v: D_R(v) \to D_R(\sigma_v)$.
\end{lemma}

Next lemma will be used to reconstruct the reduced words.

\begin{lemma}\label{unique maximal}
Suppose that $\sigma_v \in \Sigma(w, A, I)$ and $\sigma_t \in D_R(\sigma_v)$. Then there exists a unique maximal element $\sigma_u \prec \sigma_v$ such that $\sigma_u \in \Sigma(w, A, I)^{\{ \sigma_t \}}$.
\end{lemma}

As a corollary, we obtain the following theorem, which gives a necessary condition.

\begin{proposition}\label{not}
If $S(w) \cap I$ is empty but $S(w') \cap I'$ is not, then $X(w, A, I)$ and $X(w', A', I')$ are not isomorphic as algebraic varieties.
\end{proposition}

Now we compute some entries of the Cartan matrix from the cohomology ring.

\begin{lemma}\label{C1}
Let $s \in S$ and $t \in S \setminus I$ with $s \neq t$. If $st \in [t, w]^I$, then the coefficient of $\sigma_{st}$ in $\sigma_t^2$ is $- a_{st}$.
\end{lemma}

\begin{lemma}\label{C2}
Let $r, s \in S$ and $t \in S \setminus I$ with $rst \in [1, w]^I$ being a reduced word. Then the coefficient of $\sigma_{rst}$ in $\sigma_t \sigma_{st}$ is $\delta_{rt} - a_{rt} + a_{rs} a_{st}$, where $\delta_{rt} = 1$ if $r = t$, and vanishes if $r \neq t$.
\end{lemma}



\subsection{Case \texorpdfstring{$I = \{ s \}$}{I={s}}}\label{sp ca}
We further assume that $I = \{ s \}$. We are primarily interested in this situation.

First, we define the reduced word of $\sigma_v \in \Sigma$. We let $\sigma_s$ be a symbol and define
\begin{equation*}
\widetilde{\Sigma}_1 = \Sigma_1 \cup \{ \sigma_s \},
\end{equation*}
and extend the bijection $S \setminus I \to \Sigma_1$ to a bijection $S \to \widetilde{\Sigma}_1$ by sending $s$ to the symbol $\sigma_s$. We define the \emph{reduced words} of $\sigma_v \in \Sigma$ inductively. First, define the reduced word of $\sigma_1 \in H^0(X(w, A, \{ s \}))$ to be the singleton of the empty word. Next, suppose that $\sigma_v \in \Sigma$ with $\sigma_t \in D_R(\sigma_v)$. Take $\sigma_u$ to be the unique maximal element described in \Cref{unique maximal}, and assume that $\deg \sigma_v - \deg \sigma_u = 2n$ for $n \in \mathbb{N}$. Then we define the sequence
\begin{equation*}
(\sigma_{s_1}, \cdots, \sigma_{s_k}, \cdots, \sigma_t, \sigma_s, \sigma_t, \sigma_s)
\end{equation*}
is a reduced of $\sigma_v$, where $(\sigma_{s_1}, \cdots, \sigma_{s_k})$ is a reduced word of $\sigma_u$, and the remaining part $(\cdots, \sigma_t, \sigma_s, \sigma_t, \sigma_s)$ of the sequence is an alternating sequence of $\sigma_t$ and $\sigma_s$ of length $n$. We also denote $RW(\sigma_v)$ as the set of reduced words of $\sigma_v$, which is obviously nonempty. The following lemma is clear.

\begin{lemma}\label{red}
Suppose that $\sigma_v \in \Sigma$. Then the bijection $S \to \widetilde{\Sigma}_1$ induces an inclusion
\begin{align*}
RW(\sigma_u) \hookrightarrow RW(u), && (\sigma_{s_1}, \cdots, \sigma_{s_k}) \mapsto (s_1, \cdots, s_k).
\end{align*}
\end{lemma}

The above lemmas allow us to prove the following proposition.

\begin{proposition}\label{neces}
Let $A = (a_{st})_{(s, t) \in S^2}$ and $A' = (a'_{s't'})_{(s', t') \in S'^2}$ be two Cartan matrices with associated Weyl groups $W$ and $W'$, and sets of simple reflections $S$ and $S'$, respectively. Let $s \in S$ and $s' \in S'$. Take two fully supported elements $w \in W^{\{ s \}}$ and $w' \in W'^{\{ s' \}}$. If $X(w, A, \{ s \})$ and $X(w', A', \{ s' \})$ are algebraically isomorphic, then there is a bijection $\tau: S \to S'$ sending $s$ to $s'$ such that
\begin{enumerate}
\item[(a)] for some reduced word $w = s_1 \cdots s_k$, $w' = \tau(s_1) \cdots \tau(s_k)$ is also a reduced word, and
\item[(b)] for any $t_1, t_2 \in S$, $a_{t_1 t_2} = a'_{\tau(t_1) \tau(t_2)}$ whenever $t_1 t_2 \leq w$.
\end{enumerate}
\end{proposition}







\section{Constructing Isomorphism}\label{con iso}
In this section, we will study the sufficient conditions and obtain \Cref{suff} and the main theorem.

Let $A = (a_{st})_{(s, t) \in S^2}$ and $A' = (a'_{s't'})_{(s', t') \in S'^2}$ be two Cartan matrices with associated Weyl groups $W$ and $W'$, and sets of simple reflections $S$ and $S'$, respectively. Let $I \subset S$ and $I' \subset S'$. Take two fully supported elements $w \in W^I$ and $w' \in W'^{I'}$. We assume that there is a bijection $\tau: S \to S'$ sending $I$ to $I'$ such that
\begin{enumerate}
\item[(a)] for some reduced word $w = s_1 \cdots s_k$, $w' = \tau(s_1) \cdots \tau(s_k)$, and
\item[(b)] for any $t_1, t_2 \in S$, $a_{t_1 t_2} = a'_{\tau(t_1) \tau(t_2)}$ whenever $t_1 t_2 \leq w$.
\end{enumerate}

We want to show that the varieties $X(w, A, \{ s \})$ and $X(w', A', \{ s' \})$ are algebraically isomorphic. First, we have the following result.

\begin{lemma}\label{Car equ}
The bijection $\tau: S \to S'$ induces a poset isomorphism $[1, w]^I \to [1, w']^{I'}$.
\end{lemma}

In the following lemmas, we add the additional assumption that $a_{st} \leq a'_{\tau(s) \tau(t)}$ for all simple reflections $s, t \in S$. 

The bijection $\tau$ induces a bijection $\tau: \Delta \mapsto \Delta'$ between simple roots, given by $\alpha_s \mapsto \alpha_{\tau(s)}$. Let $\mathfrak{g} = \mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$ and $\mathfrak{g}' = \mathfrak{n}'^- \oplus \mathfrak{h}' \oplus \mathfrak{n}'^+$ be two complex reductive Lie algebras corresponding to $A$ and $A'$, respectively. The lemma below is \cite[Lemma 3.5 (a) and 3.8]{RS21}.

\begin{lemma}\label{varphi}
There are surjective Lie algebra homomorphisms
\begin{equation*}
\varphi^\pm: \mathfrak{n}^\pm(A) \to \mathfrak{n}^\pm(A'),
\end{equation*}
given by $e_\alpha \mapsto e'_{\tau(\alpha)}$ and $f_\alpha \mapsto f'_{\tau(\alpha)}$, respectively. The homomorphism $\varphi^+$ induces an isomorphism
\begin{equation*}
\varphi^+: \mathfrak{n}^+(A)_v \to \mathfrak{n}^+(A')_{\tau(v)}.
\end{equation*}
\end{lemma}

If $\lambda$ and $\lambda'$ are $I$ and $I'$-regular weights, then we write $V = L_\lambda$ and $V' = L_{\lambda'}$ with corresponding highest weight vectors $\omega$ and $\omega'$, respectively. The lemma below is given in \cite[Lemma 3.5 (b) and 3.9]{RS21}

\begin{lemma}\label{pi}
Let $\lambda$ and $\lambda'$ be $I$ and $I'$-regular weights, respectively. If $\lambda(h_\alpha) = \lambda'(h'_{\tau(\alpha)})$ for each $\alpha \in \Delta$, then there is a surjective $\mathfrak{n}^-$-homomorphism $\pi: V \to V'$, sending the highest weight vector $\omega$ to $\omega'$, where $V'$ is regarded as a $\mathfrak{n}^-$-module via the homomorphism $\varphi^-$. The homomorphism $\pi$ satisfies
\begin{equation*}
\pi(\exp(e) v \omega) = \exp(\varphi^+(e)) \tau(v) \omega'
\end{equation*}
for all $e \in \mathfrak{n}^+(A)$ and $v \leq w$.
\end{lemma}

\begin{remark}
The formula makes sense since the Weyl group element $v$ can be lifted to an element in the normalizer of the maximal torus, and different liftings acting on the vector $\omega$ have the same result.
\end{remark}

With the preparation above, we have the following sufficient condition.

\begin{theorem}\label{suff}
Let
\begin{align*}
A = (a_{st})_{(s, t) \in S^2} && \mbox{and} && A' = (a'_{s't'})_{(s', t') \in S'^2}
\end{align*}
be two Cartan matrices with associated Weyl groups $W$ and $W'$, and sets of simple reflections $S$ and $S'$, respectively. Let $I \subset S$ and $I' \subset S'$. Take $w \in W^I$ and $w' \in W'^{I'}$. If there is a bijection $\tau: S(w) \to S(w')$ sending $S(w) \cap I$ to $S(w') \cap I'$ such that
\begin{enumerate}
\item[(a)] for some reduced word $w = s_1 \cdots s_k$, $w' = \tau(s_1) \cdots \tau(s_k)$ is also a reduced word, and
\item[(b)] for any $t_1, t_2 \in S$, $a_{t_1 t_2} = a'_{\tau(t_1) \tau(t_2)}$ whenever $t_1 t_2 \leq w$,
\end{enumerate}
then $X(w, A, I)$ and $X(w', A', I')$ are isomorphic.
\end{theorem}

Finally, combining \Cref{neces} and \Cref{suff}, we conclude that conditions 1 and 2 in our main theorem are equivalent. The last statement in the main theorem follows from \Cref{not}.

As one may notice that the sufficient condition given in \Cref{suff} is a generalization of the one in the main theorem, one may conjecture that the necessary condition stated in \Cref{neces} can also be generalized to all equally supported pairs.

\begin{conjecture}\label{conj}
Let
\begin{align*}
A = (a_{st})_{(s, t) \in S^2} && \mbox{and} && A' = (a'_{s't'})_{(s', t') \in S'^2}
\end{align*}
be two Cartan matrices with associated Weyl groups $W$ and $W'$, and sets of simple reflections $S$ and $S'$, respectively. Let $I \subset S$ and $I' \subset S'$. Take $w \in W^I$ and $w' \in W'^{I'}$. Assume that $|S(w)| = |S(w')|$. Then the following are equivalent:
\begin{enumerate}
\item the Schubert varieties $X(w, A, I)$ and $X(w', A', I')$ are algebraically isomorphic;
\item there exists a bijection $\tau: S(w) \to S(w')$ sending $S(w) \cap I$ to $S(w') \cap I'$, such that
\begin{enumerate}
\item[(a)] for a reduced word $w = s_1 \cdots s_k$, $w' = \tau(s_1) \cdots \tau(s_k)$ is also a reduced word, and
\item[(b)] for any $t_1, t_2 \in S(w)$, $a_{t_1, t_2} = a'_{\tau(t_1), \tau(t_2)}$ whenever $t_1 t_2 \leq w$.
\end{enumerate}
\end{enumerate}
\end{conjecture}

\acknowledgements{The author thanks Weihong Xu for introducing the isomorphism problem to him, providing him with relevant materials, and guiding him in writing the thesis. He also thanks Jing-Song Huang and Rui Xiong for their instruction on representation theory.}
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography



\end{document}