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\begin{document}

\title[The Isomorphism Problem for cominuscule Schubert Varieties]
{The Isomorphism Problem for cominuscule Schubert Varieties}

\date{\today}

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\title[Cominuscule Schubert varieties]{The isomorphism problem for cominuscule Schubert varieties}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Edward Richmond\thanks{\href{mailto:edward.richmond@okstate.edu}{edward.richmond@okstate.edu}, supported by a grant from the Simons Foundation 941273}\addressmark{1},
        Mihail Țarigradschi\thanks{\href{mailto:mt994@math.rutgers.edu}{mt994@math.rutgers.edu}, supported by NSF grant MS-2152316}\addressmark{2},
        \and Weihong Xu\thanks{\href{mailto:weihong@vt.edu}{weihong@vt.edu}, supported by NSF grant MS-2152316}\addressmark{3}}

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\address{\addressmark{1}Department of Mathematics, Oklahoma State University, Stillwater, OK \\
        \addressmark{2}Department of Mathematics, Rutgers University, Piscataway, NJ \\
        \addressmark{3}Department of Mathematics, Virginia Tech, Blacksburg, VA}

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\received{14 November 2022}

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\revised{28 March 2023}

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\keywords{Schubert variety, flag variety, cominuscule}

\abstract{
        Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.
}

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\section{Introduction}

Cominuscule flag varieties correspond to algebraic varieties that admit the structure of a compact Hermitian symmetric space and have been studied extensively due their shared properties with Grassmannians \cites{Bo88,BP99,Pe09,ThYo09,Buch2013,Ro14,BCMP18,Chung,BCMP,MR4099894,buch2022positivity}. These varieties come in four infinite families and two exceptional types and are determined by a pair \((\cD,\gamma)\) of a Dynkin diagram \(\cD\) of a reductive Lie group and a cominuscule simple root \(\gamma\). See \cref{table:poset} for a classification of cominuscule flag varieties. Let \(X\) denote the cominuscule flag variety corresponding to \((\cD,\gamma)\) and \(R\) denote the root system of the Dynkin diagram \(\cD\).  Set \(\cP_X\coloneqq \{\alpha\in R: \alpha\geq\gamma\}\) with the partial order $\alpha\leq \beta$ if $\beta-\alpha$ is a non-negative sum of simple roots, and give \(\cP_X\) a labeling of long/short roots. By~\cite[Theorem 2.4]{buch2016k}, Schubert varieties in \(X\) are indexed by lower order ideals in \(\cP_X\), generalizing the fact that Schubert varieties in a Grassmannian are indexed by Young diagrams.

\input{diagrams.tex}

Our main result \cref{thm:main} is a combinatorial criterion for distinguishing isomorphism classes of Schubert varieties coming from cominuscule flag varieties.

\begin{thm}\label{thm:main}
        Let $X_\lambda\subseteq X$ and $Y_{\mu}\subseteq Y$ be cominuscule Schubert varieties indexed by lower order ideals \(\lambda\subseteq\cP_X\) and \(\mu\subseteq\cP_Y\), respectively.  Then $X_\lambda$ and $Y_{\mu}$ are algebraically isomorphic if and only if $\lambda$ and $\mu$ are isomorphic as labeled posets.
\end{thm}

For illustrative examples of \cref{thm:main}, see \cref{sec:examples}.

Since Grassmannians are cominuscule flag varieties, \cref{thm:main} extends the work of Țarigradschi and Xu in \cite{Tarigradschi-Xu22}, where they prove two Grassmannian Schubert varieties are isomorphic if and only if their Young diagrams are the same or the transpose of each other. Other related works include Richmond and Slofstra's characterization of the isomorphism classes of Schubert varieties coming from complete flag varieties in \cite{Richmond-Slofstra21} using Cartan equivalence. However, they also note that Cartan equivalence is neither necessary nor sufficient to distinguish Schubert varieties in partial flag varieties. A class of smooth Schubert varieties in type \(A\) partial flag varieties are classified by Develin, Martin, and Reiner in \cite{develin_martin_reiner_2007}. Yet many Schubert varieties are singular, with the first example being the Schubert divisor in the Grassmannian \(\Gr(2,4)\).

We discuss preliminaries in \cref{sec:prelim}, and then in \cref{sec:proof}, we prove \cref{thm:main} and illustrate it with examples. Our proof is type-independent and employs several new techniques.

\section{Examples of \cref{thm:main}}
\label{sec:examples}

For the following examples, recall that cominuscule Schubert varieties are indexed by lower order ideals in $\cP_X$. Examples of \(\cP_X\) are illustrated in \cref{table:poset}, where each element in \(\cP_X\) is drawn as a box, and boxes decorated with an ``\(s\)'' correspond to short roots.  The partial order on boxes is given by $\alpha \leq \beta$ if and only if $\alpha$ is weakly north-west of $\beta$. In particular, lower order ideals are given by subsets of boxes that are closed under moving to the north and west.

\begin{example}
        As illustrated below, transposing a Young diagram does not change the poset structure. Therefore, two Grassmannian Schubert varieties are isomorphic if their indexing Young diagrams are the transpose of each other. Geometrically, this is related to the isomorphism \(\Gr(m,m+k) \cong \Gr(k, m+k)\).
        \begin{equation*}
                \tableau{\tableausz}{
                [a]&[a]&[a]&[a]\\
                [a]&[a]\\
                [a]
                }\ \ \cong\ \
                \tableau{\tableausz}{
                [a]&[a]&[a]\\
                [a]&[a]\\
                [a]\\
                [a]
                }
        \end{equation*}
\end{example}

\begin{example}
        Using \cref{table:poset}, it is not hard to see that if a Grassmannian Schubert variety is isomorphic to a non-type \(A\) cominuscule Schubert variety, then they are both isomorphic to a projective space. Indeed, in order to fit inside a \(\cP_X\) of another type, the lower order ideal is forced to be a chain.

                {As a special case, we also see that any Schubert curve in any cominuscule flag variety is isomorphic to \(\P^1\). In fact, any Schubert curve in any flag variety is isomorphic to \(\P^1\), which follows from the more general statements that Schubert varieties are rational normal projective varieties and that \(\P^1\) is the only rational normal projective curve.}
\end{example}

\begin{example}
        The Schubert divisor in \(Q^3\) is not isomorphic to \(\P^2\), because the labeling of their posets does not match:
        \begin{equation*}
                \tableau{\tableausz}{
                        [a]&[a]s
                }\ \ \not\cong\ \
                \tableau{\tableausz}{
                        [a]&[a]
                }\ .
        \end{equation*}
        We can also see it geometrically, as the Schubert divisor in \(Q^3\) is singular.
\end{example}

\begin{example}
        The quadric \(Q^3\) embeds in \(\LG(n,2n)\) (\(n\geq 3\)) as a Schubert variety, as illustrated by
        \begin{equation*}
                \tableau{\tableausz}{
                [AA]&[AA] s &[a] s &[a] s\\
                []&[AA]&[a]s &[a] s\\
                []&[]&[a]&[a]s\\
                []&[]&[]&[a]
                }\ .
        \end{equation*}
\end{example}

\begin{example}\label{example:Q10}
        The quadric \(Q^{10}\) embeds in \(E_7/P_7\) as a Schubert variety, as illustrated by
        \begin{equation*}
                \tableau{\tableausz}{
                [AA]&[AA]&[AA]&[AA]&[AA]&[a]\\
                []&[]&[]&[AA]&[AA]&[a]\\
                []&[]&[]&[]&[AA]&[a]&[a]\\
                []&[]&[]&[]&[AA]&[a]&[a]&[a]&[a]\\
                []&[]&[]&[]&[AA]&[a]&[a]&[a]&[a]\\
                []&[]&[]&[]&[]&[]&[]&[a]&[a]\\[]&[]&[]&[]&[]&[]&[]&[]&[a]\\[]&[]&[]&[]&[]&[]&[]&[]&[a]\\[]&[]&[]&[]&[]&[]&[]&[]&[a]
                }\ .
        \end{equation*}
\end{example}

\begin{example}
        There are two non-isomorphic \(6\)-dimensional Schubert varieties in \(E_6/P_6\), given by the two order ideals illustrated below.
        \begin{equation*}
                \tableau{\tableausz}{
                [AA]&[AA]&[AA]&[AA]\\
                []&[]&[AA]&[a]&[a]&[a]\\
                []&[]&[AA]&[a]&[a]&[a]\\
                []&[]&[]&[]&[a]&[a]&[a]&[a]
                }\ \ \not\cong\ \
                \tableau{\tableausz}{
                [AA]&[AA]&[AA]&[AA]\\
                []&[]&[AA]&[AA]&[a]&[a]\\
                []&[]&[a]&[a]&[a]&[a]\\
                []&[]&[]&[]&[a]&[a]&[a]&[a]
                }
        \end{equation*}
\end{example}

\begin{example}
        While \(\cP_{\LG(n,2n)}\) and \(\cP_{\OG(n+1,2n+2)}\) are isomorphic as posets, this isomorphism does not preserve the labeling of long/short roots (see illustration below). As a result, \(\LG(n,2n)\) and \(\OG(n+1,2n+2)\) do not contain isomorphic Schubert varieties of dimension greater than one.
        \begin{equation*}
                \tableau{\tableausz}{
                [a]&[a] s &[a] s &[a] s\\
                []&[a]&[a]s &[a] s\\
                []&[]&[a]&[a]s\\
                []&[]&[]&[a]
                }\ \ \not\cong\ \
                \tableau{\tableausz}{
                [a]&[a] &[a] &[a]\\
                []&[a]&[a] &[a] \\
                []&[]&[a]&[a]\\
                []&[]&[]&[a]
                }
        \end{equation*}
\end{example}

\section{Preliminaries}\label{sec:prelim}

Let \(G\) be a complex reductive linear algebraic group. We fix subgroups \(T \subset B \subset G\), where \(T\) is a maximal torus and \(B\) is a Borel subgroup. With this setup, \(T \subset G\) determines a root system \(R\) of \(G\), with corresponding Weyl group \(W:= N(T)/T\), and \(B\) determines a set of simple roots \(\Delta\subseteq R\). The set of roots decomposes into positive and negative roots $R=R^+\sqcup R^-$, with \(R^+\) being non-negative sums of simple roots. The Weyl group \(W\) is generated by the set of simple reflections
\[
        S\coloneqq\{s_\alpha: \alpha\in\Delta\}.
\]

To each subset \(I \subseteq S\) one can associate a Weyl subgroup \(W_I := \langle s : s \in I\rangle \subseteq W \), a parabolic subgroup \(P_I = B W_I B \subseteq G\) and the corresponding (partial) flag variety \(X = G/P_I\). Schubert varieties in \(X\) are indexed by \(W^I\), the set of minimal length coset representatives of \(W/W_I\). Explicitly, for \(w\in W^I\), the Schubert variety
\[
        X_w:=\overline{BwP_I/P_I}
\]
has dimension the Coxeter length of \(w\), denoted \(\ell(w)\). Moreover, for any \(u\in W^I\), we have \(X_u\subseteq X_w\) if and only if \(u\leq w\) in Bruhat order.

From now on, \(X\) is a cominuscule flag variety. In other words, \(I = S \setminus \{s_\gamma\}\), where \(\gamma\) is a cominuscule simple root, i.e., \(\gamma\) appears with coefficient \(1\) in the highest root of \(R\). Cominuscule roots are illustrated by filled-in circles in \cref{table:poset}.

Recall that
\[
        \cP_X\coloneqq \{\alpha\in R: \alpha\geq\gamma\}
\]
inherits the usual partial order on roots, i.e., $\alpha\leq \beta$ if $\beta-\alpha$ is a non-negative sum of simple roots, and in addition, we give \(\cP_X\) a labeling of long/short roots.

In \cite{Proctor84}, Proctor proves that \(W^I\) is a distributive lattice under the induced Bruhat partial order from $W$. Birkhoff's representation theorem implies there is a bijection between $W^I$ and the set of lower order ideals in $\cP_X$. In particular, the join-irreducible elements of $W^I$ are identified with principal lower order ideals of $\cP_X$ and hence with $\cP_X$ itself. Explicitly, to each \(w\in W^I\) we associate its inversion set
\begin{equation}\label{eqn:inversion_set}
        \lambda(w)\coloneqq \{\alpha\in R^+: w.\alpha<0\},
\end{equation}
viewed as a sub-poset of \(\cP_X\). It is well known that $\ell(w)=|\lambda(w)|$. Moreover, the following proposition was proved in \cite[Theorem 2.4 and Corollary 2.6]{buch2016k}:

\begin{prop}[Buch--Samuel]\label{prop:comin_properties}
        For any \(w\in W^I\), the inversion set $\lambda(w)$ is a lower order ideal in $\cP_X$.  Moreover:
        \begin{enumerate}
                \item The map $w\mapsto \lambda(w)$ is a bijection between $W^I$ and the set of lower order ideals in $\cP_X$.
                \item For any $u\in W^I$, we have $u\leq w$ in Bruhat order if and only if $\lambda(u)\subseteq \lambda(w)$.
                \item If $\alpha\in \lambda(w)$ and $\lambda(w)\setminus\{\alpha\}$ is a lower order ideal, then $ws_\alpha\in W^I$ and $\lambda(ws_\alpha)=\lambda(w)\setminus\{\alpha\}$, where $s_\alpha\in W$ is the reflection corresponding to \(\alpha\).
        \end{enumerate}
\end{prop}

\begin{notation}
        Given a lower order ideal \(\lambda\subseteq\cP_X\), we will write \(w_\lambda\) for the element of \(W^I\) corresponding to \(\lambda\) in \cref{prop:comin_properties}. We also use $X_\lambda:=X_{w_\lambda}$ to denote the corresponding Schubert variety.
\end{notation}

In \cref{sec:converse}, we will use a map \(\delta: \cP_X\to\Delta\) defined in \cite{buch2022positivity} as follows.

\begin{defn}
        For \(\alpha\in\cP_X\), let \(\lambda_\alpha\) be the principal lower order ideal generated by \(\alpha\). Let \(\delta(\alpha)=-w_{\lambda_\alpha}.\alpha\in R^+\). Then \(s_{\delta(\alpha)}=w_{\lambda_\alpha}s_\alpha w_{\lambda_\alpha}^{-1}\) has length \(1\) \cite[Section 4.1]{buch2022positivity}. Therefore, \(\delta(\alpha)\in\Delta\).
\end{defn}

The following lemma is a restatement of \cite[Corollary 2.10]{buch2016k}. See also \cite[Section 4.1]{buch2022positivity}.

\begin{lemma}[Buch--Samuel]\label{lemma:red_decomp}
        Let \(\lambda\subseteq\cP_X\) be a lower order ideal and \(\beta_1,\beta_2,\dots,\beta_\ell\) be the boxes it contains written in an increasing order. Then \(s_{\delta(\beta_\ell)}\cdots s_{\delta(\beta_2)}s_{\delta(\beta_1)}\) is a reduced decomposition of \(w_\lambda\). Moreover, every reduced decomposition of \(w_\lambda\) can be obtained in this way.
\end{lemma}

\section{Proof of \cref{thm:main}}\label{sec:proof}

In this section, we prove each direction of \cref{thm:main} separately.

\subsection{Forward direction: the isomorphism class of \texorpdfstring{\(X_\lambda\)}{Xlambda} determines the labeled poset \texorpdfstring{\(\lambda\)}{lambda}}\label{sec:forward}

Let \(i_\lambda: X_\lambda \hookrightarrow X\) denote the embedding of a Schubert subvariety into a cominuscule flag variety \(X=G/P_I\).  The definition of the labeled poset $\lambda$ depends on the root system of the reductive group $G$ and hence on the embedding \(i_\lambda: X_\lambda \hookrightarrow X\).  The goal of this section is to show that $\lambda$ (as a labeled poset) can be constructed using only the variety structure of $X_\lambda$ and is therefore intrinsic to the isomorphism class of $X_\lambda$.  We prove the following proposition which states the ``forward" direction of \cref{thm:main}.

\begin{prop}\label{prop:forward}
        Let $X_\lambda\subseteq X$ and $Y_{\mu}\subseteq Y$ be cominuscule Schubert varieties indexed by lower order ideals \(\lambda\subseteq\cP_X\) and \(\mu\subseteq\cP_Y\), respectively. If $X_\lambda$ and $Y_{\mu}$ are algebraically isomorphic, then $\lambda$ and $\mu$ are isomorphic as labeled posets.
\end{prop}

Our primary tools come from the intersection theory of algebraic varieties (see \cite{MR1644323} for more details).  Let \(\Pic(X_\lambda)\) and \(A_*(X_\lambda)\) denote the Picard and Chow groups of $X_\lambda$.  It is well known that these groups are algebraic invariants of \(X_\lambda\). Recall that the \(k\)-th Chow group \(A_k(Z)\) of a scheme \(Z\) is the free abelian group on the \(k\)-dimensional subvarieties of \(Z\) modulo rational equivalence. When \(Z\) is a normal variety, the Picard group \(\Pic(Z)\) can be identified with the subgroup of \(A_{\dim(Z)-1}(Z)\) generated by classes of locally principal divisors (note that all Schubert varieties are normal).  Our aim is to construct the labeled poset \(\lambda\) from the intersection class map or intersection product \cite[Definition 2.3]{MR1644323}:
\[
        \Pic(X_\lambda)\times A_*(X_\lambda)\to A_*(X_\lambda).
\]
If $(\sigma,\tau)\in \Pic(X_\lambda)\times A_*(X_\lambda)$, we denote the image of the intersection product by $\sigma\cdot\tau$.  Next, we consider the effective cone of a scheme:
\begin{defn}
        Let \(Z\) be a scheme. The \emph{effective cone} in the Chow group \(A_*(Z)\) is the semigroup in \(A_*(Z)\) generated by the classes of closed subvarieties of \(Z\).
\end{defn}
Since the flag variety $X$ is cominuscule, there is a unique Schubert variety of codimension \(1\) in \(X\), called the Schubert divisor.  Its class, denoted \(D\), is the unique effective generator of the Picard group \(\Pic(X)\subseteq A_*(X)\). Recall that \(i_\lambda: X_\lambda \hookrightarrow X\) is a closed embedding of varieties.  \cref{lemma:pic} below follows from \cite[Proposition 6]{Ol88} (see also \cite[Proposition 2.2.8 part (ii)]{Br05}).

\begin{lemma}\label{lemma:pic}
        For any non-empty lower order ideal \(\lambda\subseteq \cP_X\), the map \(i_\lambda^*: \Pic(X)\to \Pic(X_\lambda)\) is an isomorphism.
\end{lemma}

Since $D$ is effective and generates $\Pic(X)$, \cref{lemma:pic} implies that \(i_\lambda^*D\) the unique effective generator of \(\Pic(X_\lambda)\).  Recall from \cref{prop:comin_properties} that we have lower order ideals $\mu\subseteq \lambda$ if and only if $w_\mu\leq w_\lambda$ in Bruhat order.  Hence, we have $\mu\subseteq \lambda$ if and only if $X_\mu\subseteq X_\lambda$.  We write \([X_\mu]\) for the class of \(X_\mu\) in \(A_*(X_\lambda)\) and in \(A_*(X)\), in an abuse of notation.  It is well known that the classes $\{[X_\mu]\}_{\mu\subseteq \lambda}$ form an integral basis of $A_*(X_\lambda)$.  \cref{lemma:schub_classes} below is a special case of \cite[Corollary of Thereom 1]{MR1299008} and allows us to identify Schubert classes (the effective cone) in \(A_*(X_\lambda)\) without referring to the embedding \(i_\lambda: X_\lambda \hookrightarrow X\) or using the $B$-variety structure of $X_\lambda$.

\begin{lemma}[Fulton--MacPherson--Sottile--Sturmfels]\label{lemma:schub_classes}
        The Schubert classes \([X_\mu]\) such that \(\mu\subseteq \lambda\) are exactly the minimal elements in the extremal rays of the effective cone in \(A_*(X_\lambda)\).
\end{lemma}

We shall see later that the poset structure of $\lambda$ can be recovered from the intersection products \(i_\lambda^*D\cdot [X_\mu]\).
Let \(j:X_\mu\hookrightarrow X_\lambda\) be an inclusion of Schubert varieties, where $\mu\subseteq\lambda$. By the definition of intersection product,
\[
        i_\lambda^*D\cdot [X_\mu]=[j^*i_\lambda^*D]=[(i_\lambda\circ j)^*D]=D\cdot [X_\mu].
\]

This implies that the product \(i_\lambda^*D\cdot [X_\mu]\) on \(X_\lambda\) can be computed on \(X\). By \cite[Example 19.1.11]{MR1644323}, \(A_*(X)\) can be identified with the homology group \(H_*(X)\).  By \cite[Proposition 19.1.2]{MR1644323} we have that the intersection product \(D\cdot [X_\mu]\) can be identified with a cap product.  Since \(X\) is a smooth complex variety, the Poincar\'e duality  further identifies the intersection product with the cup product of cohomology classes corresponding to \(D\) and \([X_\mu]\). This cup product is given by the Chevalley formula \cite[Lemma 8.1]{fulton2004quantum}. Using these identifications, we restate the Chevalley formula for cominuscule flag varieties (and hence Schubert varieties):

\begin{lemma}[Fulton--Woodward]\label{lemma:chevalley}
        Let \(X\) be a cominuscule flag variety with corresponding cominuscule simple root $\gamma$.  For any lower order ideals \(\mu\subseteq\lambda\subseteq\cP_X\), we have
        \begin{equation*}
                \label{eqn:mult_equation}
                i_\lambda^*D\cdot [X_\mu]=\sum \frac{(\gamma,\gamma)}{(\alpha,\alpha)} [X_{\mu\setminus\{\alpha\}}],
        \end{equation*}
        the sum over all positive roots \(\alpha\) such that \(\mu\setminus\{\alpha\}\) is a lower order ideal in \(\cP_X\).  Here $(\cdot,\cdot)$ denotes the usual inner product.
\end{lemma}

Observe that \cref{lemma:chevalley} reinterprets the Chevalley formula as a degree lowering operator since intersection product with divisors is a map from $A_k(X_\lambda)$ to $A_{k-1}(X_\lambda)$. This is opposite to the standard presentation of the Chevalley formula as a degree raising operator in cohomology.

\begin{example}
        By \cref{lemma:chevalley}, the following calculations hold for \(X=\LG(3,6)\).  We refer to \cref{table:poset} for the poset $\cP_X$.
        \begin{equation*}
                D\cdot\bracks[\bigg]{X_{\tableau{8}{
                [a]&[a]s&[a]s\\
                []&[a]}}}=2\bracks[\bigg]{X_{\tableau{8}{
                [a]&[a]s\\
                []&[a]}}}+\bracks[\bigg]{X_{\tableau{8}{
                                                [a]&[a]s&
                                                [a]s}}},\quad
                D\cdot\bracks[\bigg]{X_{\tableau{8}{
                [a]&[a]s\\
                []&[a]}}}=\bracks[\bigg]{X_{\tableau{8}{
                                                [a]&[a]s}}},\quad
                D\cdot\bracks[\bigg]{X_{\tableau{8}{
                                                [a]&[a]s}}}=2\bracks[\bigg]{X_{\tableau{8}{[a]}}}.
        \end{equation*}
        Note that a coefficient \(2\) occurs whenever the removed box (root) is short.
\end{example}


\begin{proof}[Proof of \cref{prop:forward}]
        \cref{lemma:pic} allows us to identify the unique effective generator \(i_\lambda^*D\) of \(\Pic(X_\lambda)\), and \cref{lemma:schub_classes} allows us to
        identify Schubert classes in \(A^*(X_\lambda)\), or equivalently, their indexing set: the set of lower order ideals in \(\cP_X\) that are contained in \(\lambda\). The Bruhat order on this set is recovered from \cref{lemma:chevalley}, and its sub-poset of join-irreducible elements recovers the poset \(\lambda\) as discussed in \cref{sec:prelim}. Finally, note that if \(\mu\) is a join-irreducible diagram contained in \(\lambda\) with (unique) maximal box \(\alpha\), then
        \[
                i_\lambda^*D\cdot[X_\mu]=\frac{(\gamma,\gamma)}{(\alpha,\alpha)}[X_{\mu\setminus\{\alpha\}}].
        \]
        Since \(\gamma\) is always long, the coefficient \((\gamma,\gamma)/(\alpha,\alpha)\) is not \(1\) if and only if \(\alpha\) is short. Hence, the labeling of the boxes in \(\lambda\) is also determined.
\end{proof}

\subsection{Converse direction: the labeled poset \texorpdfstring{\(\lambda\)}{lambda} determines the isomorphism class of \texorpdfstring{\(X_\lambda\)}{Xlambda}}\label{sec:converse}

Let \(X=G/P_I\) be a cominuscule flag variety and $\lambda\subseteq\cP_X$ be a lower order ideal. In this section, we prove that the poset $\lambda$ and its labeling of long/short roots determine the isomorphism class of $X_\lambda$. More precisely, we prove the following proposition, which states the ``converse" direction of \cref{thm:main}.

\begin{prop}\label{prop:converse}
        Let $X_\lambda\subseteq X$ and $Y_{\mu}\subseteq Y$ be cominuscule Schubert varieties indexed by lower order ideals \(\lambda\subseteq\cP_X\) and \(\mu\subseteq\cP_Y\), respectively. If $\lambda$ and $\mu$ are isomorphic as labeled posets, then $X_\lambda$ and $Y_{\mu}$ are algebraically isomorphic.
\end{prop}

Our strategy is to embed \(X_\lambda\) in a ``minimal" flag variety \(X'\) determined by the labeled poset \(\lambda\).

Recall that \(S\) is the set of simple reflections defined in \cref{sec:prelim}.

\begin{defn}
        The \emph{support} of \(\lambda\) is defined as
        \[
                S(\lambda)\coloneqq\{s\in S: s\leq w_\lambda\}.
        \]
        Equivalently, \(S(\lambda)\) is the set of simple reflections appearing in any reduced decomposition of \(w_\lambda\).
\end{defn}

Every reduced decomposition of \(w_\lambda\), and in particular, \(S(\lambda)\), can be read out from the poset \(\lambda\) \cite[Section 4]{buch2022positivity}. The variety \(X'\) is constructed using \(S(\lambda)\) as follows. Let \(G'\) be the reductive subgroup of \(P_{S(\lambda)}\) with Weyl group \(W'\coloneqq W_{S(\lambda)}\) and \(P'\coloneqq G'\cap P_I\) be the reductive subgroup of \(G'\) corresponding to \(I'\coloneqq I\cap S(\lambda)\). Set \(X'\coloneqq G'/P'\). Note that \(w_\lambda\in {W'}^{I'}\).

\cref{lemma:isom} below is a restatement of \cite[Lemma 4.8]{MR3552231}.

\begin{lemma}[Richmond--Slofstra]\label{lemma:isom}
        The inclusion \(X'\hookrightarrow X\) induces an isomorphism \(X'_{w_\lambda}\to X_\lambda\).
\end{lemma}

Let \(Y\) be another cominuscule flag variety and \(\mu\subseteq\cP_Y\) be a lower order ideal. Next, we show that a labeled poset isomorphism between \(\lambda\) and \(\mu\) induces an isomorphism between \(X'\) and \(Y'\), which restricts to an isomorphism between \(X_{\lambda}\) and \(Y_{\mu}\). We shall see that \(X'\) and \(Y'\) are cominuscule and that this isomorphism is given by an isomorphism of their Dynkin diagrams.

In the following, let $\cD_X$ be the Dynkin diagram of \(X\) with vertex set \(\Delta_X\).

\begin{defn}
        The diagram \(\cD_X^\lambda\) is defined to be the full subgraph of \(\cD_X\) with vertex set
        \[
                \Delta_X^\lambda\coloneqq\{\alpha\in\Delta_X: s_\alpha\in S(\lambda)\}.
        \]
\end{defn}


\begin{defn}\label{defn:Pdelta}
        Let a \emph{Dynkin chain} in \(\cP_X\) be a chain \(\pi\subseteq\cP_X\) such that:
        \begin{enumerate}
                \item the set \(\pi\) is a lower order ideal;
                \item the lengths of roots in \(\pi\) are weakly decreasing.
        \end{enumerate}

        The lower order ideal \(\cP_X^\Delta\subseteq\cP_X\) is defined to be the union of all Dynkin chains in \(\cP_X\).
\end{defn}

In fact, Dynkin chains in \(\cP_X\) correspond to paths in \(\cD_X\) starting from the cominuscule root~\(\gamma\). Examples of \(\cP_X^\Delta\) are illustrated in \Figure{DChains}.

\begin{figure}[!ht]
        \caption{We highlight \(\cP_X^\Delta\subset \cP_X\) with a bold border and label its boxes by the images of \(\delta\).
        }\label{fig:DChains}
        \centering

        \begin{tabular}{ccc}
                $X$        & $\cP_X$                                                                                                  & $\cD_X$                                    \\ \hline & & \\
                $\LG(4,8)$ & $\tableau{13}{
                [aHL] 4    & [aH] 3s                                                                                                  & [aH] 2s & [aHR] 1s                         \\
                []         & [a]                                                                                                      & [a]s    & [a] s                            \\
                []         & []                                                                                                       & [a]     & [a]s                             \\
                []         & []                                                                                                       & []      & [a]
                } $        & $\begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,3,4},scale=2, backwards]
                                              {C}{ooo*}
                                      \end{dynkinDiagram}$                                               \\ & & \\
                $E_6/P_1$  & \qquad  $\tableau{\tableausz}{
                [Aa]1      & [Aa]3                                                                                                    & [Aa]4   & [Aa]2                            \\
                []         & []                                                                                                       & [Aa]5   & [a]      & [a] & [a]             \\
                []         & []                                                                                                       & [Aa]6   & [a]      & [a] & [a]             \\
                []         & []                                                                                                       & []      & []       & [a] & [a] & [a] & [a]
                }$  \qquad & $\begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,3,4,5,6},scale=2]
                                              {E}{*ooooo}
                                      \end{dynkinDiagram}$
        \end{tabular}

\end{figure}

The proof of \cref{lemma:biject} and \cref{cor:restrict} below are left to the reader.

\begin{lemma}\label{lemma:biject}
        The restriction \(\delta:\cP_X^\Delta\to\Delta_X\) is a bijection.
\end{lemma}

\begin{cor}\label{cor:restrict}
        Let \(\lambda\subseteq\cP_X\) be a lower order ideal. Then \(\delta(\lambda\cap\cP_X^\Delta)=\Delta_X^\lambda\), and \(\cD_X^\lambda\) is a connected Dynkin diagram.
\end{cor}

\begin{remark}
        \cref{cor:restrict} implies that \(X'\) is the cominuscule flag variety given by \((\cD_X^\lambda, \gamma)\).
\end{remark}

The last ingredient is \cref{prop:group_restriction}, a purely combinatorial result. Geometrically, it implies that the ``minimal'' cominuscule flag varieties for Schubert varieties with isomorphic labeled posets are isomorphic.

\begin{prop}\label{prop:group_restriction}
        Let \(\lambda\subseteq\cP_X\) and \(\mu\subseteq\cP_Y\) be lower order ideals. Then every labeled poset isomorphism between \(\lambda\) and \(\mu\) induces a graph isomorphism between \(\cD_X^\lambda\) and \(\cD_Y^\mu\) that identifies reduced decompositions of \(w_\lambda\) and \(w_\mu\).
\end{prop}

\begin{proof}[Proof of \cref{prop:converse}]
        Let $\gamma_X$ and $\gamma_Y$ denote the cominuscule simple roots corresponding to the cominuscule flag varieties $X$ and $Y$.  Let $X'$ and $Y'$ denote the cominuscule flag varieties given by the pairs \( (\cD_X^\lambda, \gamma_X)\) and \((\cD_Y^\mu, \gamma_Y)\).  \cref{prop:group_restriction} implies the cominuscule flag varieties \(X'\) and \(Y'\) are isomorphic.  By \cref{lemma:isom}, this isomorphism restricts to an isomorphism between \(X_\lambda\) and \(Y_\mu\), upon identifying them with Schubert varieties in \(X'\) and \(Y'\), respectively.
\end{proof}

We illustrate the above process with \cref{example:D4} and \cref{example:E6P5} below.

\begin{example}\label{example:D4}
        \begin{equation*}
                \cP_{Q^6}=\tableau{\tableausz}{
                [a]&[a]&[a]&[]\\
                []&[a]&[a]&[a]
                }\ \ \cong\ \
                \tableau{\tableausz}{[a]&[a]&[a]\\[]&[a]&[a]\\[]&[]&[a]}=\cP_{\OG(4,8)},
        \end{equation*}
        therefore, \(Q^6\cong \OG(4,8)\). This isomorphism comes from a symmetry of the \(D_4\) Dynkin diagram:
        \begin{center}
                \begin{dynkinDiagram}[text style/.style={scale=0.9},scale=1.5]{D}{*ooo}\end{dynkinDiagram}\(\cong\ \ \)
                \begin{dynkinDiagram}[text style/.style={scale=0.9},scale=1.5]{D}{ooo*}\end{dynkinDiagram}.
        \end{center}
\end{example}

\begin{example}\label{example:E6P5}
        Let \(X = E_6/P_6\) and \(\lambda\) be the lower order ideal depicted on the left below. Then \(S(\lambda) = \{s_1, s_3, s_4, s_5, s_6\}\), where \(s_i\) is the simple reflection corresponding to the simple root labeled by \(i\). Therefore, the pair \((\cD_X^\lambda,\gamma)\) is as depicted on the right, isomorphic to that of \(\P^5\), showing \(X_{\lambda} \cong X' \cong \P^5\).

        \begin{table}[ht]
                \centering
                \begin{tabular}{c c c}
                        \(X = E_6/P_6\)                                                                              &         & \(X'\)                                 \\ & & \\
                        \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,2,3,4,5,6},scale=2]
                                {E}{ooooo*}
                        \end{dynkinDiagram} &         &
                        \begin{dynkinDiagram}[text style/.style={scale=0.9},labels={1,3,4,5,6},scale=2]
                                {A}{oooo*}
                        \end{dynkinDiagram}
                        \\ & & \\
                        \(
                        \tableau{\tableausz}{
                        [AA]                                                                                         & [AA]    & [AA]   & [a]                           \\
                                                                                                                     &         & [AA]   & [a]  & [a]  & [a]             \\
                                                                                                                     &         & [AA]   & [a]  & [a]  & [a]             \\
                                                                                                                     &         &        &      & [a]  & [a] & [a] & [a]
                        }
                        \)                                                                                           & $\cong$ &
                        \(\tableau{\tableausz}{
                        [AA]                                                                                         & [AA]    & [AA]   & [AA] & [AA]
                        }\)                                                                                                                                             \\
                \end{tabular}
        \end{table}
\end{example}

\acknowledgements{We thank Anders Buch for helpful discussions.}

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\end{document}