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\title[]{Filtered RSK and matrix Schubert varieties}
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\author[]{\texorpdfstring{Abigail Price\thanks{\href{mailto:price29@illinois.edu}{price29@illinois.edu}. Supported by an
NSF RTG in Combinatorics (DMS 1937241).}\addressmark{1}, Ada Stelzer \thanks{\href{mailto:astelzer@illinois.edu}{astelzer@illinois.edu}. Supported by a Susan C.~Morisato IGL graduate student scholarship, an NSF graduate fellowship, and an
NSF RTG in Combinatorics (DMS 1937241).}\addressmark{1}, \and Alexander Yong\thanks{\href{mailto:ayong@illinois.edu}{ayong@illinois.edu}. Supported by a Simons Collaboration grant and an
NSF RTG in Combinatorics (DMS 1937241).}\addressmark{1}}{Abigail Price, Ada Stelzer, and Alexander Yong}}

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\address{\addressmark{1}Department of Mathematics, University of Illinois Urbana--Champaign}

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\received{14 November 2024}

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\revised{31 March 2025}

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\abstract{
Matrix Schubert varieties [Fulton ’92] carry natural actions of Levi groups. Their coordinate rings are thereby Levi-representations; what is a combinatorial counting rule for the multiplicities of their irreducibles? When the Levi group is a torus, [Knutson--Miller '04] answers the question. We present a general solution, a common refinement of the multigraded Hilbert series, the Cauchy identity, and the Littlewood--Richardson rule. The proof introduces a ``filtered’’ generalization of the Robinson--Schensted--Knuth correspondence and uses the operators of 
[Kashiwara '95] and of 
[Danilov--Koshevoi '05, van Leeuwen '06]. 
}

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\keywords{matrix Schubert varieties, RSK, crystal graphs, Levi groups}

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\begin{document}

\maketitle
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\section{Introduction}\label{sec:intro}
\textit{Matrix Schubert varieties}, introduced by Fulton in \cite{Fulton:duke}, are defined as Borel-orbit closures of partial permutation matrices in the space ${\sf Mat}_{m,n}$ of $m\times n$ complex matrices. They have been extensively studied; see, e.g., \cite{Knutson, KM:adv, Knutson.Miller, KMY}. Since each matrix Schubert variety $\mathfrak{X}_w$ is stable under the Borel action, a torus $\mathbf{T}$ also acts on $\mathfrak{X}_w$ by restriction. Knutson--Miller studied the multigraded Hilbert series of $\mathfrak{X}_w$, i.e., the $\mathbf{T}$-character of $\mathfrak{X}_w$, in \cite{Knutson.Miller} and \cite{KM:adv}. For appropriate choices of $\mathbf{I}$ and $\mathbf{J}$, $\mathfrak{X}_w$ also carries an action of a Levi group $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$. We give the first  manifestly-positive combinatorial rule for computing $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-graded Hilbert series of ${\mathfrak X}_w$ for arbitrary Levi groups $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$.

 Our rule simultaneously generalizes 
 the Littlewood--Richardson rule, the classical Cauchy identity, and 
 the Knutson--Miller formula (since $\mathbf{T}$ is a Levi group). The rule is based on a new, ``filtered'' generalization of the RSK algorithm. Its proof relies on Kashiwara's crystal basis theory (\cite{Kashiwara}). In fact, the rule extends to a class of varieties we call \textit{bicrystalline}, which includes all unions and intersections of matrix Schubert varieties.

\subsection{Statement of the representation-theoretic question}
Let $\mathbf{GL} = GL_m(\mathbb{C})\times GL_n(\mathbb{C})$, let $\mathbf{B}\leq \mathbf{GL}$ denote the Borel group of pairs of invertible lower triangular matrices, and let $\mathbf{T}\leq \mathbf{GL}$ denote the torus of pairs of invertible diagonal matrices. Let $\mathbf{L}_{\mathbf{I}|\mathbf{J}} := (GL_{i_1-i_0}\times \cdots \times GL_{i_{r}-i_{r-1}})\times (GL_{j_1-j_0}\times \cdots \times GL_{j_{s}-j_{s-1}})\leq \mathbf{GL}$ be a Levi subgroup, where ${\mathbf I} = \{0 = i_0<\dots<i_r = m\}$ and ${\mathbf J} = \{0 = j_0<\dots<j_s = n\}$. The finite-dimensional, irreducible polynomial representations of $GL_m$ are the \emph{Weyl modules} $V_\lambda(m)$, where $\lambda$ is a partition with at most $m$ parts. Analogously, the finite-dimensional, irreducible polynomial representations of $\mathbf{L}_{\mathbf{I}} = GL_{i_1-i_0}\times\dots\times GL_{i_r-i_{r-1}}$ are tensor products of Weyl modules. These \emph{split-Weyl modules} are indexed by partition-tuples $\underline\lambda = (\lambda^{(1)},\dots, \lambda^{(r)})$:
\[V_{\underline\lambda}(m) := V_{\lambda^{(1)}}(i_1-i_0)\boxtimes \dots\boxtimes V_{\lambda^{(r)}}(i_r-i_{r-1}).\]
The \emph{representation ring} ${\rm Rep}(G)$ of a reductive group $G$ (such as $\mathbf{L}_{\mathbf{I}}$) is the ring of (formal differences of) isomorphism classes of finite-dimensional, complex, polynomial $G$-representations, with ring operations given by direct sum and tensor product. We denote the isomorphism class of a $G$-representation $V$ in ${\rm Rep}(G)$ by $[V]$. The classes of irreducible representations form a $\Z$-linear basis for ${\rm Rep}(G)$. Thus for any finite-dimensional, polynomial $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-representation $V$, there exist unique nonnegative integers $c^V_{\underline\lambda|\underline\mu}$ such that
\begin{equation}\label{eqn:mainq}
    [V] = \sum_{\underline\lambda|\underline\mu}c^V_{\underline\lambda|\underline\mu}[V_{\underline\lambda}(m)\boxtimes V_{\underline\mu}(n)]\text{ in }{\rm Rep}(\mathbf{L}_{\mathbf{I}|\mathbf{J}}).
\end{equation}
Let $s_\lambda(\mathbf{x})$ and $s_{\underline\lambda}(\mathbf{x})$ denote the \emph{Schur} and \emph{split-Schur} polynomials, defined as the characters of $V_\lambda(m)$ and $V_{\underline\lambda}(m)$ respectively. Then taking characters of both sides of (\ref{eqn:mainq}) yields the equivalent polynomial identity
\[{\rm char}(V) = \sum_{\underline\lambda|\underline\mu}c^V_{\underline\lambda|\underline\mu}s_{\underline\lambda}(\mathbf{x})s_{\underline\mu}(\mathbf{y}).\]
What is a positive combinatorial rule for the constants $c^V_{\underline\lambda|\underline\mu}$ in the case where $V$ is the coordinate ring of some matrix Schubert variety?\footnote{Technically, such $V$ are usually infinite-dimensional, so $[V]$ is only an element of the completion $\widehat{{\rm Rep}(\mathbf{L}_{\mathbf{I}|\mathbf{J}})}$ of ${\rm Rep}(\mathbf{L}_{\mathbf{I}|\mathbf{J}})$. This causes no concern since coordinate rings of matrix Schubert varieties are graded, with each graded piece $V_i$ a finite-dimensional $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-representation. All copies of the split-Weyl module $V_{\underline\lambda}(m)\boxtimes V_{\underline\mu}(n)$ in $V$ lie in a single $V_i$, so each $c^V_{\underline\lambda|\underline\mu}$ in (\ref{eqn:mainq}) is finite.} This extended abstract of \cite{AAA} provides a complete answer to the question.

\subsection{Main new combinatorial ingredient: filtered RSK}\label{subsec:introcomb}

We introduce the \emph{filtered RSK algorithm}. It takes a nonnegative integer matrix as input and produces a tuple of \emph{semistandard Young tableaux} (not necessarily of the same shape).

\begin{definition}\label{def:matrixwords}
    The \emph{row word} of a matrix $M = [m_{ij}]\in {\sf Mat}_{m,n}({\mathbb Z}_{\geq 0})$, denoted ${\sf row}(M)$, records $m_{ij}$ copies of the row index $i$ for each entry $m_{ij}$ of $M$. The entries are read along columns top-to-bottom, from left to right. The \emph{column word} ${\sf col}(M)$ is the row word of the transpose matrix $M^t$.
\end{definition}
\begin{example}
    If $M = \begin{bmatrix}0 & 1 \\ 2 & 3\end{bmatrix}$ then ${\sf row}(M) = 221222$ and ${\sf col}(M) = 211222$.
\end{example}

\begin{definition}\label{def:filtration}
    For an integer sequence $\mathbf{I} = \{0=i_0<\dots<i_r = m\}$, the \emph{${\mathbf I}$-filtration} of a word $w$ on the alphabet $[m]$ is a tuple of words ${\sf filter}_{{\mathbf I}}(w) = (w^{(1)},\dots, w^{(r)})$. Each $w^{(k)}$ is the subword of $w$ consisting of all letters in the interval $[i_{k-1}+1, i_k]$.
\end{definition}

Let $\sim_K$ denote the Knuth equivalence relation on words, and let ${\sf word}(T)$ denote the reading word of a semistandard tableau $T$. Then each Knuth equivalence class contains ${\sf word}(T)$ for a unique tableau $T$. This tableau $T$ can be constructed from any word in its equivalence class via the row-insertion or jeu-de-taquin algorithms, see, e.g., \cite{Fulton}.

\begin{maindef}[Filtered RSK]\label{alg:filteredrsk}
    Let $M\in {\sf Mat}_{m,n}({\mathbb Z}_{\geq 0})$ and fix two integer sequences 
    \begin{equation}\label{eqn:twointseq}
    {\mathbf I} = \{0 = i_0<\dots<i_r = m\} \text{ \  and \ } {\mathbf J} = \{0 = j_0<\dots<j_s = n\}.
    \end{equation}
    Define ${\sf filterRSK}_{{\mathbf I}|{\mathbf J}}(M)= (\underline{P}|\underline{Q}):= (P^{(1)},\dots,P^{(r)}|Q^{(1)},\dots,Q^{(s)})$,
    where $P^{(a)}$ and $Q^{(b)}$ are the unique SSYT 
    with ${\sf word}(P^{(a)})\sim_K {\sf filter}_{{\mathbf I}}({\sf row}(M))^{(a)}$ and ${\sf word}(Q^{(b)})\sim_K {\sf filter}_{{\mathbf J}}({\sf col}(M))^{(b)}$, respectively.
\end{maindef}

We conflate monomials ${\sf m} = \prod_{i, j}z_{ij}^{m_{i,j}}$ with their exponent vectors $M = [m_{i,j}]\in{\sf Mat}_{m, n}(\Z_{\geq 0})$. This makes sense of notation such as ${\sf filterRSK}_{\mathbf{I}|\mathbf{J}}({\sf m})$ for a monomial ${\sf m}$.

\begin{example}
Let ${\mathbf I} = \{0,1,3\} = {\mathbf J}$ and let  $A = \begin{bmatrix}
        0 & 2 & 0 \\
        2 & 1 & 0 \\
        2 & 0 & 0
    \end{bmatrix}$. Then ${\sf row}(M)= 2233112$ and ${\sf col}(M) = 2211211$, so \[\ytableausetup{aligntableaux=center, boxsize=1em}
    {\sf filter}_{\mathbf{I}}({\sf row}(M)) = (11,22332) \text{\ \  and \ \ } {\sf filter}_{\mathbf{J}}({\sf col}(M)) = (1111,222).\] Now, consider the tuple of tableaux
    \[(P^{(1)},P^{(2)}|Q^{(1)},Q^{(2)})=\left(\ \begin{ytableau}1 & 1\end{ytableau}, \begin{ytableau}2 & 2 & 2 & 3\\ 3\end{ytableau}\ \bigg\vert \ 
        \begin{ytableau}1 & 1 & 1 & 1\end{ytableau}, \begin{ytableau}2 & 2 & 2\end{ytableau}\ \right).\]
    The reading words of these tableaux are $(11,32223|1111,222)$. Since $32223\sim_K 22332$, ${\sf filterRSK(M)} = (P^{(1)},P^{(2)}|Q^{(1)},Q^{(2)})$. 
\end{example}

\begin{definition}\label{def:highestweight}
    The \textit{highest-weight tableau} of \emph{shape} $\lambda$ and content $[a, b]$ is the tableau $T_{\lambda, [a, b]}$ of shape $\lambda$ taking the constant value $a-1+i$ on each row $i$. 
    For a given $\mathbf{I}$, the \emph{highest-weight tableau-tuple} $T_{\underline\lambda}$ of \emph{shape} $\underline\lambda$
    has components $T_{\underline\lambda}^{(k)} := T_{\lambda^{(k)}, [i_{k-1}+1, i_k]}$.
\end{definition}

\Cref{thm:main1} below foreshadows our main result, \Cref{thm:main2}:
\begin{theorem}\label{thm:main1}
    Let $\C[{\sf Mat}_{m, n}]$ be the ring of polynomial functions on $m\times n$ matrices, viewed as an $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-representation. A positive combinatorial rule for the coefficients $c^{\C[{\sf Mat}_{m, n}]}_{\underline\lambda|\underline\mu}$ in (\ref{eqn:mainq}) is then: 
    \[c^{\C[{\sf Mat}_{m, n}]}_{\underline\lambda|\underline\mu}=\#\{M\in{\sf Mat}_{m, n}(\Z_{\geq 0}): {\sf filterRSK}_{{\mathbf I}|{\mathbf J}}(M)=
    (T_{\underline\lambda}|T_{\underline\mu})\}.\]
\end{theorem}
When $\mathbf{I}=\{0, m\}$ and $\mathbf{J}=\{0, n\}$, ${\sf filterRSK}_{\mathbf{I}|\mathbf{J}}$ is the classical RSK correspondence. In this special case, \Cref{thm:main1} is a restatement of Howe duality, and the corresponding character formula is the classical Cauchy identity:

\begin{equation}\label{eqn:Cauchy}
\prod_{\substack{1\leq i\leq m \\ 1\leq j\leq n}} \frac{1}{1-x_i y_j} = \sum_{\lambda} s_{\lambda}(x_1,\ldots,x_m)
s_{\lambda}(y_1,\ldots,y_n).
\end{equation}

We sketch the proof of \Cref{thm:main1} in \Cref{subsec:crystalops}; it involves van Leeuwen and Danilov--Koshevoi's ``pull-back'' of Kashiwara's crystal operators to bicrystal operators on ${\sf Mat}_{m, n}(\Z_{\geq 0})$. In the next section, we generalize \Cref{thm:main1} by replacing ${\sf Mat}_{m,n}$ with an arbitrary matrix Schubert variety.

\subsection{Matrix Schubert varieties and the main theorem}
The Borel group $\mathbf{B}$ acts on ${\sf Mat}_{m,n}$ on the right by 
$(b_1,b_2)\cdot M=b_1^{-1}M (b_2^{-1})^{T}$. This convention yields a left action of the torus $\mathbf{T}\leq\mathbf{B}$ on the coordinate ring $\C[{\sf Mat}_{m, n}]$ and ensures that the multigrading induced by this $\mathbf{T}$-action is positive. We recall the definition of matrix Schubert varieties, following Fulton's construction in \cite{Fulton:duke}.

\begin{definition}\label{def:partialperm}
    A \emph{partial permutation matrix} $M_w\in{\sf Mat}_{m, n}$ is a matrix with at most one $1$ in each row and column and $0$s everywhere else. The indexing \emph{partial permutation} is a function $w:[m]\to[n]\cup\{\infty\}$, where $w(i) = j\in[n]$ if $M_w$ has a $1$ in position $(i, j)$ and $w(i) = \infty$ if $M_w$ has no $1$ in row $i$.
\end{definition}

\begin{theorem}[{\cite[Lemma 3.1]{Fulton:duke}}]\label{thm:Mar6abc}
    Every $\mathbf{B}$-orbit in ${\sf Mat}_{m, n}$ contains a unique $M_w$.
\end{theorem}

\begin{definition}[{\cite[Proposition 3.3]{Fulton:duke}}]\label{thm:MSV}
   The \emph{matrix Schubert variety} ${\mathfrak X}_w\subseteq{\sf Mat}_{m, n}$ is the Zariski closure of the orbit $\mathbf{B}\cdot M_w$. Each ${\mathfrak X}_w$ is irreducible.
\end{definition} 

\begin{definition}
    The \emph{row descent set} of $M_w$, denoted $\desc_{\sf row}(w)$, consists of row indices $i$ such that $w(i) > w(i+1)$ (where $\infty > k$ for all $k\in[n]$). The \emph{column descent set} of $M_w$, denoted $\desc_{\sf col}(w)$, is the row descent set of the transpose matrix $M_w^t$.
\end{definition}

\begin{remark}\label{remark:wbar}
    By \cite[Equation 3.4]{Fulton:duke}, every  $\mathfrak{X}_w\subseteq {\sf Mat}_{m, n}$ is equal (up to a product with affine space) to some $\mathfrak{X}_{\overline{w}}\subseteq {\sf Mat}_{m+n, m+n}$ indexed by a full permutation $\overline{w}\in\mathfrak{S}_{m+n}$.
    The row and column descent sets of $w$ are then the descent sets of $\overline{w}$ and $\overline{w}\inv$ respectively.
\end{remark}

\begin{proposition}\label{prop:leviacts}
    Let $\mathfrak{X}_w\subseteq{\sf Mat}_{m, n}$ be a matrix Schubert variety. If $\desc_{\sf row}(w)\subseteq\mathbf{I}$ and $\desc_{\sf col}(w)\subseteq\mathbf{J}$, then $\mathfrak{X}_w$ is stable under the action of the Levi group $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$ on ${\sf Mat}_{m, n}$.
\end{proposition}
\begin{proof}
    The argument is straightforward; see \cite[Proposition~6.9]{AAA}.
\end{proof}
When $\mathfrak{X}_w$ is stable under the action of $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$, the coordinate ring $\C[\mathfrak{X}_w]$ is an $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-representation. Now, the Gr\"obner basis methods of Knutson--Miller \cite{Knutson.Miller, KM:adv} identify a set of monomials that form a basis for $\C[\mathfrak{X}_w]$ as a vector space. This basis, denoted ${\sf Std}_\prec(\mathfrak{X}_w)$, is called the set of \emph{standard monomials} of $\mathfrak{X}_w$. We recall an explicit description of ${\sf Std}_\prec(\mathfrak{X}_w)$ in \Cref{subsec:MSVstdmono}. We now state our main theorem.

\begin{MainThm}\label{thm:main2}
    Let $\mathfrak{X}_w\subseteq{\sf Mat}_{m, n}$ be a matrix Schubert variety and let $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$ be a Levi group with $\mathbf{I}\supseteq\desc_{{\sf row}}(w)$ and $\mathbf{J}\supseteq\desc_{{\sf col}}(w)$. Then the coefficients in the expression (\ref{eqn:mainq}) for the class of $\C[\mathfrak{X}_w]$ in ${\rm Rep}(\mathbf{L}_{\mathbf{I}|\mathbf{J}})$ are given by the positive combinatorial rule
    \begin{equation}\label{eqn:MSVrule}
        c^{\C[\mathfrak{X}_w]}_{\underline\lambda|\underline\mu} = \#\{{\sf m}\in {\sf Std}_\prec(\mathfrak{X}_w) :  {\sf filterRSK}_{{\mathbf I}|{\mathbf J}}({\sf m})=
        (T_{\underline\lambda}|T_{\underline\mu})\}.
    \end{equation}
\end{MainThm}

We remark that this result is strongest when $\mathbf{I}$ and $\mathbf{J}$ are chosen minimally. In \Cref{sec:MSVs}, we sketch the proof of \Cref{thm:main2}, showing how it follows from \Cref{thm:main1} and a compatibility between the Knutson--Miller description of ${\sf Std}_\prec(\mathfrak{X}_w)$ and the bicrystal operators of van Leeuwen and Danilov--Koshevoi. Our method of proof in fact extends to give an analogue of \Cref{thm:main2} for a new class of varieties we call \emph{bicrystalline}. All unions and intersections of matrix Schubert varieties are bicrystalline; in particular, by \Cref{thm:borelunion}, every $\textbf{B}$-stable variety is bicrystalline.

\Cref{thm:main2} generalizes the study of $\mathbf{GL}$-stable varieties $\mathfrak{X}\subset{\sf Mat}_{m, n}$, which are exactly the \emph{classical determinantal varieties} $\mathfrak{X}_k$ of rank $\leq k$ matrices. The $\mathbf{T}$-characters of these varieties have been long studied (see, e.g., \cite{Abhyankar, dCP, DRS, Galigo, Ghorpade, Herzog.Trung}).

\begin{example}
Let ${\mathfrak X}\subseteq{\sf Mat}_{4, 4}$ be the space of $4\times 4$ matrices of rank $\leq 3$ whose northwest $1\times 1$ submatrix has rank $0$. By a result of Fulton stated below as \Cref{theorem:Fultonsresult}, $\mathfrak{X}$ is the matrix Schubert variety
$\mathfrak{X}_w\subseteq{\sf Mat}_{4, 4}$ for $w = 213\infty$ (\Cref{def:partialperm} explains this notation). Now, $\mathfrak{X}$ is stable under the action of the Levi group $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$ with ${\bf I} = \{0, 1, 4\} = {\bf J}$:  applying row operations to the first row (respectively, column) and the second through fourth rows (respectively, columns) of a matrix $M\in {\mathfrak X}$ returns matrices in 
${\mathfrak X}$. 

The Knutson--Miller Gr\"obner basis theorem \cite[Theorem B]{Knutson.Miller} restated here as \Cref{thm:KMgrob} shows that \[A = \begin{bmatrix}
    0 & 0 & 1 & 0\\
    0 & 2 & 0 & 0\\
    1 & 0 & 0 & 0\\
    0 & 0 & 0 & 0
\end{bmatrix} \text{\  \ and \ \ } B = \begin{bmatrix}
    0 & 1 & 0 & 0\\
    1 & 0 & 1 & 0\\
    0 & 1 & 0 & 0\\
    0 & 0 & 0 & 0
\end{bmatrix}\] are (exponent vectors of) standard monomials of $\C[{\mathfrak X}]$. 
Since ${\sf row}(A)={\sf col}(A)=3221$ and ${\sf row}(B)={\sf col}(B)=2132$, applying ${\sf filterRSK}_{{\mathbf I}|{\mathbf J}}$ to either matrix yields
\[(T_{\underline\lambda}|T_{\underline\mu}) = \left(\ \begin{ytableau}1\end{ytableau}, \begin{ytableau}2 & 2\\ 3\end{ytableau}\  \bigg\vert \ 
        \begin{ytableau}1\end{ytableau}, \begin{ytableau}2 & 2\\ 3\end{ytableau}\  \right).\] 
In fact, $A$ and $B$ are the only two such standard monomials, so $c_{\underline{\lambda}|\underline{\mu}}^{\C[\mathfrak X]} = 2$. The character formula for ${\mathbb C}[{\mathfrak X}_{213\infty}]$ implied by \Cref{thm:main2} begins:
\ytableausetup{boxsize=0.25em,aligntableaux=center}
\begin{multline*} 
s_{(\emptyset,\emptyset)}({\bf x})s_{(\emptyset,\emptyset)}({\bf y}) + s_{(\emptyset,\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{1})}({\bf y}) + s_{(\ydiagram{1},\emptyset)}({\bf x})s_{(\emptyset,\ydiagram{1})}({\bf y}) + s_{(\emptyset,\ydiagram{1})}({\bf x})s_{(\ydiagram{1},\emptyset)}({\bf y}) +\\ s_{(\emptyset,\ydiagram{1,1})}({\bf x})s_{(\emptyset,\ydiagram{1,1})}({\bf y}) +
s_{(\ydiagram{1},\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{1,1})}({\bf y}) + s_{(\emptyset,\ydiagram{2})}({\bf x})s_{(\emptyset,\ydiagram{2})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{2})}({\bf y}) +\\ s_{(\ydiagram{2},\emptyset)}({\bf x})s_{(\emptyset,\ydiagram{2})}({\bf y}) + s_{(\emptyset,\ydiagram{1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1})}({\bf y}) + s_{(\emptyset,\ydiagram{2})}({\bf x})s_{(\ydiagram{1},\ydiagram{1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1})}({\bf y}) + \\s_{(\emptyset,\ydiagram{2})}({\bf x})s_{(\ydiagram{2},\emptyset)}({\bf y}) + s_{(\emptyset,\ydiagram{1,1,1})}({\bf x})s_{(\emptyset,\ydiagram{1,1,1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{1,1})}({\bf x})s_{(\emptyset,\ydiagram{1,1,1})}({\bf y}) +
s_{(\emptyset,\ydiagram{2,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1})}({\bf y}) +\\ s_{(\ydiagram{1},\ydiagram{1,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{2})}({\bf x})s_{(\emptyset,\ydiagram{2,1})}({\bf y}) + s_{(\ydiagram{2},\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{2,1})}({\bf y}) + s_{(\emptyset,\ydiagram{1,1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1})}({\bf y}) +\\
s_{(\emptyset,\ydiagram{2,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{2})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1})}({\bf y}) + s_{(\emptyset,\ydiagram{3})}({\bf x})s_{(\emptyset,\ydiagram{3})}({\bf y}) +\\ s_{(\ydiagram{1},\ydiagram{2})}({\bf x})s_{(\emptyset,\ydiagram{3})}({\bf y}) +
s_{(\ydiagram{2},\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{3})}({\bf y}) + s_{(\ydiagram{3},\emptyset)}({\bf x})s_{(\emptyset,\ydiagram{3})}({\bf y}) + s_{(\emptyset,\ydiagram{2,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2})}({\bf y}) + \\s_{(\ydiagram{1},\ydiagram{1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2})}({\bf y}) +
s_{(\emptyset,\ydiagram{3})}({\bf x})s_{(\ydiagram{1},\ydiagram{2})}({\bf y})+ s_{(\ydiagram{1},\ydiagram{2})}({\bf x})s_{(\ydiagram{1},\ydiagram{2})}({\bf y}) + s_{(\ydiagram{2},\ydiagram{1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2})}({\bf y}) +\\ s_{(\emptyset,\ydiagram{2,1})}({\bf x})s_{(\ydiagram{2},\ydiagram{1})}({\bf y}) +
s_{(\emptyset,\ydiagram{3})}({\bf x})s_{(\ydiagram{2},\ydiagram{1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{2})}({\bf x})s_{(\ydiagram{2},\ydiagram{1})}({\bf y}) + s_{(\emptyset,\ydiagram{3})}({\bf x})s_{(\ydiagram{3},\emptyset)}({\bf y}) +\\ s_{(\emptyset,\ydiagram{2,1,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1,1})}({\bf y}) + \displaybreak[3]
s_{(\ydiagram{1},\ydiagram{1,1,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1,1})}({\bf y}) +s_{(\ydiagram{1},\ydiagram{2,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1,1})}({\bf y}) + s_{(\ydiagram{2},\ydiagram{1,1})}({\bf x})s_{(\emptyset,\ydiagram{2,1,1})}({\bf y}) + \\s_{(\emptyset,\ydiagram{2,1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1,1})}({\bf y}) + s_{(\ydiagram{1},\ydiagram{2,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{1,1,1})}({\bf y}) +
s_{(\emptyset,\ydiagram{2,2})}({\bf x})s_{(\emptyset,\ydiagram{2,2})}({\bf y}) +s_{(\ydiagram{1},\ydiagram{2,1})}({\bf x})s_{(\emptyset,\ydiagram{2,2})}({\bf y}) + \\s_{(\ydiagram{2},\ydiagram{2})}({\bf x})s_{(\emptyset,\ydiagram{2,2})}({\bf y}) +s_{(\emptyset,\ydiagram{3,1})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) +
s_{(\ydiagram{1},\ydiagram{2,1})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) + s_{(\ydiagram{2},\ydiagram{1,1})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) + \\s_{(\ydiagram{1},\ydiagram{3})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) +s_{(\ydiagram{2},\ydiagram{2})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) +
s_{(\ydiagram{3},\ydiagram{1})}({\bf x})s_{(\emptyset,\ydiagram{3,1})}({\bf y}) + s_{(\emptyset,\ydiagram{2,1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2,1})}({\bf y}) + \\s_{(\ydiagram{1},\ydiagram{1,1,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2,1})}({\bf y}) + s_{(\emptyset,\ydiagram{2,2})}({\bf x})s_{(\ydiagram{1},\ydiagram{2,1})}({\bf y})+ s_{(\emptyset,\ydiagram{3,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2,1})}({\bf y})
+ 2s_{(\ydiagram{1},\ydiagram{2,1})}({\bf x})s_{(\ydiagram{1},\ydiagram{2,1})}({\bf y})+\cdots.
\end{multline*}
\ytableausetup{boxsize=normal,aligntableaux=top}
\end{example}

\begin{remark}[\Cref{thm:main2} and Littlewood--Richardson coefficients]
Let ${\bf I}=\{0<t<m\}$, ${\bf J}=\{0,n\}$, and $V = \C[{\sf Mat}_{m, n}]$. In this case, \Cref{thm:main1} shows that the coefficients $c^{V]}_{\underline\lambda|\underline\mu}$ in (\ref{eqn:mainq}) are exactly the \emph{Littlewood--Richardson coefficients} $c_{\lambda^{(1)}, \lambda^{(2)}}^\mu$ (in their coproduct role in the
Hopf algebra of symmetric functions).
Thus \Cref{thm:main1} and \Cref{thm:main2} are generalized Littlewood--Richardson rules. 
\end{remark}

\section{Proof of Main \texorpdfstring{\Cref{thm:main2}}{Theorem}} \label{sec:MSVs}
\subsection{Crystal operators and bicrystalline varieties}\label{subsec:crystalops}
We briefly review crystal operators on words and matrices, define bicrystalline varieties, and explain how \Cref{thm:main1} yields combinatorial rules for the coefficients $c^V_{\underline\lambda|\underline\mu}$ in (\ref{eqn:mainq}) whenever $V = \C[\mathfrak{X}]$ for a bicrystalline variety $\mathfrak{X}$. Let $w = w_1w_2\dots w_N$ be a word on the alphabet $[n]$ and fix $i\in [n-1]$. Kashiwara (\cite{Kashiwara}) defined crystal operators on words:

\begin{definition}
    The \textit{ith bracket operator} ${\sf bracket}_i$ sends $w$ to a word on the alphabet $\{(, )\}$ by recording a ``$)$'' for each $i$ and a ``$($'' for each $i+1$ (maintaining order and ignoring all other letters in $w$).
\end{definition}
    
\begin{definition}\label{def:wordcrystal}
    Let $w_e = i+1$ and $w_f = i$ be the letters of $w$ associated to the leftmost unmatched ``$($'' and rightmost unmatched ``$)$'' of ${\sf bracket}_i(w)$ respectively. The \textit{crystal raising operator} $e_i$ sends $w$ to the word obtained by changing $w_e$ to $i$. Analogously, the \textit{crystal lowering operator} $f_i$ sends $w$ to the word obtained by changing $w_f$ to $i+1$. If no such letters $w_e$ or $w_f$ exist, the operators output the special symbol $\varnothing$.
\end{definition}

Danilov--Koshevoi and van Leeuwen \cite{bicrystal2, bicrystal1} pulled these crystal operators back to commuting operators on ${\sf Mat}_{m, n}$ via the row and column words of \Cref{def:matrixwords}:

\begin{definition}\label{def:feb11bicrystal}
    Let $M = [m_{i, j}]\in {\sf Mat}_{m,n}({\mathbb Z}_{\geq 0})$, and let $(i+1, b)$ and $(i, a)$ be the coordinates of the entries of $M$ yielding the letters of ${\sf row}(M)$ altered by $e_i$ and $f_i$ respectively. The \textit{row crystal raising operator} $e_i^{\sf row}$ sends $M$ to the matrix $e_i^{\sf row}(M)$ obtained by subtracting one from $m_{i+1, b}$ and adding one to $m_{i, b}$. Similarly, the \textit{row crystal lowering operator} $f_i^{\sf row}$ sends $M$ to the matrix $f_i^{\sf row}(M)$ obtained by subtracting one from $m_{i, a}$ and adding one to $m_{i+1, a}$. When $e_i({\sf row}(M)) = \varnothing$ or $f_i({\sf row}(M)) = \varnothing$, we define the corresponding row bicrystal operators to output the special symbol $\varnothing$ instead of a matrix.
    The \textit{column bicrystal operators} $e_j^{\sf col}$ and $f_j^{\sf col}$ are defined analogously using ${\sf col}(M)$ (or using transposes: $e_j^{\sf col}(M) = (e_j^{\sf row}(M^t))^t$ and $f_j^{\sf col}(M) = (f_j^{\sf row}(M^t)^t)$).
\end{definition}

Identifying  ${\sf m} = \prod_{i, j}z_{ij}^{m_{i, j}}$ with its exponent vector $M = [m_{i, j}]\in {\sf Mat}_{m, n}(\Z_{\geq 0})$, we obtain bicrystal operators on monomials. This permits a proof-sketch of \Cref{thm:main1} based on a non-obvious compatibility between (filtered) RSK and the bicrystal operators.

\begin{proof}[Proof of \Cref{thm:main1} (sketch)]
    We first consider the case where $\mathbf{L}_{\mathbf{I}|\mathbf{J}} = \mathbf{GL}$, for which the argument appears in \cite{bicrystal2, bicrystal1}.
    Since the set $\mathcal{M}$ of all monomials is a $\mathbf{T}$-graded basis for $\C[{\sf Mat}_{m,n}]$ and all irreducible $\mathbf{T}$-representations are $1$-dimensional, the $\mathbf{T}$-character of $\C[{\sf Mat}_{m, n}]$ is the sum of the $\mathbf{T}$-weights of all monomials. Now, we have the expression
    \[\mathcal{M} = \bigsqcup_{{\sf m}\in\mathcal{M}}\mathcal{C}_{\sf m},\] 
    where the union is over monomials ${\sf m}$ such that ${\sf RSK}({\sf m})$ is a pair of highest-weight tableaux, and $\mathcal{C}_{\sf m}$ is the set of monomials ${\sf m}'$ connected to ${\sf m}$ by a sequence of the bicrystal operators.
    If ${\sf RSK}({\sf m}) = (T_\lambda|T_\lambda)$, then the sum over the $\mathbf{T}$-weights of the monomials in $\mathcal{C}_{\sf m}$ is the character of the irreducible $\mathbf{GL}$-representation $V_\lambda(m)\boxtimes V_\lambda(n)$.
    This fact can be proved by showing that the row and column crystal operators in \Cref{def:feb11bicrystal} commute with each other, and thus, via RSK, commute with the usual crystal operators on pairs of semistandard Young tableaux (see \cite{BS} for the definition of these operators).
    Our proof of \Cref{thm:main1} in \cite{AAA} is analogous; we replace each $\mathcal{C}_{\sf m}$ with a ``filtered'' version $\mathcal{C}_{\sf m}^{\mathbf{I}|\mathbf{J}}$ connected by the bicrystal operators $e_i^{\sf row}, f_i^{\sf row}, e_j^{\sf col}, f_j^{\sf col}$ with $i\notin\mathbf{I}$, $j\notin\mathbf{J}$. We then show that these operators commute with the usual crystal operators on a product of tableau crystal graphs via ${\sf filterRSK}$.
    Thus, the sum over the $\mathbf{T}$-weights of the monomials in any $\mathcal{C}_{\sf m}^{\mathbf{I}|\mathbf{J}}$ is the character of some irreducible representation of ${\bf L}_{{\bf I}|{\bf J}}$, proving the theorem. 
\end{proof}

Given any variety $\mathfrak{X}\subseteq{\sf Mat}_{m, n}$ and a choice of term order $\prec$, there exists a set ${\sf Std}_\prec(\mathfrak{X})$ of monomials called \emph{standard monomials} which form a vector space basis for $\C[\mathfrak{X}]$. The set ${\sf Std}_\prec(\mathfrak{X})$ can be computed algorithmically using Gr\"obner bases. The next definition concerns a compatibility between standard monomials and the bicrystal operators.

\begin{definition}
${\mathfrak X}\subseteq {\sf Mat}_{m,n}$ is ${\bf L}_{{\mathbf I}|{\mathbf J}}$-\emph{bicrystal closed} if there exists a term order $\prec$ such that $e_i^{\sf row}({\sf m}),f_i^{\sf row}({\sf m}), e_j^{\sf col}({\sf m}), f_j^{\sf col}({\sf m}) \in  {\sf Std}_\prec(\mathfrak{X})\cup \{\varnothing\}$ for ${\sf m} \in {\sf Std}_\prec(\mathfrak{X}), i\not\in \mathbf{I}$, and $j\not\in\mathbf{J}$.
\end{definition}
    
\begin{definition} $\mathfrak{X}\subseteq{\sf Mat}_{m, n}$ is ${\mathbf L}_{{\mathbf I}|{\mathbf J}}$-\emph{bicrystalline} if
it is $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-stable and $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystal closed.
\end{definition}

\begin{corollary}\label{cor:bicrystalline}
    If $V = \C[\mathfrak{X}]$ for a $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline variety $\mathfrak{X}\subseteq{\sf Mat}_{m, n}$, then the coefficients $c^V_{\underline\lambda|\underline\mu}$ in (\ref{eqn:mainq}) are computed by the combinatorial rule
    \begin{equation}\label{eqn:bicrystalline}
        c^{\C[\mathfrak{X}]}_{\underline\lambda|\underline\mu} = \#\{{\sf m}\in {\sf Std}_\prec(\mathfrak{X}) :  {\sf filterRSK}_{{\mathbf I}|{\mathbf J}}({\sf m})=
  (T_{\underline\lambda}|T_{\underline\mu})\}.
  \end{equation}
\end{corollary}

\Cref{cor:bicrystalline} is an immediate consequence of the proof of \Cref{thm:main1} and the definition of $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline varieties. Comparing (\ref{eqn:MSVrule}) and (\ref{eqn:bicrystalline}), we see that showing $\mathfrak{X}_w$ is $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline will suffice to prove \Cref{thm:main2}. 

\subsection{Matrix Schubert varieties}\label{subsec:MSVs}
We now turn to proving \Cref{thm:main2}. Equations for the matrix Schubert varieties $\mathfrak{X}_w\subseteq{\sf Mat}_{m, n}$ were given in \cite{Fulton:duke}. We recall the standard permutation combinatorics needed.

The \emph{graph} of a partial permutation matrix $M_w$ is an $m\times n$ grid with a $\bullet$ symbol in the entries where $M_w$ has a $1$ and blank spaces elsewhere. The \emph{Rothe diagram} of $M_w$, denoted $D(w)$, consists of all boxes in $[m]\times[n]$ not weakly below or right of a $\bullet$ in the graph of $M_w$. 
The \emph{essential set} $E(w)$ of $M_w$ is comprised of the maximally southeast boxes of each connected component of $D(w)$, i.e.,
$E(w)=\{(i,j)\in D(w): (i,j+1), (i+1,j)\not\in D(w)\}$. The \emph{rank function} for $M_w$ is denoted $r_w:[m]\times[n]\to \Z_{\geq 0}$; it maps $(i, j)$ to the rank of the northwest $i\times j$ submatrix of $M_w$.
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.5]
    \fill [gray, opacity  = 0.25] (0,4) rectangle (5,3);
    \fill [gray, opacity  = 0.25] (1,2) rectangle (5,0);
    \fill [gray, opacity  = 0.25] (6,2) rectangle (9,1);
    \draw (0,4)--(9,4)--(9,0)--(0,0)--(0,4);
    \draw (1,0) -- (1,4);
    \draw (2,0) -- (2,4);
    \draw (3,0) -- (3,4);
    \draw (4,0) -- (4,4);
    \draw (5,0) -- (5,4);
    \draw (6,0) -- (6,4);
    \draw (7,0) -- (7,4);
    \draw (8,0) -- (8,4);
    \draw (0,0) -- (9,0);
    \draw (0,1) -- (9,1);
    \draw (0,2) -- (9,2);
    \draw (0,3) -- (9,3);
 
    \node at (-8,2) {$\left[\begin{matrix}
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\
\end{matrix}\right]$};

    \node at (5.5,3.5) {$\bullet$};
    \node at (0.5,2.5) {$\bullet$};
    \node at (6.5,0.5) {$\bullet$};
    \draw[line width = 0.35mm] (5.5,0) -- (5.5,3.5) -- (9,3.5);
    \draw[line width = 0.35mm] (0.5,0) -- (0.5,2.5) -- (9,2.5);
    \draw[line width = 0.35mm] (6.5,0) -- (6.5,0.5) -- (9,0.5);
    \end{tikzpicture}
    \caption{The partial permutation matrix $M_w$ for $w = {61\infty 7}$ and its Rothe diagram; the boxes of $D(w)$ are shaded.
    \label{fig:61infinity7}}
\end{figure}
\begin{example}
\Cref{fig:61infinity7} depicts $M_w\in{\sf Mat}_{4, 9}$ and $D(w)$ for $w=61\infty 7$. Here, $\desc_{\sf row}(w)=\{1,3\}$ and $\desc_{\sf col}(w)=\{5\}$. The associated permutation $\overline{w}\in\mathfrak{S}_{4+9}$ from Remark~\ref{remark:wbar} is
${\overline w}=61(10)7234589(11)(12)(13)$.
\end{example}

We now state a concrete description of matrix Schubert varieties.
Write $\C[{\sf Mat}_{m, n}]=\C[z_{ij}:1\leq i\leq m,1\leq j\leq n]$, where $z_{ij}$ is the $(i,j)$-coordinate function.

\begin{theorem}[{\cite[Proposition~3.3]{Fulton:duke}}]\label{theorem:Fultonsresult}
    ${\mathfrak X}_w\subset {\sf Mat}_{m, n}$ is the set of $m\times n$ matrices $M$ such that the rank of the northwest $i\times j$ submatrix of $M$ has rank at most $r_w(i, j)$.

    Let $Z=[z_{ij}]_{1\leq i\leq m, 1\leq j\leq n}$ be the generic $m\times n$ matrix and set $Z_{ij}$ to be the northwest $i\times j$ submatrix of $Z$.
    Then the defining ideal of $\mathfrak{X}_w\subset{\sf Mat}_{m, n}$ is the \emph{Schubert determinantal ideal}
    \begin{equation}\label{eqn:thegenerators}
        I_w := I(\mathfrak{X}_w) = \langle \text{rank $r_w(i, j)+1$ minors of $Z_{ij}:$ $(i,j)\in E(w)$}\rangle.
    \end{equation}
\end{theorem}

By \Cref{prop:leviacts}, $\C[\mathfrak{X}_w]$ is a $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-representation for any ${\bf I}\supseteq \desc_{{\sf row}}(w)$ and ${\bf J}\supseteq \desc_{{\sf col}}(w)$. We now check that 
${\mathfrak X}_w$ is ${\bf L}_{{\bf I}|{\bf J}}$-crystal closed in order to compute the $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-graded Hilbert series of $\C[\mathfrak{X}_w]$.

\subsection{Standard monomials of \texorpdfstring{$\mathfrak{X}_w$}{matrix Schubert varieties} and proof of \texorpdfstring{\Cref{thm:main2}}{the Main Theorem}}\label{subsec:MSVstdmono}
Fix an \emph{antidiagonal term order} $\prec$ on $S=\C[z_{11},\dots, z_{mn}]$, i.e., one that picks the
antidiagonal term of any minor of $Z$. One example is pure lexicographic order obtained by setting $z_{ab}\succ z_{cd}$ if $a<c$, or if $a=c$ and $b>d$. This is Knutson--Miller's Gr\"obner basis theorem:

\begin{theorem}[{\cite[Theorem~B]{Knutson.Miller}}]\label{thm:KMgrob}
Fulton's generators (\ref{eqn:thegenerators}) for $I_w$ form a Gr\"obner basis with respect to $\prec$.
\end{theorem}

Identifying monomials with nonnegative integer matrices as before, 
\Cref{thm:KMgrob} yields the following description of the standard monomials of $\mathfrak{X}_w$:

\begin{corollary}\label{cor:groebner}
    A monomial ${\sf m}$ lies in ${\sf Std}_\prec(\mathfrak{X}_w)$ if and only if for any Fulton generator $g$ of $I_w$ from
    (\ref{eqn:thegenerators}), the product of the entries of ${\sf m}$ along the main antidiagonal of $g$ is $0$.
\end{corollary}

The main result, \Cref{thm:main2}, follows immediately from \Cref{thm:MSVLtype} below.

\begin{theorem}\label{thm:MSVLtype}
    Let $\mathfrak{X}_w\subseteq{\sf Mat}_{m, n}$ and suppose $\mathbf{I}\supseteq\desc_{{\sf row}}(w)$ and $\mathbf{J}\supseteq\desc_{{\sf col}}(w)$. Then
    $\mathfrak{X}_w$ is ${\bf L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline.
\end{theorem}
\begin{proof}
By \Cref{prop:leviacts}, ${\mathfrak X}_w$ is $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-stable. It remains to show that $\mathfrak{X}_w$ is $\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-bicrystal closed. We focus first on the row operators $e_i^{\sf row}$ and $f_i^{\sf row}$. 
Since $e_i^{\sf row}$ and $f_i^{\sf row}$ are inverses whenever their outputs are not $\varnothing$, the 
$\mathbf{L}_{{\mathbf I}|{\mathbf J}}$-bicrystal closed claim follows from the (slightly stronger)
statement that if ${\sf m}\not\in {\sf Std}_\prec(\mathfrak{X}_w)$ (identified with a matrix in ${\sf Mat}_{m,n}({\mathbb Z}_{\geq 0})$), then $e_i^{\sf row}({\sf m})\not\in {\sf Std}_\prec(\mathfrak{X}_w)$ for all $i$ and $f_i^{\sf row}({\sf m})\not\in {\sf Std}_\prec(\mathfrak{X}_w)$ for $i\not\in\desc_{{\sf row}}(w)$. We prove this latter statement. 

Since ${\sf m}\not\in {\sf Std}_\prec(\mathfrak{X}_w)$, by \Cref{cor:groebner}, there exists a Fulton generator $g$ for $I_w$ such that the product of all entries of ${\sf m}$ along the antidiagonal $A_g$ of $g$ is nonzero. 
Let $R\subseteq [m]$ and $C\subseteq [n]$ be the row and column indices, respectively, of the minor $g$.

First, we argue that $e_i^{\sf row}({\sf m})\not\in {\sf Std}_\prec(\mathfrak{X}_w)$ by constructing a Fulton generator $g'$ for $I_w$ such that the product of all entries of $e_i^{\sf row}({\sf m})$ 
along $A_{g'}$ is positive.

\medskip
\noindent
{\sf Case e1:} ($i,i+1\not\in R$) Here, $e_i^{\sf row}$ does not affect any of the entries of $A_g$ in ${\sf m}$. Take $g' = g$.

\medskip
\noindent
{\sf Case e2:} ($i\in R, i+1\not\in R$) All entries of $A_g$ in $e_i^{\sf row}({\sf m})$ are only larger (by at most $1$)
in comparison to the same entry in ${\sf m}$, so we may again take $g' = g$.

\medskip
\noindent
{\sf Case e3:} ($i\not\in R, i+1\in R$) If $e_i^{\sf row}$ does not affect the entry of $A_g$ in row $i+1$,
take $g'=g$. Otherwise, we may take $g'$ to be the minor defined by row indices $R' = (R\setminus\{i+1\})\cup \{i\}$ and column indices $C$ (which is also a Fulton generator for $I_w$ by (\ref{eqn:thegenerators})).

\medskip\noindent
{\sf Case e4:} ($i,i+1\in R$) Let $b$ be as in \Cref{def:feb11bicrystal}. If $m_{i+1, b}$ does not lie on the antidiagonal $A_g$ or $m_{i+1, b}\geq 2$, then we may take $g' = g$. Otherwise, since we assume $e_i^{\sf row}({\sf m})\neq \varnothing$, the entry ${\sf m}_{i+1,b}=1$ corresponds to an unmatched
``$($'' in ${\sf bracket}_i({\sf row}({\sf m}))$. Let $b'$ be the first column to the right of $b$ in $C$, so $(i,b')\in A_g$. None of the ``$)$'' in 
${\sf bracket}_i({\sf row}({\sf m}))$ associated to ${\sf m}_{i,b'}>0$ match with this aforementioned ``$($''. In particular, the leftmost ``$)$'' associated to ${\sf m}_{i,b'}$ matches with a ``$($'' associated
to ${\sf m}_{i+1,c'}>0$ for some $b<c'<b'$. Take $g'$ to be the minor defined by row indices $R$ and column indices $C' = (C\setminus\{b\})\cup\{c'\}$ (which is also a Fulton generator for $I_w$ by (\ref{eqn:thegenerators})).

We show that $f^{\sf row}_i({\sf m})\not\in {\sf Std}_\prec(\mathfrak{X}_w)$ when $i\not\in\desc_{{\sf row}}(w)$ by constructing a Fulton generator $g''$ for $I_w$ such that the product of entries of $f^{\sf row}_i({\sf m})$ along $A_{g''}$ is positive.

\medskip
\noindent
{\sf Case f1:} ($i,i+1\not\in R$) Same argument as {\sf Case e1}, take $g'' = g$.

\medskip
\noindent
{\sf Case f2:} ($i\in R, i+1\not\in R$) If $f_{i}^{\sf row}$ does not affect the entry of $A_g$
in row $i$, let $g''=g$. Otherwise, let $g''$ be the minor that uses the rows $R''=(R\setminus \{i\})\cup \{i+1\}$ and columns $C$ (which is also a Fulton generator of $I_w$ provided $i\not\in\desc_{{\sf row}}(w)$ by (\ref{eqn:thegenerators})).

\medskip
\noindent
{\sf Case f3:} ($i\not\in R, i+1\in R$) Same argument as {\sf Case e2}, take $g'' = g$. 

\medskip
\noindent
{\sf Case f4:} ($i,i+1\in R$) We use left-right ``flipped'' version of {\sf Case e4}. Let $a$ be as in \Cref{def:feb11bicrystal}. If $m_{i, a}$ does not lie on  $A_g$ or $m_{i, a}\geq 2$, then we may take $g'' = g$. Otherwise, since we assume $f_i^{\sf row}({\sf m})\neq \varnothing$, the entry ${\sf m}_{i,a}=1$ corresponds to an unmatched
``$)$'' in ${\sf bracket}_i({\sf row}({\sf m}))$. Let $a'$ be the first column to the left of $a$ in $C$, so $(i+1, a')\in A_g$. None of the ``$($'' in 
${\sf bracket}_i({\sf row}({\sf m}))$ associated to ${\sf m}_{i+1,a'}>0$ match with this aforementioned ``$)$''. In particular, the rightmost ``$($'' associated to ${\sf m}_{i+1,a'}>0$ matches with a ``$)$'' associated
to ${\sf m}_{i,c''}>0$ for some $a'<c''<a$. Take $g''$ to be the minor defined by row indices $R$ and column indices $C'' = (C\setminus \{a\})\cup\{c''\}$ (which is a Fulton generator of $I_w$ by (\ref{eqn:thegenerators})).

    The statements for $f_j^{\sf col}$ and $e_j^{\sf col}$ hold by taking transposes of all matrices involved.
\end{proof}

\begin{proof}[Proof of \Cref{thm:main2}]
    Let $\mathfrak{X}_w\subseteq{\sf Mat}_{m, n}$, let $\mathbf{I}\supseteq\desc_{\sf row}(w)$, and let $\mathbf{J}\supseteq\desc_{\sf col}(w)$. Then \Cref{thm:MSVLtype} shows that $\mathfrak{X}_w$ is $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline. Thus \Cref{cor:bicrystalline} applies, giving the desired combinatorial rule for $c^{\C[\mathfrak{X}_w]}_{\underline\lambda|\underline\mu}$.
\end{proof}

In fact, \Cref{thm:MSVLtype} extends to show that any $\mathbf{B}$-stable $\mathfrak{X}\subseteq {\sf Mat}_{m,n}$ is bicrystalline. The following corollary of \Cref{thm:Mar6abc} is known. \begin{corollary}\label{thm:borelunion}
    Any $\mathbf{B}$-stable variety $\mathfrak{X}\subseteq{\sf Mat}_{m, n}$ is a finite union of matrix Schubert varieties.
\end{corollary}
\begin{proof}
There are finitely many $M_w\in{\sf Mat}_{m, n}$. Apply \Cref{thm:Mar6abc}
and \Cref{thm:MSV}.
\end{proof}

The standard monomials of any $\mathbf{B}$-stable variety $\mathfrak{X}\subset{\sf Mat}_{m, n}$ are described in terms of the standard monomials of matrix Schubert varieties. This follows immediately from work of Knutson on Frobenius splittings \cite{Knutson}, although it is not explicitly stated there.\footnote{See also \cite{Bertiger} for Bertiger's construction of a Gr\"obner basis for arbitrary $\mathbf{B}$-stable varieties.}

\begin{proposition}[cf.~\cite{Knutson}]\label{thm:stdmonounion}
    Let $\mathfrak{X}\subseteq{\sf Mat}_{m, n}$ be a $\mathbf{B}$-stable variety. Write $\mathfrak{X} = \bigcup_{i=1}^k \mathfrak{X}_{w^{(i)}}$ as a union of matrix Schubert varieties (by \Cref{thm:borelunion}). Then the set of standard monomials for $\mathfrak{X}$, with respect to $\prec$, is 
${\sf Std}_{\prec}({\mathfrak X})=\bigcup_{i=1}^k {\sf Std}_{\prec}(\mathfrak{X}_{w^{(i)}})$.    
\end{proposition}

When each component $\mathfrak{X}_{w^{(i)}}$ of a $\mathbf{B}$-stable variety $\mathfrak{X}$ is $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline for some $\mathbf{I}$ and $\mathbf{J}$, \Cref{thm:stdmonounion} implies that $\mathfrak{X}$ is $\mathbf{L}_{\mathbf{I}|\mathbf{J}}$-bicrystalline as well. Thus \Cref{cor:bicrystalline} applies to $\mathfrak{X}$, giving a combinatorial rule for its ${\bf L}_{\mathbf{I}|\mathbf{J}}$-graded Hilbert series via ${\sf filterRSK}$.

\acknowledgements{We thank Andrew Hardt for a helpful conversation about \cite{KM:adv} and Shiliang Gao for pointing us to the motivating reference \cite{Sturmfels} on standard monomials and Hilbert series.}

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