%% 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]{FPSAC2021}
\articlenumber{90}
%% 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}

%=======================================================
\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{graphicx}
\usepackage{shuffle}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[bbgreekl]{mathbbol}
\usepackage{mathdots}
\usepackage{ytableau}
\usepackage[enableskew,vcentermath]{youngtab}
\usepackage{tikz-cd}
\usepackage{enumerate}

%\usepackage[all]{xy}
%\input xy \xyoption{frame}
%\xyoption{dvips}

\usepackage[colorinlistoftodos]{todonotes}
\usetikzlibrary{chains,scopes}
\usetikzlibrary{shapes.geometric,positioning}

%\usepackage{caption}\captionsetup[figure]{font=footnotesize}

\DeclareSymbolFontAlphabet{\mathbb}{AMSb}
\DeclareSymbolFontAlphabet{\mathbbl}{bbold}

\newcommand{\SSYT}{\ensuremath\mathrm{SSYT}}
\newcommand{\SYT}{\ensuremath\mathrm{SYT}}
\newcommand{\RPP}{\ensuremath\mathrm{RPP}}
\newcommand{\MRPP}{\ensuremath\mathrm{MRPP}}



\numberwithin{equation}{section}
\allowdisplaybreaks[1]
%\UseCrayolaColors

%=========================================================================
% theorem declarations
\theoremstyle{definition}
\newtheorem* {theorem*}{Theorem}
\newtheorem* {corollary*}{Corollary}
\newtheorem* {conjecture*}{Conjecture}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{thmdef}[theorem]{Theorem-Definition}
\newtheorem{propdef}[theorem]{Proposition-Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{problem}[theorem]{Problem}
\theoremstyle{definition}
\newtheorem{observation}[theorem]{Observation}
\newtheorem* {claim}{Claim}
\newtheorem* {example*}{Example}
\newtheorem* {fact}{Fact}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{definition}
\newtheorem* {notation}{Notation}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem*{remark*}{Remark}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem {example}[theorem]{Example}
\theoremstyle{definition}
\newtheorem* {examples}{Examples}
\theoremstyle{definition}
\newtheorem {exercise}[theorem]{Exercise}
\theoremstyle{definition}
\newtheorem* {remarks}{Remarks}
%\xyoption{dvips}





\newcommand{\CC}{\mathbb{C}}
\newcommand{\NN}{\mathbb{N}}	
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathcal{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\I}{\mathcal{I}}
\newcommand{\K}{\mathcal{K}}

\newcommand{\Hs}{\mathcal{H}}

\newcommand{\RP}{\mathcal{RP}}

\newcommand{\FP}{\mathcal{FP}}
\newcommand{\RFP}{\mathcal{FP}}
%\newcommand{\NP}{\mathcal{NP}}
\newcommand{\NFP}{\FP^+}%{\mathcal{NFP}}

\DeclareMathOperator{\word}{word}
\DeclareMathOperator{\init}{in}
\DeclareMathOperator{\Ess}{Ess}
\DeclareMathOperator{\dom}{dom}
\DeclareMathOperator{\rad}{rad}
\newcommand{\eqdef}{\overset{\text{def}}{=}}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Sym}{Sym}

\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sort}{sort}
\DeclareMathOperator{\link}{lk}
\DeclareMathOperator{\del}{del}
\DeclareMathOperator{\pf}{pf}
\DeclareMathOperator{\sgn}{sgn}
\newcommand{\ch}{\operatorname{char}}
\newcommand{\setsgn}[2]{\sgn\!\tbinom{#1}{#2}}



\newcommand{\Dbot}{D_{\text{bot}}}

\newcommand{\Hilb}{\mathcal{H}}
\newcommand{\fkG}{\mathfrak{G}}

\newcommand{\Umat}{\mathcal{U}}
\newcommand{\SSU}{\mathcal{U}^{\text{ss}}}
\newcommand{\SSu}{u^{\text{ss}}}



\newcommand{\openX}{\mathring{X}}
\newcommand{\openSSX}{\mathring{X}^{\text{ss}}}
\newcommand{\SSX}{X^{\text{ss}}}
\newcommand{\SSM}{\operatorname{Mat}^{\text{ss}}}
\newcommand{\SX}{SX}
\newcommand{\SM}{SM}
\newcommand{\subword}{\Sigma}

\newcommand{\fkS}{\mathfrak{S}}
\DeclareMathOperator{\Sp}{Sp}
\renewcommand{\O}{\operatorname{O}}

\newcommand{\FPF}{\operatorname{FPF}}
\renewcommand{\ss}{\operatorname{ss}}
\newcommand{\SSI}{I^{\text{ss}}}
\newcommand{\SSD}{D^{\text{ss}}}
\newcommand{\SSr}{{\bf r}^{\text{ss}}}
\newcommand{\SSJ}{J^{\text{ss}}}
\newcommand{\biwords}{\mathcal{BW}}
\newcommand{\twist}{\textsf{twist}}

\def\r{{\bf r}}
\def\s{{\bf s}}

\def\D{\mathsf{D}}
\def\SD{\mathsf{SD}}

\def\Gfpf{\mathfrak{G}^{\mathsf{Sp}}}

\def\cP{\mathcal{P}}

\def\untwist{\mathsf{untwist}}

%\newcommand{\rtriang}{\raisebox{-0.5pt}{\tikz{\draw (0,.25) -- (.25,.25) -- (0,0) -- (0,.25);}}\hspace{-0.5mm}}
\newcommand{\ltriang}{\raisebox{-0.5pt}{\tikz{\draw (0,0) -- (.25,0) -- (0,.25) -- (0,0);}}}
%\newcommand{\ltriang}{\Delta}
\newcommand{\ltriangeq}{\ltriang}
\newcommand{\ltriangneq}{\ltriang^{\!\!\neq}}

\def\quand{\quad\text{and}\quad}
\def\DesR{\mathrm{Des}_R}
\def\DesL{\mathrm{Des}_L}

\def\be{\begin{equation}}
\def\ee{\end{equation}}

\def\fpf{\mathsf{FPF}}
\def\ellfpf{\ell_{\fpf}}
\newcommand{\iDbotfpf}{\hat D_{\text{bot}}^{\fpf}}
\newcommand{\Ifpf}{\mathcal{I}^{\fpf}}

\def \bfr{{\r}}
\def\cM{\mathcal{M}}
\def\OC{\mathsf{OC}}
\def\dbl{\mathsf{dbl}}

\def\dbackslash{\backslash\hspace{-1mm}\backslash}

\def\cX{\mathcal{X}}

\newcommand{\J}[1]{\mathsf{Ideal}(#1)}

\def\deltafpf{\delta_{\fpf}}
\def\mon{\mathsf{mon}}


\numberwithin{equation}{section}
\allowdisplaybreaks[1]
%\UseCrayolaColors

% I believe this has something to do with footnotes
%\makeatletter
%\renewcommand{\@makefnmark}{\mbox{\textsuperscript{}}}
%\makeatother

%% define your title in the usual way
\title[Skew-symmetric matrix Schubert varieties]{Gr\"obner geometry for skew-symmetric matrix Schubert varieties}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{
Eric Marberg\thanks{\href{mailto:eric.marberg@gmail.com}{eric.marberg@gmail.com}. Eric Marberg was partially supported by RGC Grant ECS 26305218.}\addressmark{1}
\and Brendan Pawlowski\thanks{\href{mailto:br.pawlowski@gmail.com}{br.pawlowski@gmail.com}.}\addressmark{2}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology
\\
\addressmark{2}Department of Mathematics, University of Southern California}
%
%\author[Optional shorter names]{Eric Marberg\thanks{\href{mailto:eric.marberg@gmail.com}{eric.marberg@gmail.com}. Eric Marberg was partially supported by Hong Kong RGC Grant ECS 26305218.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}
%
%%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}

%% 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{Matrix Schubert varieties are the orbit closures of $B\times B$ acting on all $n\times n$ matrices, where $B$ is the group of invertible  lower triangular matrices. We define skew-symmetric matrix Schubert varieties to be the orbit closures of $B$ acting on all $n \times n$ skew-symmetric matrices.
In analogy with Knutson and Miller's work on matrix Schubert varieties, we describe a natural generating set for the prime ideals of these varieties. We then compute a related Gr\"obner basis. Using these results, we identify a primary decomposition for the corresponding initial ideals involving certain fpf-involution pipe dreams, analogous to the pipe dreams of Bergeron and Billey. We show that these initial ideals are the Stanley--Reisner ideals of shellable simplicial complexes. As an  application, we give a geometric proof of an explicit generating function for symplectic Grothendieck polynomials.}

%% 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, Gr\"obner bases, Grothendieck polynomials, simplicial complexes}

%% 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, ie. [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]{biblatex}
\addbibresource{90Pawlowski.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}
Let $\KK$ be an algebraically closed field and write $B_n \subseteq \GL_n := \GL_n(\KK)$ for the Borel group of $n \times n$ invertible lower triangular matrices over $\KK$. An $n\times n$ matrix $A$ is \emph{skew-symmetric} if $A_{ij} = -A_{ji}$ and $A_{ii} = 0$ for all $i,j \in [n]$; note that the second condition is redundant if $\ch(\KK) \neq 2$. 
Let $\SSM_n := \SSM_n(\KK) $ denote the set of such matrices.

Consider the $B_n$-action on $\SSM_n$ defined by $g \cdot A = g A g^T$. The orbits for this action are \emph{skew-symmetric matrix Schubert cells}, and their Zariski closures are the \emph{skew-symmetric matrix Schubert varieties}.  We write $\SSX_A$ for the closure of the $B_n$-orbit of $A \in \SSM_n$.




%Let $\NN = \{1,2,3,\dots\}$ denote the set of positive integers.
% Let $u_{ij}$ for $i,j \in [n]:= \{1,2,\dots,n\}$ be commuting indeterminates and 
% write $\Umat$ for the $n\times n$ matrix with $u_{ij}$ in position $(i,j)$.
% We identify the coordinate ring $\KK[\Mat_{n \times n}]$ with $\KK[u_{ij} : 1 \leq i, j \leq n]$. Given a closed subvariety $X \subseteq \Mat_{n \times n}$, let $I(X) \subseteq \KK[\Mat_{n \times n}]$ be the ideal of polynomials vanishing on $X$. If $A$ is an $n\times n$ matrix and $R, C \subseteq  \{1,2,\ldots,n\}$, then we write $A_{RC}$ for the submatrix $[A_{ij}]_{(i,j) \in R \times C}$. Finally, let $X_A$ denote the closure of the $B_n \times B_n$-orbit of  $A \in \Mat_{n \times n}$.
%  Each orbit closure $X_A$ is a subvariety, and one can describe the 
% ideal $I(X_A)$ explicitly as follows.
%  %

%  \begin{theorem}[{\cite[Thms. A and B]{KnutsonMiller}}] \label{thm:KM-ideal}
%     For each $A \in \Mat_{n \times n}$,  the collection of minors $\det(\Umat_{RC})$, for all $R \subseteq [i]$ and $C \subseteq [j]$ with $|R| = |C| = \rank A_{[i][j]}+1$ for some $i,j \in [n]$, generates the (prime) ideal $I(X_A)$ and forms a Gr\"obner basis with respect to any \emph{antidiagonal term order}.
%     % one where the leading term of any minor $\det(\Umat_{RC})$ is the product of the antidiagonal entries of $\Umat_{RC}$.
%  \end{theorem}
 
%  Here, a term order on the polynomial ring $\KK[\Mat_{n \times n}]=\KK[u_{ij} : 1 \leq i, j \leq n]$ 
%  is \emph{antidiagonal} if the leading term of any minor $\det(\Umat_{RC})$ is the product of the antidiagonal entries of $\Umat_{RC}$.
%   The prototypical example of an antidiagonal term order on $\KK[\Mat_{n \times n}]$ is  \emph{(graded) reverse lexicographic order}, whose definition is reviewed in \S \ref{subsec:init}, along with the definitions of \emph{term order} and \emph{Gr\"obner basis}. The assertion in Theorem~\ref{thm:KM-ideal} that the minors $\det(\Umat_{RC})$ generate $I(X_A)$ is originally due to Fulton \cite[Prop. 3.3]{FultonEssentialSet}.
%    Knutson and Miller \cite{KnutsonMiller} reprove this result using different techniques to obtain the stronger statement that 
%    these minors form a Gr\"obner basis. % for $I(X_A)$



% The \emph{(type A) Schubert varieties} are the $B_n$-orbit closures on $\GL_n/B_n$. Their inverse images under the quotient $\GL_n \to \GL_n/B_n$ are the matrix Schubert varieties contained in $\GL_n\subset \Mat_{n\times n}$. These two families of varieties are closely related, but working in the matrix setting can offer an advantage: matrix Schubert varieties are affine subvarieties of an affine space, and one can apply tools of commutative algebra that are unavailable in the projective variety $\GL_n/B_n$. Knutson and Miller \cite{KnutsonMiller} used such an approach to shed light on combinatorial formulas for the well-known \emph{Schubert polynomials} representing cohomology classes of Schubert varieties; we briefly outline their results.



%  If a subvariety $X \subseteq \Mat_{n \times n}(\CC)$ is invariant under the left $B_n$-action, then it defines a class $[X]_{B_n}$ in the equivariant cohomology ring $H_{B_n}^*(\Mat_{n \times n}(\CC)) \simeq \ZZ[x_1, \ldots, x_n]$. The polynomial $[X]_{B_n}$ can also be computed algebraically as the \emph{multidegree} of the ideal $I(X)$, and this definition works more generally over any algebraically closed field $\KK$. 
 
%  Identify a permutation $w \in S_n$ with the $n \times n$ permutation matrix with 1 in each position $(i, w(i))$ and 0 everywhere else. The $B_n \times B_n$-orbits contained in $\GL_n \subseteq \Mat_{n \times n}$ are precisely the closures of the matrix Schubert varieties $\{X_w : w \in S_n\}$. For simplicity, we state the next two theorems only for those orbits; both statements can be generalized to all matrix Schubert varieties without serious difficulty.
% When we write $\init(I(X_w))$ we mean the initial ideal under any fixed antidiagonal term order.

%  \begin{theorem}[{\cite[Thms. A and B]{KnutsonMiller}}]\label{thm:KM-primary-decomp} 
% If $w \in S_n$ then $[X_w]_{B_n}$ is equal to the polynomial $\fkS_w$,
% and the initial ideal of $I(X_w)$ has primary decomposition
%     $\init(I(X_w))=\bigcap_{D \in \RP(w)} (u_{ij} : (i,j) \in D)$,
%     where $\RP(w)\subseteq \binom{[n]\times [n]}{\ell(w)}$ is the set of \emph{(reduced) pipe dreams} for $w$ (see \S\ref{subsec:pipe-dreams}),
%     and 
%      $(u_{ij} : (i,j) \in D)$ denotes the ideal of $\KK[\Mat_{n \times n}]$ generated by $u_{ij}$ 
% for all $(i,j) \in D$.
% \end{theorem}


% As an application of this theorem, by using an algebraic interpretation of $[X_w]_{B_n}$ as the multidegree of  $I(X_w)$, Knutson and Miller 
% provide a geometric proof of 
% the Billey-Jockusch-Stanley formula \cite[Thm. 1.1]{BJS} for
% the Schubert polynomial $\fkS_w$. 
% Knutson and Miller extract more detailed information about the ideals $I(X_w)$, such as their $K$-polynomials, 
% from this related result:



% \begin{theorem}[\cite{SubwordComplexes}] \label{thm:subword-complexes} 
% For each $w \in S_n$, 
% the ideal $\init(I(X_w))$ is square-free and equal to the Stanley--Reisner ideal of a 
%  shellable simplicial complex.
% \end{theorem}



Identify the coordinate ring $\KK[\SSM_n]$ with $\KK[u_{ij} : n \geq i > j \geq 1]$, and write $\SSU$ for the $n \times n$ skew-symmetric matrix with $\SSU_{ij} = u_{ij} = -\SSU_{ji}$ for $i > j$
and $\SSU_{ii} = 0$ for all $i$. If $M$ is an $n\times n$ matrix and $R, C \subseteq [n]$, we write $M_{RC}$ for the submatrix of $M$ in rows $R$ and columns $C$. One has $M \in \SSX_A$ if and only if $\rank M_{[i][j]} \leq \rank A_{[i][j]}$ for all $i,j \in [n]$. Accordingly, $\SSX_A$ is the zero locus of the family consisting of all minors $\det(\SSU_{RC})$ for all $R \subseteq [i]$ and $C \subseteq [j]$ with $|R| = |C| = \rank A_{[i][j]}+1$ for some $i,j\in [n]$.
These polynomials do not always generate the prime ideal $I(\SSX_A)$ of the variety $\SSX_A$, however.
There is nevertheless a natural generating set for $I(\SSX_A)$, which we describe as follows:

\begin{theorem}[See Theorem~\ref{ssi-thm}] \label{thm:fpf-ideal-intro} For each $A \in \SSM_n$, the collection of  Pfaffians $\pf(\SSU_{RR})$, as $R$ ranges over all even sized subsets of $[n]$ such that $R \subseteq [i]$ and $|R \cap [j]| > \rank A_{[i][j]}$ for some $i,j \in [n]$ with $i\geq j$, generate the (prime) ideal $I(\SSX_A)$.
\end{theorem}

We are also interested in the problem of describing Gr\"obner bases for the ideals $I(\SSX_A)$ with respect to the \emph{(graded) reverse lexicographic order} on $\KK[\SSM_n]$. Recall that a generating set $S$ of an ideal $I$ is a \emph{Gr\"obner basis} of $I$ if the leading terms of the polynomials in $S$ generate the \emph{initial ideal} generated by the leading terms of all elements of $I$. These definitions are reviewed more carefully in \S \ref{subsec:init}.

The generating set  in the preceding theorem is generally \emph{not} a Gr\"obner basis for $I(\SSX_A)$; see Example~\ref{ex:not-grobner}. Our second main result resolves this problem. Specifically, in 
Theorem~\ref{ss-main-thm3} we show that a Gr\"obner basis for $I(\SSX_A)$ with respect to the reverse lexicographic term order is provided by the Pfaffians of the
block diagonal matrices  \[\left[ \begin{array}{ll} \SSU_{CC} & \SSU_{CR} \\ \SSU_{RC} & 0
    \end{array}  \right]\]
    for certain subsets $R,C\subseteq [n]$.
Experimental evidence suggests that these Pfaffians may also form a Gr\"obner basis for other so-called \emph{antidiagonal} term orders, but this more general claim does not follow from our present methods.



Suppose $n$ is even and $z \in S_n$ is a fixed-point-free involution,
that is, a permutation with $z(z(i)) = i \neq z(i)$ for all $i \in [n]$.
Associated to such a permutation is 
a set $\FP(z)$ of \emph{fpf-involution pipe dreams}, whose elements are certain subsets
of $\{ (i,j) \in [n] \times [n] : i > j\}$; see Definition~\ref{fp-def}
for the full details.
We let $\SSX_z := \SSX_A$ where $A$ is the skew-symmetric $n\times n$ matrix with
 $A_{ij} = 1$ if $z(j) = i<j =z(i)$ and $A_{ij} = -1$ if $z(j) = i>j = z(i)$.
Whenever we write $\init(I(\SSX_z))$, we mean the initial ideal under the reverse lexicographic term order defined in Example~\ref{lex-term2}.



The varieties $\SSX_z$ as $z$ ranges over the fixed-point-free involutions in $S_n$ are exactly the $B_n$-orbit closures in $\SSM_n \cap \GL_n$,
which is nonempty only if $n$ is even. For simplicity, we only consider these varieties here, but our results can be extended to the full family of skew-symmetric matrix Schubert varieties in $\SSM_n$ for any $n$.



\begin{theorem}[See Theorem~\ref{thm:primary-decomposition-ss}] \label{thm:fpf-primary-decomp-intro} For each fixed-point-free involution $z \in S_n$,
the initial ideal of $I(\SSX_z)$ has primary decomposition $\init(I(\SSX_z))=\bigcap_{D \in \RFP(z)} (u_{ij} : (i,j) \in D)$,
where $(u_{ij} : (i,j) \in D)$ denotes the ideal in $\KK[\SSM_n]$ generated by $u_{ij}$ 
for all $(i,j) \in D$.
 \end{theorem}


 If a subvariety $X \subseteq \SSM_n(\CC)$ is invariant under the left $B_n$-action, then it defines a class $[X]_{B_n}$ in the equivariant cohomology ring $H_{B_n}^*(\SSM_n(\CC)) \simeq \ZZ[x_1, \ldots, x_n]$. The polynomial $[X]_{B_n}$ can also be computed algebraically as the \emph{multidegree} of the ideal $I(X)$, and this definition works more generally over any algebraically closed field $\KK$. 

Using Theorem~\ref{thm:fpf-primary-decomp-intro},
one can show that $[\SSX_z]_{B_n} = \sum_{D \in \RFP(z)} \prod_{(i,j) \in D} (x_i + x_j)$ under the identification $H_{B_n}^*(\SSM_n) \simeq \ZZ[x_1, \ldots, x_n]$. This formula was proven combinatorially in \cite{HMPdreams}. It was also shown in \cite{HMPdreams} that the polynomials $[\SSX_z]_{B_n}$ are the same as the \emph{fpf-involution Schubert polynomials} introduced by Wyser and Yong \cite{WyserYongOSp}, which represent the ordinary cohomology classes the orbit closures of the symplectic group $\Sp_n(\CC)$ acting on $\GL_n(\CC)/B_n$. Briefly, the connection to our situation is that $\Sp_n(\CC)$-orbits on $\GL_n(\CC)/B_n$ are in bijection with $B_n$-orbits on $\GL_n(\CC)/\Sp_n(\CC)$, which can be identified with $\SSM_n(\CC) \cap \GL_n(\CC)$.




Finally, we relate the initial ideal of $I(\SSX_z)$ to the geometry of simplicial complexes. We also use the next result to give a new geometric proof of 
a combinatorial formula \cite[Thm. 4.5]{MacdonaldNote}  for 
the $B_n$-equivariant $K$-theory representative of $\SSX_z$; see Theorem~\ref{4.5-thm}.

\begin{theorem}[See Theorem~\ref{thm:SR-ideal}]
 \label{thm:fpf-subword-complexes-intro}
For each fixed-point-free involution
 $z \in S_n$, the ideal $\init(I(\SSX_z))$ is square-free and equal to the Stanley--Reisner ideal of a shellable simplicial complex. 
 %generated by the facets $\{\{(i,j) \in [n] \times [n] : i > j\} \setminus D : D \in \RFP(z))\}$, which is shellable.
\end{theorem}


One can equally well consider the orbits of $B_n \times B_n$ acting on the space of all $n \times n$ matrices by the formula $(g,h) \cdot A = gAh^t$. The closures of these orbits are called \emph{matrix Schubert varieties}. These varieties have been studied by Fulton \cite{FultonEssentialSet} and Knutson and Miller \cite{KnutsonMiller} among others, who proved results analogous to those described above. Indeed, our main goal in this work was to reproduce their results in the skew-symmetric setting.



Ideals generated by Pfaffians of a generic skew-symmetric matrix have been well-studied \cite{DeNegri,DeNegriSbarra,HerzogTrung,JonssonWelker,RaghavanUpadhyay}, and there is some overlap between our results and prior work. De Negri and Sbarra \cite{DeNegriSbarra} consider a family of ideals which, translated into our language, turns out to be a subfamily of the ideals $I(\SSX_A)$ for $A \in \SSM_n$. They observe that the Pfaffian generators of Theorem~\ref{thm:fpf-ideal-intro} need not form a Gr\"obner basis with respect to an antidiagonal term order, and proceed to classify the ideals which do enjoy this property. Raghavan and Upadhyay \cite{RaghavanUpadhyay} study the same family of ideals, computing their initial ideals and realizing the latter as Stanley--Reisner ideals of shellable complexes, as we do in Theorems~\ref{thm:fpf-primary-decomp-intro} and \ref{thm:fpf-subword-complexes-intro}. Although similar in form, their results are in fact quite different from ours, because they use term orders which are far from antidiagonal.


%The techniques we use to prove Theorems~\ref{thm:fpf-ideal-intro} and \ref{thm:fpf-primary-decomp-intro} differ from those in \cite{KnutsonMiller}, and in fact lead to new proofs of Theorems~\ref{thm:KM-ideal} and \ref{thm:KM-primary-decomp}. We discuss these applications in Section~\ref{new-proofs-sect}. Knutson and Miller use a method in \cite{KnutsonMiller} called \emph{Bruhat induction}, which is an induction on weak Bruhat order on $S_n$ leveraging divided difference recurrences for Schubert classes. We instead induct on (strong) Bruhat order and use the transition recurrences of Lascoux and Sch\"utzenberger from \cite{LascouxSchutzenbergerTransitions}. A similar inductive approach to studying matrix Schubert varieties appears in \cite{HPW}.




% The questions studied here are equally interesting to consider for 
% \emph{symmetric matrix Schubert varieties}, where $\SSM_n$ is replaced by the space of symmetric matrices in $\Mat_{n\times n}$. 
% A number of technical difficulties arise in that setting, however.
% For example, the expected analogue of
% Theorem~\ref{thm:fpf-primary-decomp-intro} no longer gives a pure primary decomposition into ideals of the same dimension, and this poses a fundamental obstruction to the techniques of Knutson and Miller from \cite{KnutsonMiller}. 
% We hope that a variation of our new inductive approach can
% be adapted in future work to avoid these issues.


In this extended abstract we have omitted most proofs and focused on new results. However, the techniques we use to prove the results above differ from those of Knutson and Miller, and a secondary goal of this work was to use these techniques to give new proofs of their results; see \cite{MP2020}, of which this abstract is a condensed version. 





\acknowledgements{
We thank Zach Hamaker for a much improved proof of a key lemma and for many other helpful conversations. We are also grateful to Allen Knutson for explaining his perspective on Schubert transition recurrences, which inspired some of the main techniques used to prove the results described here.}



\section{Preliminaries}
\label{subsec:init}

Throughout, we write $\NN = \{1,2,3,\dots\}$ 
for the set of natural numbers and $[n] = \{ i \in \ZZ : 0 < i \leq n\}$ for the first $n$
positive integers. We write $\Ifpf_n = \{w \in S_n : w^2 = 1, w(i) \neq i \text{ for $i \in [n]$}\}$ for the set of fixed-point-free involutions in the symmetric group $S_n$. This set is nonempty only if $n$ is even, in which case it is the $S_n$-conjugacy class
of $(1,2)(3,4)\cdots(n-1,n)$.


Suppose $x_1,x_2,\dots,x_N$ are commuting variables, and consider the polynomial ring $\KK[{\bf x}] := \KK[x_1,x_2,\dots,x_N]$ over a field $\KK$ (for now arbitrary, but in later sections algebraically closed). A \emph{term order} on the polynomial ring $\KK[{\bf x}] := \KK[x_1,x_2,\dots,x_N]$ is a total order on the set of  all monomials,
such that $1$ is the unique minimum and such that if $\mon_1, \mon_2, \mon_3$ are monomials and $\mon_1 \leq \mon_2$, 
then $\mon_1  \mon_3 \leq \mon_2 \mon_3$. 


\begin{example} \label{lex-term1}
The \emph{lexicographic term order} on $\KK[{\bf x}]$ declares
that  $x_1^{a_1}\cdots x_N^{a_N} \leq x_1^{b_1} \cdots x_N^{b_N}$ whenever $(a_1, \ldots, a_N) \leq (b_1, \ldots, b_N)$ in lexicographic order.
The \emph{(graded) reverse lexicographic term order} declares that 
$x_1^{a_1}\cdots x_N^{a_N} \leq x_1^{b_1} \cdots x_N^{b_N}$ whenever $\sum_i a_i \leq \sum_i b_i$ and $(a_N, \ldots, a_1) \geq (b_N, \ldots, b_1)$
in lexicographic order; note the double reversal. 
%Both orders are term orders on $\KK[{\bf x}]$.
\end{example}



Fix a term order and suppose $f = \sum_{\mon} c_\mon \cdot \mon \in \KK[{\bf x}]$ where the sum is over monomials $\mon$ and each $c_\mon \in \KK$.  
If $f$ is nonzero, then 
its \emph{initial term} (or \emph{leading term}) is the maximal monomial $\mon$ such that $c_\mon \neq 0$.
If $f=0$ then its initial term is also defined to be zero. In either case, we write $\init(f)$ for the corresponding initial term.

The \emph{initial ideal} of an ideal $I$ in $\KK[{\bf x}]$ is then $\init(I) := \KK\text{-}\mathrm{span}\{\init(f) : f \in I\}$. This abelian group is itself an ideal
 in $\KK[{\bf x}]$.
A \emph{Gr\"obner basis} $G$ for an ideal $I\subseteq \KK[{\bf x}]$, relative to a fixed term order,
is a generating set  whose set of initial terms $\{ \init(g) :g \in G\}$ generates $\init(I)$.






\begin{example}\label{lex-term2}
In our applications, we will usually take $x_1,x_2,\dots,x_N$ to be 
either the commuting variables $u_{ij}$ indexed by all positions $(i,j) \in [n]\times [n]$ for some $n$,
or the subset of these variables indexed by positions strictly below the main diagonal. Unless otherwise mentioned, we equip $\KK[u_{ij} : i,j \in [n]]$ and its subrings with the (graded) reverse lexicographic order 
from Example~\ref{lex-term2}
with $u_{ij}$ identified with $x_{n(i-1) + j}$.


This means that we order the variables $u_{ij}$ lexicographically, so that $u_{ij} < u_{i'j'}$
if $i<i'$ or if $i=i'$ and $j<j'$.
Then, we declare that $\mon_1 < \mon_2$  if either $\deg (\mon_1)< \deg (\mon_2)$
or $\deg (\mon_1 )= \deg (\mon_2)$ and the following holds: there is some variable $u_{ij}$ whose exponent $e_1$ in $\mon_1$
differs from its exponent $e_2$ in $\mon_2$, and 
when $u_{ij}$ is the (lexicographically) largest such variable one has $e_1> e_2$.
If $\mon_1$ and $\mon_2$ are both square-free of the same degree,
then we have $\mon_1 < \mon_2$ if and only there is some variable $u_{ij}$ that does not divide both monomials, and 
the largest such variable divides $\mon_1$ but not $\mon_2$.
\end{example}


%We note a few basic facts in the generic setting of $\KK[{\bf x}]$ with a fixed term order.
%
%\begin{lemma}\label{lem:finite-chains} Every strictly descending chain of monomials in a term order on $\KK[{\bf x}]$ is finite. \end{lemma} 
%%    \begin{proof}
%%    Since $1 \leq \mon$ for any monomial,  it follows that $\mon_1 \leq \mon_2$ whenever 
%%    $\mon_1, \mon_2 \in \KK[{\bf x}]$ are monomials with $\mon_1 \mid \mon_2
%%    $. %This means that in a strictly descending chain of monomials in $\KK[{\bf x}]$, no element can divide any later element.
%%        Dickson's lemma asserts that if $v_1, v_2, \ldots$ is an infinite sequence in $\NN^N$, then there are indices $i < j$ such that $v_i \leq v_j$ component-wise. Applying this to the exponent vectors of an infinite sequence of monomials $\mon_1, \mon_2, \ldots \in \KK[{\bf x}]$ shows that there are always 
%%        indices $i < j$ with $\mon_i \mid \mon_j$, so an infinite sequence is not strictly descending.
%%    \end{proof}
%
%
%    \begin{proposition} \label{prop:init-facts} Suppose $I$ and $J$ are ideals in $\KK[{\bf x}]$. Fix a term order on $\KK[{\bf x}]$. Then:
%        \begin{enumerate}[(a)]
%            \item If $I \subseteq J$, then $\init(I) \subseteq \init(J)$.
%            \item It holds that $\init(I)+\init(J) \subseteq \init(I+J)$.
%            \item It holds that $\init(I \cap J) \subseteq \init(I) \cap \init(J)$.
%            \item If $I \subseteq J$ and $\init( I) = \init (J)$, then $I = J$.
%        \end{enumerate}
%    \end{proposition}
%%        \begin{proof}
%%        Part (a) is clear. It implies $\init(I) \subseteq \init(I+J)$ and $\init(J) \subseteq \init(I+J)$, so part (b) follows. Similarly, part (a) implies $\init(I \cap J) \subseteq \init(I)$ and $\init(I \cap J) \subseteq \init(J)$, so part (c) follows.
%%    
%%       For part (d), suppose the hypotheses hold and $f \in J$. Since $\init(f) \in \init (I)$, there is some $g \in I$ with $\init(g) = \init(f)$. For some $c \in \KK$, $cf-g$ will therefore have a smaller initial term than $f$. Since $cf-g \in J$, we can iterate this process. Lemma~\ref{lem:finite-chains} implies that the process must terminate, and when it does we will have written $f$ as a linear combination of elements of $I$, so $J \subseteq I$.
%%        \end{proof}
%
%
%
%\begin{lemma} \label{lem:plus-vs-intersection} Suppose $I$ and $J$ are homogeneous ideals in $\KK[{\bf x}]$. 
%If $\init(I+J) = \init(I) + \init(J)$ relative to a given term order on $\KK[{\bf x}]$, then $\init(I \cap J) = \init(I) \cap \init(J)$. \end{lemma}
%
%%    \begin{proof}
%%        The \emph{Hilbert series} of a graded vector space $V = \bigoplus_{n \geq 0} V_n$ with each $\dim V_n$ is finite is the generating function $\Hilb(V) = \sum_{n \geq 0} \dim(V_n)t^n \in \ZZ[[t]]$. 
%%       A homogeneous ideal $I$ in $\KK[{\bf x}]$ is a graded vector space of finite graded dimension, as is its initial ideal,
%%       and it holds that $\Hilb(\init (I)) = \Hilb(I)$ \cite[Theorem 15.3]{Eisenbud}. 
%%        Since 
%%        $\init(I \cap J) \subseteq \init(I) \cap \init(J)$ by Proposition~\ref{prop:init-facts}, we 
%%        may consider the quotient ${\init(I) \cap \init(J)}/{\init(I \cap J)}$,
%%        which is itself a graded vector space.
%%        
%%        If $0 \to U \to V \to W \to 0$ is a short exact sequence of degree-preserving linear maps between graded vector spaces, then $\Hilb(U) = \Hilb(V) - \Hilb(W)$. 
%%        Using this fact with the exact sequences
%%        $ 0 \to A \hookrightarrow B \to B /A \to 0$ (when $A\subseteq B$) and
%%        $0 \to A \cap B \xrightarrow{x \mapsto (x,-x)} A \oplus B \to A + B \to 0,$ it is straightforward to check that
%%        $  \Hilb\left(\tfrac{\init(I) \cap \init(J)}{\init(I \cap J)}\right) = \Hilb(I\oplus J)  - \Hilb(I \cap J) - \Hilb(\init( I) + \init (J))$.
%%        If $\init(I + J) = \init(I) + \init(J)$, then the last term becomes 
%%        $\Hilb(\init(I+J)) = \Hilb(I+J) = \Hilb(I\oplus J) - \Hilb(I\cap J) $ and the whole expression simplifies to zero,
%%%        \begin{align*}
%%%            \Hilb\left(\tfrac{\init(I) \cap \init(J)}{\init(I \cap J)}\right) &= \Hilb(\init(I) \cap \init(J)) - \Hilb(\init(I \cap J))\\
%%%            &=  \Hilb(\init(I) \oplus \init(J)) - \Hilb(\init (I) + \init (J)) - \Hilb(\init(I \cap J))\\
%%%            &= \Hilb(\init( I)) + \Hilb(\init (J)) - \Hilb(\init (I) + \init (J)) - \Hilb(\init(I \cap J))\\
%%%            &= \Hilb(I) + \Hilb(J) - \Hilb(\init( I) + \init (J)) - \Hilb(I \cap J)\\
%%%            &= \Hilb(I) + \Hilb(J) - \Hilb(\init(I+J)) - \Hilb(I \cap J) & \quad\text{[by hypothesis]}\\
%%%            &= \Hilb(I) + \Hilb(J) - \Hilb(I+J) - \Hilb(I \cap J)\\
%%%            &= 0. & \quad\text{[by exactness]}
%%%        \end{align*}
%%%        This 
%%        which can only happen if $\init(I) \cap \init(J) = \init(I \cap J)$.
%%    \end{proof}
%    
% 
%
% Given an element $f \in \KK[{\bf x}]$, let $(f)$ denote the principal ideal it generates.
%
%    \begin{lemma} \label{lem:hilbert} Suppose $J \subseteq I$ are homogeneous ideals in $\KK[{\bf x}]$. Let $f \in \KK[{\bf x}]$ be a non-constant homogeneous polynomial such that $I \cap (f) = fI$ and $I + (f) = J + (f)$. Then $I = J$. \end{lemma}
%    
%
%%\begin{proof} Write $I = \bigoplus_{n \geq 0} I_n$ where $I_n$ is the set of homogeneous elements of $I$ with degree $n$.
%%Write
%% $J = \bigoplus_{n \geq 0} J_n$ where $J_n$ is defined similarly. Choose $i \in I_n$ for some $n$, and assume by induction that $I_m = J_m$ for $m < n$. By hypothesis we can write $i = j + fg$ for some $j \in J$ and $g \in \KK[{\bf x}]$. 
%% The homogeneous parts of $j$ and $fg$ outside degree $n$ must 
%%sum to zero. As $f$ is homogeneous, we can therefore assume that $j$ is homogeneous of degree $n$ and that $g$ is homogeneous of degree $n-\deg(f)$.
%%Now we have $fg = i-j \in I \cap (f) = fI$, so $g \in I_{n-\deg f}$. Since $f$ is non-constant,
%%we may assume by induction that $g \in J_{n-\deg f}$, and hence $i = j + fg \in J_n$.
%%\end{proof}








\section{Skew-symmetric matrix Schubert varieties}\label{ss-sect}

Fix a positive even integer $n$ and an algebraically closed field $\KK$. Recall that $\SSM_n$ denotes the set of $n \times n$ skew-symmetric matrices over $\KK$.

    
\begin{definition} \label{defn:SSX}
    Given an involution $z \in \Ifpf_n$, 
    the associated \emph{$n\times n$ skew-symmetric matrix Schubert cell} $\openSSX_z$
    and
     \emph{$n\times n$ skew-symmetric matrix Schubert variety} $\SSX_{z}$ 
    are 
    \begin{align*}
    &\openSSX_z = \{ A \in \SSM_n : \text{$\rank A_{[i][j]} = \rank z_{[i][j]}$ for $i,j \in [n]$}\},
    \\
    &\SSX_z = \{ A \in \SSM_n : \text{$\rank A_{[i][j]} \leq \rank z_{[i][j]}$ for $i,j \in [n]$}\}.
\end{align*}
\end{definition}

Here we identify $z \in \Ifpf_n \subseteq S_n$ with its permutation matrix, so that $\rank z_{[i][j]}$ is the cardinality of $\{z(1), \ldots, z(i)\} \cap [j]$. These definitions would still make sense if $z$ were an arbitrary permutation of $\NN$,
but then it could happen that $\openSSX_z =\emptyset$. We require $z \in \Ifpf_n$ to exclude this degenerate case.
%When discussing these spaces, we consider $n$ to be fixed.

Skew-symmetric matrix  Schubert cells arise as the orbits of a certain group action.
 Specifically,
observe that the general linear group $\GL_n$ acts on $\SSM_n$ by $g: A\mapsto gAg^T$. Given any permutation $w$ of $\NN$, let $\ss_{n}(w)$ be the $n \times n$ matrix
 whose entry in position $(i,j)$
 is $1$ if $w(j) = i<j =w(i)$, $-1$ if $w(j) = i>j = w(i)$, and $0$ otherwise; see Example~\ref{not-prime-ex}. The next theorem shows that Definition~\ref{defn:SSX} is equivalent to the definition of $\SSX_z$ given in the introduction.


% Observe that if $z \in \I_n$ then $\ss_{n}(z) = \ss_{n}(\FPF_n(z))$.

 

 \begin{theorem}[\cite{Cherniavsky}]
 \label{ssx-thm}
  Suppose $z\in \Ifpf_n$.
 Then $\openSSX_z$ is the $B_n$-orbit of the skew-symmetric matrix $\ss_{n}(z)$.
 Moreover, $\SSX_z$ is the Zariski closure of $\openSSX_z$ and is an irreducible variety.
\end{theorem}



\subsection{Pfaffian generators for prime ideals}

As in the introduction, we identify the coordinate ring $\KK[\SSM_n]$ with $\KK[u_{ij} : i,j\in [n], i>j]$
where $u_{ij}$ represents the function $A \mapsto A_{ij}$.
If $A$ is a matrix and $I$ and $J$ are subsets of indices,
then we write $A_{IJ}:= [A_{ij}]_{(i,j) \in I \times J}$ for the corresponding $|I|\times |J|$ submatrix. 
We often apply this notation to the 
$n\times n$ skew-symmetric matrix of variables $\SSU$ with entries defined by
\begin{equation}
\label{SSU-eq}
  \SSU_{ij} = \begin{cases} -u_{ji} & \text{if $i < j$} \\ u_{ij} & \text{if $i > j$} \\ 0 & \text{if $i = j$.} \end{cases}
\end{equation}

%If $A$ is an $m \times n$ matrix and $I \subseteq [m]$ and $ J \subseteq [n]$ are subsets, then 
%we write \[A_{IJ} := [A_{ij}]_{(i,j) \in I \times J}\]
%for the corresponding submatrix. 
If $A$ is a matrix then $\rank A \leq r$ if and only if all size $r+1$ minors of $A$ vanish.
Hence,  $\SSX_z$ is the zero locus of the ideal in 
$\KK[u_{ji} : i,j\in [n], i>j]$ 
 generated by all size $\rank z_{[i][j]} + 1$ minors of $\SSU_{[i][j]}$ for  
   $(i,j) \in [n] \times [n]$.
 %
 This ideal is often \emph{not} prime, however.
 
\begin{example}\label{not-prime-ex}
Take $n=6$ and  let $z = (1,2)(3,5)(4,6) \in \Ifpf_6$, so
\[
\ss_n(z) = \left[\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 & 0\\
-1 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1\\
0 & 0 & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & 0 & 0\\
\end{smallmatrix}\right].
\]
As $\rank z_{[4][3]} = 2$, the ideal described above contains the four $3 \times 3$ minors in $\SSU_{[4][3]}$, one of which is
    \begin{equation*}
    \begin{aligned}
        \det ( \SSU_{\{1,2,4\},\{1,2,3\}}) =  \det \left[ \begin{smallmatrix}
            0 & -u_{21} & -u_{31}\\
            u_{21} & 0  & -u_{32}\\
            %u_{31} & u_{32} & 0\\
            u_{41} & u_{42} & u_{43} \end{smallmatrix}\right] 
            &
            %= u_{21}^2 u_{43} - u_{21}u_{31}u_{42} + u_{21}u_{32}u_{41} \\& 
            = u_{21}(u_{32}u_{41} - u_{31}u_{42} + u_{21}u_{43}).
            \end{aligned}
        \end{equation*}
        However, one can check that the single condition $\rank A_{[4][3]} \leq 2$ implies all the others defining $\SSX_z$, and that accordingly the ideal under consideration is generated by the four $3 \times 3$ minors in $\SSU_{[4][3]}$. In particular, it is generated by homogeneous degree $3$ polynomials, so cannot contain either factor $u_{21}$ or $u_{32}u_{41} - u_{31}u_{42} + u_{21}u_{43}$. (Since $u_{21}$ is nonzero on the matrix $\ss(z) \in \SSX_z$, the other factor must lie in the prime ideal of the irreducible variety $\SSX_z$. In fact, it generates it; see Example~\ref{ex:pf-gens}.)

% Suppose $z \in \Ifpf_6$ is the involution with  
%    \begin{equation*}
%       z = (1,2)(3,5)(4,6) = \begin{array}{cccccc}
%            0 & 1 & 0 & 0 & 0 & 0\\  
%            1 & 0 & 0 & 0 & 0 & 0\\
%            0 & 0 & 0 & 0 & 1 & 0\\
%            0 & 0 & \boxed{0} & 0 & 0 & 1\\
%            0 & 0 & 1 & 0 & 0 & 0\\
%            0 & 0 & 0 & 1 & 0 & 0\\
%        \end{array}
%    \end{equation*}
%    where we have boxed the positions in $\SSD(z)$. The only essential rank condition says that $\rank A_{[4][3]} \leq 2$, so the ideal described above is generated by the four $3 \times 3$ minors in $\SSU_{[4][3]}$. However, one of these minors is 
%    \begin{equation*}
%    \begin{aligned}
%        \det \left[ \begin{smallmatrix}
%            0 & -u_{21} & -u_{31}\\
%            u_{21} & 0  & -u_{32}\\
%            %u_{31} & u_{32} & 0\\
%            u_{41} & u_{42} & u_{43} \end{smallmatrix}\right] 
%            &= u_{21}^2 u_{43} - u_{21}u_{31}u_{42} + u_{21}u_{32}u_{41} 
%            \\& = u_{21}(u_{21}u_{43} - u_{31}u_{42} + u_{32}u_{41}).
%            \end{aligned}
%        \end{equation*}
%        The ideal is generated by homogeneous polynomials of degree $3$, so by degree considerations it cannot contain either factor of the polynomial above, and therefore is not prime.
\end{example}

In this section, we identify a different set of polynomials which turn out to generate the prime ideal $I(\SSX_z)$. 
The key idea in our construction is to replace minors of matrices by Pfaffians.
Recall that the \emph{Pfaffian} of a skew-symmetric $n\times n$ matrix $A$ is
\be\label{pf-def-eq}
 \pf (A)
= \sum_{z \in \Ifpf_n} (-1)^{\ellfpf(z)} \prod_{z(i)<i \in [n] } A_{z(i),i}
\ee
where once again $\ellfpf(z) = |\SSD(z)|$. For example, we have
$
\pf( \SSU_{[2][2]}) = \pf \left[\begin{smallmatrix} 0 & - u_{21} \\ u_{21} & 0 \end{smallmatrix}\right] = -u_{21}
$.
%and
%\[
%\pf(\SSU_{[4][4]}) =\pf \left[\begin{smallmatrix}
% 0 & - u_{21} & -u_{31} & -u_{41} \\ 
% u_{21} & 0 & -u_{32} & -u_{42} \\
% u_{31} & u_{32} & 0 & -u_{43} \\
% u_{41} & u_{42} & u_{43} & 0 \end{smallmatrix}\right]  = u_{32} u_{41} - u_{31} u_{42} + u_{21} u_{43}.
%\]
If $n$ is odd then
the outer summation in \eqref{pf-def-eq} is empty so $\pf(A) = 0$. This is consistent with the well-known fact that
$ \pf (A)^2 = \det (A)$, which is zero if $A$ is skew-symmetric of odd size.


 

\begin{theorem}\label{ssi-thm} Given $z \in \Ifpf_n$, the prime ideal $I(\SSX_z)$ in 
$\KK[\SSM_n] = \KK[u_{ij} : i,j \in [n], i>j]$ is generated by the Pfaffians $\pf (\SSU_{RR})$ for all nonempty sets $R \subseteq [n]$ of even size 
for which there exist 
 indices $i,j \in [n]$ with $i\geq j$ such that $R \subseteq [i]$ and $|R \cap [j]| > \rank z_{[i][j]}$.  \end{theorem}



\begin{example} \label{ex:pf-gens} Take $n=6$ and  let $z = (1,2)(3,5)(4,6) \in \Ifpf_6$ as in Example~\ref{not-prime-ex}. As mentioned in that example, $\SSX_z$ is defined by the single rank condition $\rank A_{[4][3]} \leq 2$, and similarly one can deduce that it is only necessary to consider $(i,j) = (4,3)$ in Theorem~\ref{ssi-thm}. That is, $I(\SSX_z)$ is generated by $\pf(\SSU_{RR})$ where $R \subseteq [4]$ has even size and $|R \cap [3]| > \rank z_{[4][3]} = 2$, i.e., by the single Pfaffian 
\begin{equation*}
  \pf(\SSU_{\{1,2,3,4\},\{1,2,3,4\}}) =  \pf \left[\begin{smallmatrix}
        0 & -u_{21} & -u_{31} & -u_{41} \\ 
        u_{21} & 0 & -u_{32} & -u_{42} \\ 
        u_{31} & u_{32} & 0 & -u_{43} \\ 
        u_{41} & u_{42} & u_{43} & 0 \\ 
    \end{smallmatrix}\right] = u_{32}u_{41} - u_{31}u_{42} + u_{21}u_{43}.
\end{equation*}
To verify Theorem~\ref{ssi-thm} in this case, observe that the Pfaffian above is a square root of $\det(\SSU_{\{1,2,3,4\},\{1,2,3,4\}})$, so it vanishes on $A \in \SSM_6$ if and only if $\rank A_{[4][4]} \leq 3$. Because a skew-symmetric matrix has even rank, this is equivalent to $\rank A_{[4][3]} \leq 2$. Therefore the ideal $(u_{21}u_{43} - u_{31}u_{42} + u_{41}u_{32})$ does have zero locus $\SSX_z$, and it is prime because it is generated by an irreducible polynomial.
\end{example}

\subsection{Gr\"obner bases} \label{grob-sect}
The generating set of $I(\SSX_z)$ given in Theorem~\ref{ssi-thm} need \emph{not} be a Gr\"obner basis.
\begin{example} \label{ex:not-grobner}
    Let $n = 6$ and $z = (1,2)(3,6)(4,5)$. Using Theorem~\ref{ssj-thm}, one computes that $\init(I(\SSX_z)) = (u_{32}u_{41}, u_{32}u_{51}, u_{31}u_{42}u_{51})$. Theorem~\ref{ssi-thm} implies that $I(\SSX_z)$ is generated by the two Pfaffians $g := \pf(\SSU_{\{1,2,3,4\},\{1,2,3,4\}}) = u_{32}u_{41} - u_{31}u_{42} + u_{21}u_{43}$ and $h := \pf(\SSU_{\{1,2,3,5\},\{1,2,3,5\}}) = u_{32}u_{51} - u_{31}u_{52} + u_{21}u_{53}$. Their initial terms are the first two generators of $\init(I(\SSX_z))$, but the last monomial generator is evidently not in the ideal they generate, nor can it be the initial term of any Pfaffian of a submatrix of $\SSU$ (but it is the initial term of $u_{41}h - u_{51}g$).
\end{example}



Given sets $A = \{a_1<\dots<a_q\}$ and $B = \{b_1 > \dots > b_q\}$,
we define 
\[ f_{AB} :=  \pf \left[ \begin{array}{ll} \SSU_{BB} & \SSU_{B,A\ominus B} \\ \SSU_{A\ominus B, B} & 0
    \end{array}  \right] \in \KK[u_{ij} :i,j \in \NN, i> j] 
    \]
    where 
    $A \ominus B$ is the set of $a \in A$ for which no $b$ exists with \[(a,b),(b,a) \in \{(a_1,b_1),\dots,(a_q,b_q)\}.\]
\begin{theorem}\label{ss-main-thm3}
    If $z \in \Ifpf_n$ then 
    the elements $f_{AB}$, as $(i,j)$ ranges over all pairs in $[n]\times [n]$ with $i \geq j$ and $(A,B)$ ranges over all pairs in
    $ \tbinom{[i]}{q}\times \tbinom{[j]}{q} $ for $q = \rank z_{[i][j]}+1$,
    form a 
    Gr\"obner basis for $I(\SSX_z) $ with respect to the reverse lexicographic term order.
\end{theorem}
For instance, if $z = (1,2)(3,6)(4,5)$ as in Example~\ref{ex:not-grobner}, then one of the pairs arising in Theorem~\ref{ss-main-thm3} is $(A = \{1,4,5\}, B = \{1,2,3\})$. One computes that $f_{AB}$ is then $u_{51}g - u_{41}h = -u_{31}u_{42}u_{51} + u_{21}u_{43}u_{51} + u_{31}u_{41}u_{52} - u_{21}u_{41}u_{53}$, whose initial term is the monomial generator which was missing in Example~\ref{ex:not-grobner}.



\subsection{Initial ideals}\label{mon-sect}

From Theorem~\ref{ss-main-thm3} on Gr\"obner bases we can deduce an explicit formula for the initial ideal of $I(\SSX_z)$. Given sets $A = \{a_1 < \cdots < a_q\}, \{b_1 > \cdots > b_q\} \subseteq [n]$, let $u_{AB} = \prod_{i=1}^q u_{a_i b_i}$. Define $\phi : \KK[u_{ij} : i,j \in [n]] \to \KK[u_{ij} : 1 \leq j \leq i \leq n]$ to be the ring homomorphism with $u_{ii} \mapsto 0$ and $u_{ij}, u_{ji} \mapsto u_{ij}$ if $i > j$. Finally, let $\SSu_{AB}$ be the squarefree radical of $\phi(u_{AB})$, where we take the radical of $0$ to be $0$.


\begin{theorem}\label{ssj-thm}
Given $z \in \Ifpf_n$, the initial ideal of $I(\SSX_z)$ under reverse lexicographic term order is generated by all monomials of the form $\SSu_{AB}$,
where  $(A, B) \in \binom{[i]}{q}\times \binom{[j]}{q}$
for some $(i,j) \in [n]\times [n]$ with $i\geq j$ and $q = \rank z_{[i][j]}+1$.
\end{theorem}

It is not necessarily the case that $u_{AB}$ is the leading term in the polynomial $f_{AB}$ defined in \ref{grob-sect}. However, this does hold if there are no indices $i < j$ with $b_i > a_j > a_i > b_j$, and given any $(A,B)$, one can find $(A',B')$ with this property and such that $u_{A'B'} = u_{AB}$.

%
%It can be shown that in the definition of $\SSJ_z$ above, one only needs those generators corresponding to $(i,j) \in \Ess(\SSD(z))$. We will not prove this fact since we do not need it for any arguments, but we do assume it in the next example.

\begin{example} \label{ex:SSJ-generators}
    Take $n=6$ and let $z = (1,2)(3,5)(4,6) \in \Ifpf_6$ as in Examples~\ref{not-prime-ex} and \ref{ex:pf-gens}. As mentioned in those examples, $\SSX_z$ is defined by the single rank condition $\rank A_{[4][3]} \leq 2$, and similarly one can deduce that it is only necessary to consider $(i,j) = (4,3)$ and $q = 3 = \rank z_{[4][3]} + 1$ in Theorem~\ref{ssj-thm}. Thus, the ideal $\init(I(\SSX_z))$ is generated by
$\SSu_{AB}$ where $A$ ranges over the $3$-element subsets of $\{1,2,3,4\}$ and $B=\{1,2,3\}$. The
relevant monomials % $u_{AB}$, $\phi(u_{AB})$, and $\SSu_{AB}$
 are listed below:
    \begin{equation*}
        \begin{array}{r|r|r|r|r}
            A & B & u_{AB} & \phi(u_{AB}) &  \SSu_{AB}\\
            \hline
            \{1,2,3\} & \{3,2,1\} & u_{13}u_{22}u_{31} & 0  & 0\\
            \{1,2,4\} & \{3,2,1\} & u_{13}u_{22}u_{41} & 0 & 0\\
            \{1,3,4\} & \{3,2,1\} & u_{13}u_{32}u_{41} & u_{31}u_{32}u_{41} & u_{31}u_{32}u_{41}\\
            \{2,3,4\} & \{3,2,1\} & u_{23}u_{32}u_{41} & u_{32}^2 u_{41} & u_{32}u_{41} \\
        \end{array}
    \end{equation*}
 Theorem~\ref{ssj-thm} now asserts that $\init(I(\SSX_z)) = (u_{32}u_{41})$ is the ideal generated by the rightmost column. Evidently this is indeed the initial ideal of $I(\SSX_z) = (u_{32}u_{41} - u_{31}u_{42} + u_{21}u_{43})$ as computed in Example~\ref{ex:pf-gens}.
\end{example}



\section{Stanley--Reisner ideals and K-polynomials}\label{tra-sect}

\subsection{Pipe dreams and Stanley--Reisner ideals}
\label{subsec:pipe-dreams}

In this subsection, we derive an alternate expression for $\init(I(\SSX_z))$ as an intersection of prime monomial ideals indexed by the \emph{involution pipe dreams} corresponding to $z$, as introduced in \cite{HMPdreams}. Define the \emph{pipe dream reading word} of a finite set $D \subset \NN \times \NN$ to be the word $\word(D)$ whose letters list the numbers $i+j-1$ as $(i,j)$ ranges over all elements of $D$ in the order that makes $(i,-j)$ increase lexicographically
(i.e., which reads the rows in order, but going right to left).
For example, the pipe dream reading word of $D = \{(1,4),(1,3),(2,6),(5,5),(5,4),(5,3)\}$ is ${\bf 437987}$.

\begin{definition} \label{fp-def}
A \emph{reduced fpf-involution word} for $z \in \Ifpf_n$ is a sequence of positive integers $i_1i_2\cdots i_l$ of shortest possible length such that
\begin{equation*}
    z= s_{i_l}\cdots s_{i_2}s_{i_1}(1,2)(3,4)\ldots (n-1,n) s_{i_1}s_{i_2}\cdots s_{i_l},
\end{equation*}
where $s_i \in S_n$ is the transposition $(i,i+1)$.

A set $D \subseteq \ltriang_n := \{(i,j) \in [n] \times [n] : i > j\}$ is a \emph{reduced (fpf-involution) pipe dream} of $z$ if the pipe dream reading word of $D$ is a reduced fpf-involution word of $z$. Let $\FP(z)$ denote the set of reduced fpf-involution pipe dreams of $z$.
\end{definition}
Bergeron and Billey associated a set of \emph{reduced pipe dreams} to any permutation \cite{bergeron-billey}, and our fpf-involution pipe dreams are closely related:  $D \in \FP(z)$ if and only if $D$ is the intersection of $\ltriang_n$ with a pipe dream of $z$ which is symmetric about its main diagonal \cite[Theorem 3.12]{HMPdreams} 
%The set $\FP(z)$ where $z = \FPF_6((1,2)(3,6)(4,5))$ is given in Example~\ref{ex:subword-ss} below.


    Given a set $D \subseteq \ltriang_n $,
    we write $ (u_{ij} : (i,j) \in D)$ to denote the ideal in
    the coordinate ring $\KK[\SSM_n] = \KK[u_{ij} : i,j \in [n], i>j]$ generated by $u_{ij}$ for all $(i,j) \in D$.

\begin{theorem} \label{thm:primary-decomposition-ss}
Let $z \in \Ifpf_n$.
Then $\init(I(\SSX_z)) = \bigcap_{D \in \FP(z)} (u_{ij} : (i,j) \in D) \subseteq \KK[\SSM_n]$. \end{theorem}

\begin{example}
    Let $z = (1,2)(3,5)(4,6)$. Then $z = s_4 (1,2)(3,4)(5,6) s_4$, the only reduced fpf-involution word of $z$ is ${\bf 4}$, and the two elements of $\FP(z)$ are $\{(4,1)\}$ and $\{(3,2)\}$. Theorem~\ref{thm:primary-decomposition-ss} then says that $\init(I(\SSX_z)) = (u_{41}) \cap (u_{32}) = (u_{41}u_{32})$, in agreement with Example~\ref{ex:SSJ-generators}.
\end{example}


The \emph{Stanley--Reisner ideal} of a simplicial complex $\Delta$ with vertex set $[N]$ is the ideal in $\KK[x_1, \ldots, x_N]$
generated by the elements $\prod_{v \in E} x_v $ for all  $E \subseteq [N]$ with $E \notin \Delta$. Any ideal generated by squarefree monomials is a Stanley--Reisner ideal, and $\init(I(\SSX_z))$ has this property by Theorem~\ref{ssj-thm}. The next theorem identifies it as the Stanley--Reisner ideal of an explicit complex; it is not hard to deduce from Theorem~\ref{thm:primary-decomposition-ss}.



\begin{definition}
    The \emph{fpf-subword complex} associated to $z \in \Ifpf_n$ and a subset $Q \subseteq \ltriang_n$ is the simplicial complex with vertices $Q$ and faces
\begin{equation*}
    \subword(z,Q) = \{S \subseteq Q : \text{$Q \setminus S$ contains an fpf-involution pipe dream for $z$} \}.
\end{equation*}
\end{definition}

\begin{theorem} \label{thm:SR-ideal}
If $z \in \Ifpf_n$ then
    $\init(I(\SSX_z))$ is the Stanley--Reisner ideal of $\subword(z,\ltriang_n)$. Moreover, the complex $\subword(z,Q)$ is vertex-decomposable (hence shellable) for any $Q \subseteq \ltriang_n$.
\end{theorem}


\subsection{$K$-polynomials}
\label{subsec:K}
We now describe how the results of the preceding subsection lead to a new proof of a combinatorial formula for the torus-equivariant $K$-theory class of $\SSX_z$ from \cite{MacdonaldNote}. Our approach is modeled after Knutson and Miller's study of subword complexes and the resulting combinatorial formulas for Grothendieck polynomials \cite{SubwordComplexes}. 


Suppose $R = \KK[u_1, u_2,\ldots, u_N]$ is a polynomial ring that is graded by a (multiplicative) free abelian group $G$ in the sense that each variable $u_i$ is assigned a degree $\deg(u_i) \in G$, a monomial $u_1^{a_1} \cdots u_N^{a_N}$ has degree $\deg(u_1)^{a_1} \cdots \deg(u_N)^{a_N}$, and a polynomial is homogeneous if its terms all have the same degree. If $I \subseteq R$ is a homogeneous ideal (that is, generated by homogeneous elements), then one can also speak of degrees and homogeneous elements in $R/I$. The \emph{K-polynomial} of $I$ is then the following formal $\ZZ$-linear combination of elements of $G$:
\begin{equation*}
    \K(I) = \prod_{i=1}^N (1 - \deg(u_i)) \sum_{g \in G} \dim_{\KK}((R/I)_g) g,
\end{equation*}
where $(R/I)_g$ is the subspace of degree $g$ homogeneous elements in $R/I$. See \cite[\S 8]{MillerSturmfels} for a more careful definition, including conditions guaranteeing that this formal generating function is well-defined.


\begin{example}
    Suppose $R = k[x_1, x_2]$, $G =  \ZZ t_1 \oplus \ZZ t_2$, $\deg x_i = t_i$, and $I = (x_1^2)$. 
    A homogenous basis for $R/I$ is  $\{x_1^i x_2^j + I : 0 \leq i \leq 1, 0 \leq j\}$, so 
$
        \Hilb(R/I) = \sum_{j=0}^{\infty} (1 + t_1) t_2^j = \frac{1+t_1}{1-t_2}
$
    and $\K(I) = \tfrac{1+t_1}{1-t_2}(1-t_1)(1-t_2) = 1-t_1^2$.
\end{example}


Let $G$ be the multiplicative abelian group freely generated by $a_1,a_2,\dots,a_n$. We give the coordinate ring $\KK[\SSM_n] = \KK[u_{ij} : n \geq i > j \geq 1]$ a $G$-grading by setting
 $\deg (u_{ij}) = a_i a_j$. These degrees arise as torus weights induced from the action of the torus $T_n \subseteq B_n$ of diagonal matrices on $\SSM_n$. Since a skew-symmetric matrix Schubert variety $\SSX_z$ is $B_n$-stable, it is $T_n$-stable, which implies that $I(\SSX_z)$ is homogeneous under this grading.




\begin{definition} The \emph{symplectic Grothendieck polynomial} associated to $z \in \Ifpf_n$ is the $K$-polynomial 
$\Gfpf_z := \K(I(\SSX_z))$. \end{definition}

Although \emph{a priori} one only knows that  $\Gfpf_z \in \ZZ[[a_1,a_2,\dots]]$, it turns out that this formal generating function always has only finitely many nonzero terms. The functions $\Gfpf_z$ 
 are the same as the polynomials $\Upsilon^K_{\pi,(\GL_{2n},\Sp_{2n})}$ given in \cite{WyserYongOSp} %,where they are denoted $\Upsilon^K_{\pi,(\GL_{2n},\Sp_{2n})}$, 
as representatives for the ordinary $K$-theory classes of the $\Sp_n(\CC)$-orbit closures on $\GL_n(\CC)/B_n$.
Symplectic Grothendieck polynomials also appear in \cite{MP2019, MP2019b} (after making the change of variables $a_i \mapsto 1- x_i$)
as representatives for the $T$-equivariant $K$-theory classes of the varieties $\SSX_z$.
The latter polynomials (in $\ZZ[x_1,x_2,\dots]$) have well-defined ``stable limits''
converging to symmetric functions that expand positively in terms of Ikeda and Naruse's $K$-theoretic Schur $P$-functions (with parameter $\beta=-1$); see \cite[Thm. 1.9]{Mar} and \cite[Thm. 1.6]{MP2019}.
 
 
\begin{example} \label{ex:groth1-ss} 
Let $D$ be a subset of $\ltriang_n = \{ (i,j) \in [n] \times [n] : i > j\}$.
As in the proof of Theorem~\ref{thm:primary-decomposition-ss}, 
write $\J{D} = (u_{ij} : (i,j) \in D ) \subset \KK[u_{ij} : n \geq i>j\geq 1] =: R.$
The set of monomials 
$ \prod_{(i,j) \in D^c} u_{ij}^{m_{ij}}$ where $D^c:= \ltriang_n \setminus D$ descends to a basis for $R/\J{D}$. Accordingly,
    \begin{align*}
        \K(\J{D}) &= \prod_{n \geq i > j \geq 1 } (1-a_i a_j) \prod_{(i,j)  \in D^c} \sum_{m_{ij} \geq 0} (a_i a_j)^{m_{ij}}  = \prod_{(i,j) \in D} (1-a_i a_j).
    \end{align*}
If $z \in \Ifpf_n$ is \emph{fpf-dominant} in the sense that $\RFP(z)$ has a unique element $D$, then $I(\SSX_z) = \J{D}$ by Theorem~\ref{thm:primary-decomposition-ss}, and
   $
        \Gfpf_z = \prod_{(i,j) \in D} (1-a_i a_j),
   $
 which recovers the skew-symmetric half of \cite[Thm. 3.8]{MP2019}.  
    \end{example}

    


If $z \in \{1\} \sqcup \Ifpf_n$ and $s = s_i = (i,i+1) \in S_n$, then we define 
\[ z \ast s = \begin{cases} 1 & \text{if $(i,i+1)$ is a cycle of $z$} \\
z & \text{if }z=1\text{ or }z(i) > z(i+1) \\
szs&\text{otherwise}.
\end{cases}
\]
Note that if $z \in \Ifpf_n$ then either $z \ast s \in \Ifpf_n$ or $z\ast s = 1$, but we always have $1 \ast s = 1$.
The operation $\ast$ extends to a right action of the \emph{$0$-Hecke monoid} of $S_n$ but not to a group action. If $i_1  i_2\cdots i_l$ is a word, define $\deltafpf(i_1 i_2 \cdots i_l) = (\cdots (( 1_{\fpf} \ast s_{i_1}) \ast s_{i_2}) \cdots) \ast s_{i_l}$.
%Recall the notion of the pipe dream reading word of a finite set from  Section~\ref{subsec:pipe-dreams}.

\begin{definition}
An \emph{extended (fpf-involution) pipe dream} for $z \in \Ifpf_n$ is a subset $D \subseteq  \ltriang$
whose pipe dream reading word $i_1i_2\cdots i_l$ satisfies $z = \deltafpf(i_1 i_2 \cdots i_l)$.
Let $\NFP(z)$ be the set of all extended pipe dreams for $z$.
\end{definition}

 The set  $\NFP(z)$ is called $\mathsf{InvDreams}(z)$ in \cite{MacdonaldNote}.
%Each $D \in \NFP(z)$ has $|D|\geq \ellfpf(z)$, with equality if and only if  
%$D \in \FP(z)$  \cite[Thm. 3.12]{HMPdreams}.
%The containment $\FP(z) \subset \NFP(z)$ is strict unless $z =1_\fpf$.
The next theorem can be deduced from Theorem~\ref{thm:SR-ideal} and a more detailed analysis of the fpf subword complex $\subword(z,\ltriang_n)$ using the techniques of Knutson and Miller in \cite[\S 4]{SubwordComplexes}.
\begin{theorem}[{\cite[Thm. 4.5]{MacdonaldNote}}]\label{4.5-thm} Let $z \in \Ifpf_n$. Then
    \begin{equation*}
        \Gfpf_z = \sum_{D \in \NFP(z)} (-1)^{|D|-\ellfpf(z)} \prod_{(i,j) \in D} (1 - a_i a_j).
    \end{equation*}
\end{theorem}


\printbibliography
 



%\begin{thebibliography}{99}


% \bibitem{BagnoCherniavsky}
% E. Bagno and Y. Cherniavsky, 
% Congruence $B$-orbits and the Bruhat poset of involutions of the symmetric group, 
% \emph{Discrete Math.} \textbf{312} (2012), 1289--1299.
 

%  \bibitem{bergeron-billey} N. Bergeron and S. Billey, RC-graphs and Schubert polynomials, \emph{Experiment. Math.} \textbf{2} (1993), no. 4, 257--269.
 
%     \bibitem{BJS} S. C. Billey, W. Jockusch, and R. P. Stanley, Some combinatorial properties of Schubert polynomials, \emph{J. Algebraic Combin.} \textbf{2} (1993), 345--374.

% \bibitem{Cherniavsky} Y. Cherniavsky, 
% On involutions of the symmetric group and congruence $B$-orbits of anti-symmetric matrices, 
% \emph{J. Algebra Comput.} \textbf{21} (2011), 841--856.



% \bibitem{DeNegri} E. De Negri, Pfaffian ideals of ladders, \emph{J. Pure Appl. Algebra} \textbf{125} (1998), 141--153. 

% \bibitem{DeNegriSbarra} E. De Negri and E. Sbarra, Gr\"obner bases of ideals cogenerated by Pfaffians, \emph{J. Pure Appl. Algebra} \textbf{215} (2011), 812--821. 

%     \bibitem{FultonEssentialSet} W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, \emph{Duke Math. J.} \textbf{65} (1991), 381--420.

% \bibitem{HMPdreams} Z. Hamaker, E. Marberg, and B. Pawlowski, Involution pipe dreams, preprint (2019), {\tt arXiv: 1911.12009}.


% \bibitem{HerzogTrung} J. Herzog and N. V. Trung, Gr\"obner bases and multiplicity of determinantal and Pfaffian ideals, \emph{Adv. Math.} \textbf{96} (1992), 1--37.


% \bibitem{JonssonWelker} J. Jonsson and V. Welker, A spherical initial ideal for Pfaffians, \emph{Illinois J. Math.} \textbf{51} (2007), 1397--1407.

% %\bibitem{Knutson} A. Knutson. Schubert polynomials, pipe dreams, equivariant classes, and a co-transition
% %formula, preprint (2019), arXiv:1909.13777.

%  \bibitem{KnutosonMiller} A. Knutson and E. Miller, Gr\"obner geometry of Schubert polynomials, \emph{Ann. Math.} \textbf{161} (2005), 1245--1318.



%  \bibitem{SubwordComplexes} A. Knutson and E. Miller, Subword complexes in Coxeter groups, \emph{Adv. Math.} \textbf{184} (2004), 161--176.

%  %\bibitem{LascouxSchutzenberger} A. Lascoux and M-P. Sch\"utzenberger, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une vari\'et\'e de drapeaux, \emph{C. R. Acad. Sci. Paris S\'er. I Math.} \textbf{295} (1982), 629--633.


%  %\bibitem{LascouxSchutzenbergerTransitions} A. Lascoux and M-P. Sch\"utzenberger, Schubert polynomials and the Littlewood--Richardson rule, \emph{Lett. Math. Phys.} \textbf{10} (1985), 111--124.
 
%  \bibitem{Mar} E. Marberg, A symplectic refinement of shifted Hecke insertion, \emph{J. Combin. Theory Ser. A} \textbf{173} (2020), 105216.

% \bibitem{MP2019} E. Marberg and B. Pawlowski, $K$-theory formulas for orthogonal and symplectic orbit closures, \emph{Adv. Math.} \textbf{372} (2020), 107299.

% \bibitem{MP2019b} E. Marberg and B. Pawlowski, On some properties of symplectic Grothendieck polynomials, \emph{J. Pure Appl. Algebra} \textbf{225} (2021), no. 1, 106463.


% \bibitem{MP2020} E. Marberg and B. Pawlowski, Gr\"obner geometry for skew-symmetric matrix Schubert varieties, preprint (2020), {\tt arXiv:2010.00653}.

% \bibitem{MacdonaldNote} E. Marberg, and B. Pawlowski, Principal specializations of Schubert polynomials in classical types, 
% \emph{Algebr. Comb.}, to appear. %preprint (2020), {\tt arXiv: 2002.00303}.

% \bibitem{MillerSturmfels} E. Miller and B. Sturmfels, \emph{Combinatorial commutative algebra}, vol. 227, Springer Science \& Business Media, 2004.

% \bibitem{RaghavanUpadhyay} K. N. Raghavan and S. Upadhyay, Initial ideals of tangent cones to Schubert varieties in orthogonal Grassmannians, \emph{J. Combin. Theory Ser. A} \textbf{116} (2009), 663--683. 


%  \bibitem{WyserYongOSp} B. Wyser and A. Yong, Polynomials for symmetric orbit closures in the flag variety, \emph{Transform. Groups} \textbf{22} (2017), 267--290.

% \end{thebibliography}

\end{document}
