%% 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{5}
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\usepackage{latexsym,mathtools}
\usepackage{wrapfig}
\usepackage{ytableau}
\usepackage{tikz}
\usetikzlibrary{calc, shapes, backgrounds,arrows,positioning,plotmarks}
\tikzset{>=stealth',
  head/.style = {fill = white, text=black},
  plaque/.style = {draw, rectangle, minimum size = 10mm}, 
  pil/.style={->,thick},
  junct/.style = {draw,circle,inner sep=0.5pt,outer sep=0pt, fill=black}
  }

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{examplex}[theorem]{Example}
\newenvironment{example}
  {\pushQED{\qed}\renewcommand{\qedsymbol}{$\diamondsuit$}\examplex}
  {\popQED\endexamplex}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\newcommand{\gr}{\ensuremath{\mathrm{Gr}}}
\newcommand{\flags}{\ensuremath{\mathcal{F}}}
\newcommand{\GL}{\ensuremath{GL}}

\newcommand{\downshift}{\downarrow \!}
\newcommand{\rightshift}{\rightarrow \!}

\DeclareMathOperator{\codim}{codim}

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

\newcommand{\blank}{\phantom{2}}
\newcommand{\id}{\ensuremath{\mathrm{id}}}
\newcommand{\pt}{\ensuremath{\mathrm{pt}}}
\newcommand{\bbb}{\ensuremath{\mathsf{b}}}
\newcommand{\xx}{\ensuremath{\mathbf{x}}}
\newcommand{\Ess}{\ensuremath{\mathrm{Ess}}}
\newcommand{\Dom}{\ensuremath{\mathrm{Dom}}}
\newcommand{\Esshat}{\ensuremath{\mathrm{Ess}^\prime}}
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\newcommand{\reg}{\ensuremath{\mathrm{reg}}}

\newcommand{\poly}{\ensuremath{\mathrm{Poly}}}

\newcommand{\Schub}{\ensuremath{\mathfrak{S}}}
\newcommand{\Groth}{\ensuremath{\mathfrak{G}}}
\newcommand{\key}{\ensuremath{\mathfrak{D}}}


\newcommand{\bpd}[1]{{\sf BPD}(#1)}
\newcommand{\init}[1]{{\tt in}(#1)}
\newcommand{\rank}[1]{{\tt rank}(#1)}

%\usepackage{lipsum}

%% define your title in the usual way
\title[Gr\"obner geometry of Schubert polynomials through ice]{Gr\"obner geometry of Schubert polynomials through ice} 

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Z. Hamaker, O. Pechenik, and A. Weigandt]{
Zachary Hamaker\thanks{\href{mailto:zhamaker@ufl.edu}{zhamaker@ufl.edu}}\addressmark{1}, Oliver Pechenik\thanks{\href{mailto:oliver.pechenik@uwaterloo.ca}{oliver.pechenik@uwaterloo.ca} Oliver Pechenik  was partially supported by a Mathematical Sciences Postdoctoral Research Fellowship (\#1703696) from the National Science
Foundation.}\addressmark{2}, \and Anna Weigandt\thanks{\href{mailto:weigandt@umich.edu}{weigandt@umich.edu}}\addressmark{3}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Florida, Gainesville, FL 32601, USA \\ \addressmark{2}Department of Combinatorics \& Optimization, University of Waterloo, Waterloo, ON N2L3G1, Canada \\ \addressmark{3}Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA}

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

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%\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{The geometric naturality of Schubert
polynomials and their combinatorial pipe dream representations was established by
Knutson and Miller (2005) via antidiagonal Gr\"obner degeneration of matrix
Schubert varieties. We consider instead diagonal Gr\"obner degenerations. In
this dual setting, Knutson, Miller, and Yong (2009) obtained alternative combinatorics
for the class of ``vexillary'' matrix Schubert varieties. We
initiate a study of general diagonal degenerations, relating them to a neglected formula of
Lascoux (2002) in terms of the $6$-vertex ice model (recently rediscovered by
Lam, Lee, and Shimozono (2018) in the guise of ``bumpless pipe dreams'').}

%% put your French abstract here, or comment this out if you don't have one
%\resume{Je suis une pomme de terre.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Schubert polynomial, bumpless pipe dream, matrix Schubert variety}

%% 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{05Hamaker.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}
\label{s:intro}

Let $\flags_n$ be the complex flag variety, the parameter space for complete flags of nested vector subspaces of $\mathbb{C}^n$.
The Schubert cell decomposition of $\flags_n$ yields a distinguished $\mathbb{Z}$-linear basis for the cohomology ring $H^\star(\flags_n)$.
On the other hand, Borel \cite{Borel} presented this ring as
$
H^\star(\flags_n) \cong \mathbb{Z}[x_1, \dots, x_n] / I,
$
where $I$ is the ideal generated by the nonconstant elementary symmetric polynomials. 

It is natural to desire polynomial representatives for the Schubert basis with respect to this presentation. Building on work of Bernstein, Gelfand, and Gelfand \cite{Bernstein.Gelfand.Gelfand}, Lascoux and Sch\"utzenberger \cite{Lascoux.Schutzenberger} introduced \emph{Schubert polynomials}.
These are combinatorially well-adapted coset representatives for images of Schubert cohomology classes under the Borel isomorphism. 
In fact, Lascoux and Sch\"utzenberger  introduced more general \emph{double Schubert polynomials} that represent Schubert classes in the $T$-equivariant cohomology of $\flags_n$ (where $T \subset \GL_n(\mathbb{C})$ is the group of invertible diagonal matrices).

Since their introduction, (double) Schubert polynomials have become central objects in algebraic combinatorics. 
%They have also been interpreted through the geometry of degeneracy loci and used to unify many classical results in that area \cite{Fulton, Fulton.Pragacz}. 
Knutson and Miller \cite{Knutson.Miller} gave a  geometric justification for the naturality of Schubert polynomials by Gr\"obner degeneration of certain affine varieties.
Moreover, they recovered aspects of the combinatorics of Schubert polynomials through this geometry, including identifying irreducible components of the degeneration with the  \emph{pipe dreams} of earlier combinatorial formulas \cite{Bergeron.Billey,Fomin.Kirillov}. This explicit degeneration demonstrates the geometric naturality of pipe dream combinatorics.

Lascoux \cite{Lascoux:ice} introduced an alternate combinatorial model for (double) Schubert polynomials using  states of the square-ice (``$6$-vertex'') model from statistical physics. 
Recently, Lam, Lee, and Shimozono~\cite{Lam.Lee.Shimozono} rediscovered this Schubert polynomial model and gave a cleaner description in terms of \emph{bumpless pipe dreams}.
The connection between \cite{Lam.Lee.Shimozono, Lascoux:ice} is detailed in~\cite{Weigandt}. 

Although both ordinary pipe dreams and bumpless pipe dreams compute the same double Schubert polynomials and appear superficially similar, they compute these polynomials in fundamentally different ways. In particular, (except in trivial cases) no weight-preserving bijection exists between these two sets. In light of this fact, the geometric content of bumpless pipe dreams and Lascoux's ice formula remains unclear.

\begin{example}
Let $w$ be the permutation $2143 \in S_4$. The three ordinary pipe dreams
\[\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (0,4.5)--(1.5,4.5)--(1.5,5);
\draw[thick,rounded corners] (0,3.5)--(.5,3.5)--(.5,5);
\draw[thick,rounded corners] (0,2.5)--(1.5,2.5)--(1.5,3.5)--(2.5,3.5)--(2.5,4.5)--(3.5,4.5)--(3.5,5);
\draw[thick,rounded corners](0,1.5)--(.5,1.5)--(.5,3.5)--(1.5,3.5)--(1.5,4.5)--(2.5,4.5)--(2.5,5);
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (0,4.5)--(1.5,4.5)--(1.5,5);
\draw[thick,rounded corners] (0,3.5)--(.5,3.5)--(.5,5);
\draw[thick,rounded corners] (0,2.5)--(.5,2.5)--(.5,3.5)--(2.5,3.5)--(2.5,4.5)--(3.5,4.5)--(3.5,5);
\draw[thick,rounded corners](0,1.5)--(.5,1.5)--(.5,2.5)--(1.5,2.5)--(1.5,4.5)--(2.5,4.5)--(2.5,5);
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (0,4.5)--(1.5,4.5)--(1.5,5);
\draw[thick,rounded corners] (0,3.5)--(.5,3.5)--(.5,5);
\draw[thick,rounded corners] (0,2.5)--(.5,2.5)--(.5,3.5)--(1.5,3.5)--(1.5,4.5)--(3.5,4.5)--(3.5,5);
\draw[thick,rounded corners](0,1.5)--(.5,1.5)--(.5,2.5)--(1.5,2.5)--(1.5,3.5)--(2.5,3.5)--(2.5,5);
\end{tikzpicture}
\]
 for this permutation
present the corresponding double Schubert polynomial as
\[\mathfrak S_w= (x_1-y_1)(x_3-y_1)+(x_1-y_1)(x_2-y_2)+(x_1-y_1)(x_1-y_3).\]
There are also three bumpless pipe dreams
\[
\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (.5,1)--(.5,3.5)--(4,3.5);
\draw[thick,rounded corners] (1.5,1)--(1.5,4.5)--(4,4.5);
\draw[thick,rounded corners] (2.5,1)--(2.5,1.5)--(4,1.5);
\draw[thick,rounded corners] (3.5,1)--(3.5,2.5)--(4,2.5);
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (.5,1)--(.5,2.5)--(2.5,2.5)--(2.5,3.5)--(4,3.5);
\draw[thick,rounded corners] (1.5,1)--(1.5,4.5)--(4,4.5);
\draw[thick,rounded corners] (2.5,1)--(2.5,1.5)--(4,1.5);
\draw[thick,rounded corners] (3.5,1)--(3.5,2.5)--(4,2.5);
\end{tikzpicture}
\hspace{3em}
\begin{tikzpicture}[x=0.8em,y=0.8em]
\draw[step=1,gray, thin] (0,1) grid (4,5);
\draw[color=black, thick](0,1)rectangle(4,5);
\draw[thick,rounded corners] (.5,1)--(.5,3.5)--(4,3.5);
\draw[thick,rounded corners] (1.5,1)--(1.5,2.5)--(2.5,2.5)--(2.5,4.5)--(4,4.5);
\draw[thick,rounded corners] (2.5,1)--(2.5,1.5)--(4,1.5);
\draw[thick,rounded corners] (3.5,1)--(3.5,2.5)--(4,2.5);
\end{tikzpicture}\]
for $w$. These give a presentation of the same double Schubert polynomial as
\[\mathfrak S_w= (x_1-y_1)(x_3-y_3)+(x_1-y_1)(x_2-y_1)+(x_1-y_1)(x_1-y_2).\]
Note that although these expressions are necessarily equal, this equality is only apparent after significant factoring and reorganizing. In particular, there is no weight-preserving way to match up the terms of the two summations.
\end{example}

In Lie-theoretic terms, one may identify $\flags_n$ with the homogeneous space $\GL_n(\mathbb{C}) /B$, where $B$ denotes the Borel subgroup of invertible upper triangular matrices.
Pulling back a Schubert cell in $\flags_n$ to $\GL_n(\mathbb{C})$, we may then consider its closure in the affine space of all $n \times n$ complex matrices.
Fulton \cite{Fulton} showed that these \emph{matrix Schubert varieties} are irreducible, gave set-theoretic defining equations for them, and showed that these equations define reduced schemes. 
The key observation of Knutson and Miller is that these \emph{Fulton generators} form a Gr\"obner basis under any \textit{antidiagonal} term order
(that is, any term order under which the initial term of each minor of a generic matrix is the product of the entries along its main antidiagonal).

It is at least as natural to consider the dual notion of \textit{diagonal} term orders (that is, term orders where initial terms of minors are products along main diagonals). For example, much of the commutative algebra literature on determinantal ideals and generalizations focuses on this case (e.g., \cite{Sturmfels, Gonciulea.Miller, Bruns.Conca}).
Indeed, Knutson and Miller first tried unsuccessfully to carry out their program in this context before they realized that the antidiagonal term orders were more amenable to their approach.

The geometry of diagonal degenerations, in fact, is more complicated than the antidiagonal case.
In general, the Fulton generators are \emph{not} a Gr\"obner basis with respect to diagonal term orders. In \cite{Knutson.Miller.Yong}, it was shown that Fulton generators are diagonal Gr\"obner exactly for the class of matrix Schubert varieties called \emph{vexillary}. For general matrix Schubert varieties, the diagonal Gr\"obner degenerations can even fail to be reduced. Moreover, in the nonreduced case, different diagonal term orders can yield distinct scheme structures on the limiting space of the degeneration.



In this paper, an extended abstract of \cite{Hamaker.Pechenik.Weigandt}, we return to the diagonal setting. Despite the additional geometric complication, we propose that diagonal Gr\"obner degenerations naturally give rise to bumpless pipe dreams in an exactly analogous fashion to how antidiagonal degenerations yield ordinary pipe dreams.
Our main conjecture is:
\begin{conjecture}
\label{c:main}
	Let $\init{X_w}$ be the  Gr\"obner degeneration of a matrix Schubert variety with respect to any diagonal term order.
	The irreducible components of $\init{X_w}$, counted with multiplicities, naturally correspond to the bumpless pipe dreams for the permutation $w$.
\end{conjecture}

In particular, Conjecture~\ref{c:main} implies that, although different choices of diagonal term orders may yield degenerations to distinct schemes, the reduced irreducible components of the degeneration and their multiplicities do not depend on such a choice.
The vexillary case of Conjecture~\ref{c:main} follows from \cite{Knutson.Miller.Yong} and results in~\cite{Weigandt}.
Our main result is to prove Conjecture~\ref{c:main} for a larger class of permutations, called \emph{banner permutations}, extending the vexillary case. 
For these permutations, we are able to exhibit explicit diagonal Gr\"obner bases by modifying the Fulton generators in an appropriate fashion.
\begin{theorem}
\label{t:main}
	If $w$ is a banner permutation, then the CDG generators for $X_w$ are a diagonal Gr\"obner basis. The irreducible components of $\init{X_w}$, counted with multiplicities, naturally correspond to the bumpless pipe dreams for the permutation $w$. (Precise definitions of \emph{banner permutations} and \emph{CDG generators} appear in Section~\ref{s:cdg}.)
\end{theorem}

The recursive arguments in \cite{Knutson.Miller} rely on developing the combinatorics of a new \emph{mitosis} recursion for ordinary pipe dreams. In contrast, bumpless pipe dreams appear well-adapted to the simpler and more classical \emph{transition} formula of Lascoux and Sch\"utzenberger \cite{Lascoux.Schutzenberger:tree}. Our proof of Theorem~\ref{t:main} relies heavily on this latter recursion. Recently, Knutson \cite{Knutson:cotransition} has developed a dual notion of \emph{cotransition}, which can be used to simplify antidiagonal arguments of \cite{Knutson.Miller} in a similar fashion to the arguments here~\cite[Section 4]{marberg2020gr}.


We believe Theorem~\ref{t:main} holds in more generality than proved in this paper (see Conjecture~\ref{conj:patterns}) and we hope that Theorem~\ref{t:main} can be extended using similar techniques. However, we do not know a description of diagonal Gr\"obner bases in the most general case. Indeed, since different choices of diagonal term order can lead to different initial ideals, it is not guaranteed that there exists an explicit uniform description of Gr\"obner bases for all diagonal orders. Nonetheless, Conjecture~\ref{c:main} is supported by calculations in such cases. 
By computer, we have systematically verified Conjecture~\ref{c:main} through the symmetric group $S_7$ for one choice of diagonal term order, as well as in a variety of other experiments for larger permutations and for other diagonal term orders.


%\ 
%
%\noindent \textbf{Organization:} 
%In Section~\ref{s:background}, we recall necessary background. In Section~\ref{s:cdg}, we introduce generators  for Schubert determinantal ideals, which we call the \emph{CDG generators}, a modification of the more standard Fulton generators. Section~\ref{s:block} introduces a block construction for partial permutations and develops its combinatorics. In Section~\ref{s:monomial}, we  introduce \emph{block predominant permutations}, establish a recurrence for certain monomial ideals constructed from CDG generators, and apply this recurrence to prove Theorem~\ref{t:main}. 

\section{Background}
\label{s:background}
%\subsection{Combinatorics of permutations}
\begin{wrapfigure}{r}{.15\textwidth} \centering \vspace{-1em}$
  \begin{tikzpicture}[x=0.8em,y=0.8em]
      \draw[color=black, thick](0,1)rectangle(5,6);
     \filldraw[color=black, fill=gray!30, thick](0,5)rectangle(1,6);
     \filldraw[color=black, fill=gray!30, thick](0,4)rectangle(1,5);
      \filldraw[color=black, fill=gray!30, thick](2,5)rectangle(3,6);
     \filldraw[color=black, fill=gray!30, thick](1,5)rectangle(2,6);
     \filldraw[color=black, fill=gray!30, thick](2,2)rectangle(3,3);
     \draw[thick,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick,ForestGreen] (5,4.5)--(1.5,4.5)--(1.5,1);
     \draw[thick,ForestGreen] (5,3.5)--(.5,3.5)--(.5,1);
     \draw[thick,ForestGreen] (5,2.5)--(4.5,2.5)--(4.5,1);
     \draw[thick,ForestGreen] (5,1.5)--(2.5,1.5)--(2.5,1);
     \filldraw [black](3.5,5.5)circle(.1);
     \filldraw [black](1.5,4.5)circle(.1);
     \filldraw [black](.5,3.5)circle(.1);
     \filldraw [black](4.5,2.5)circle(.1);
     \filldraw [black](2.5,1.5)circle(.1);
     \end{tikzpicture}.
$
\end{wrapfigure}
%Define  $[n] = \{1,2,\dots,n\}$ and let $S_n$ denote the symmetric group on $[n]$.
%Each permutation $w \in S_n$ is determined by its one-line notation $w_1 w_2 \dots w_n$ where $w_i = w(i)$.
The \textbf{Rothe diagram} of a permutation $w \in S_n$ is
$
D_w = \{(i,j) \in [n] \times [n]: w(i) > j, w^{-1}(j) > i\}.
$
We visualize $D_w$ by placing $\bullet$ in $(i,w(i))$ for each $i \in [n]$, then drawing lines below and to the right of each $\bullet$.
Then $D_w$ is the complement of the marked boxes.
For example, $D_{42153}$ is $\{(1,1),(1,2),(1,3),(2,1),(4,3)\}$, which can be visualized as shown.

The \textbf{essential set} $\Ess(w)$ of $w$ is the maximally southeast cells in each connected component of $D_w$.  
%For instance, $\Ess(42153)=\{(1,3),(2,1),(4,3)\}$.
The $i$th \textbf{row} of $D_w$ is
$
\{j\in [n]: (i,j) \in D_w\}.
$
%For example, $42153$ is not vexillary, as neither of rows $2$ and $4$ is contained in the other.
The \textbf{Lehmer code} of $w$ is $c(w) = (c_1,\dots,c_n)$ where $c_i$ is the cardinality of the $i$th row of $D_w$. 
We say $w$ is \textbf{dominant} if $c(w)$ is nonincreasing; $w$ is \textbf{vexillary} if its rows are totally ordered by inclusion. 
%For example, $c(42153) = (3,1,0,1,0)$, so $42153$ is not dominant.
To each $w$, we associate a \textbf{rank function} $r_w : [n] \times [n] \to \mathbb{Z}$, where 
$
r_w(i,j) = \# \{k \leq i : w(k) \leq j\}.
$
For $v,w \in S_n$, we say $v \leq w$ in \textbf{Bruhat order} if $r_v(i,j) \geq r_w(i,j)$ for all $i,j \in [n]$.
We write $\lessdot$ for the covering relation in Bruhat order.

A {\bf partial permutation} is a $0$--$1$ matrix with at most one $1$ in each row and column. The definitions of the above notions naturally extend to partial permutations.   Let $M_{m,n}$ denote the set of $m \times n$ matrices over $\mathbb{C}$ and define $M_n \coloneqq M_{n,n}$. An $m\times n$ partial permutation $w\in M_{m,n}$ can be (uniquely) completed to a permutation matrix $\widetilde{w}\in M_{\max{\{m,n\}}}$.  This completion respects diagrams and essential sets.

%\subsection{Matrix Schubert varieties}
Let $Z = (z_{ij})_{i \in [m], j\in [n]}$ be a matrix of distinct indeterminates and let $R = \mathbb{C}[Z]$.
We identify $M_{m,n}$ with the $mn$-dimensional affine space $\Spec R$.
For $A \in M_n$ and $I,J \subset [n]$, let $A_{I,J} = (a_{ij})_{i\in I, j\in J}$.
Then the \textbf{matrix Schubert variety} for $w \in S_n$ is the affine variety
$
X_w = \left\{A \in M_n: \rank{A_{[i],[j]}} \leq r_w(i,j)\ \mbox{for all}\ i,j \in [n]\right\},
$
and the \emph{Schubert determinantal ideal} $I_w$ is the prime ideal associated to $X_w$, so
$
X_w \cong \Spec R/I_w
$
as reduced schemes.
%Let 
%$
%I_w = \left\langle (r_w(i,j)+1)\text{-size minors in } Z_{[i],[j]} : i,j \in [n] \right\rangle
%$
%be the {\bf Schubert determinantal ideal.}
%It is easy to see that $X_w$ is the vanishing locus of $I_w$. 
Fulton \cite[Proposition~3.3]{Fulton} showed that $I_w$ is generated by $r_w(i,j)+1$-size minors in $Z_{[i],[j]}$.
In fact, he described a smaller generating set for $I_w$:
\begin{equation}\label{eq:fulton}
	I_w = \left\langle (r_w(i,j)+1)\text{-size minors in } Z_{[i],[j]} : (i,j) \in \Ess(w) \right\rangle.
\end{equation}
The minors in Equation~\eqref{eq:fulton} are called the \textbf{Fulton generators} of $I_w$.

For example, suppose $w = 42153$. Then the Fulton generators of $I_w$ are 
\begin{equation}\label{eq:Fulton_ex}
z_{11}, z_{12}, z_{13}, z_{21}, \begin{vmatrix}
z_{11} & z_{12} & z_{13} \\
z_{21} & z_{22} & z_{23} \\
z_{31} & z_{32} & z_{33}
\end{vmatrix},
\begin{vmatrix}
z_{11} & z_{12} & z_{13} \\
z_{21} & z_{22} & z_{23} \\
z_{41} & z_{42} & z_{43}
\end{vmatrix},
\begin{vmatrix}
z_{11} & z_{12} & z_{13} \\
z_{31} & z_{32} & z_{33} \\
z_{41} & z_{42} & z_{43}
\end{vmatrix},
\begin{vmatrix}
z_{21} & z_{22} & z_{23} \\
z_{31} & z_{32} & z_{33} \\
z_{41} & z_{42} & z_{43}
\end{vmatrix}.
\end{equation}


Following Equation~\eqref{eq:fulton}, we also define matrix Schubert varieties in $M_{m,n}$, indexed by partial permutations. 
% See \cite[Chapter~15]{miller2004combinatorial} for more details.

%\subsection{Bumpless pipe dreams}

Following \cite{Lam.Lee.Shimozono}, a {\bf bumpless pipe dream} is a tiling of the $n\times n$ grid with the six tiles
\begin{equation}
\label{eqn:pipes}
\raisebox{-.5em}{
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\draw[thick, rounded corners,ForestGreen] (.5,1)--(.5,1.5)--(1,1.5);
	\end{tikzpicture}
	\hspace{2em}
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\draw[thick, rounded corners,ForestGreen] (.5,2)--(.5,1.5)--(0,1.5);
	\end{tikzpicture}
	\hspace{2em}
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\draw[thick, rounded corners,ForestGreen] (0,1.5)--(1,1.5);
	\draw[thick, rounded corners,ForestGreen] (.5,1)--(.5,2);
	\end{tikzpicture}
	\hspace{2em}
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\end{tikzpicture}
	\hspace{2em}
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\draw[thick, rounded corners,ForestGreen] (0,1.5)--(1,1.5);
	\end{tikzpicture}
	\hspace{2em}
	\begin{tikzpicture}[x=1.5em, y=1.5em]
	\draw[color=black, thick](0,1)rectangle(1,2);
	\draw[thick, rounded corners,ForestGreen] (.5,1)--(.5,2);
	\end{tikzpicture}}
\end{equation}
so that there are $n$ pipes which
start at the right edge of the grid,
	end at the bottom of the grid, and
	 pairwise cross at most once.

If $P$ is a bumpless pipe dream, we define a permutation $w$ by setting $w(i)$ to be the column where pipe $i$ exits (labeling rows top to bottom).
Write $\bpd{w}$ for the set of bumpless pipe dreams for $w$.  The ${\bf diagram}$ of $P$ is $D(P):=\{(i,j): (i,j) \, \text{is a blank tile in} \, P\}.$ Each bumpless pipe dream has weight
$\displaystyle \wt(P)=\prod_{(i,j)\in D(P)}(x_i-y_j)$.
The double Schubert polynomial $\mathfrak S_w(\mathbf x;\mathbf y)$ can be expressed as a sum over bumpless pipe dreams:
\begin{theorem}[{\cite[Theorem 5.13]{Lam.Lee.Shimozono}}]
	\label{thm:BPDformula}
$ \mathfrak S_w(\mathbf x;\mathbf y)=\sum_{P\in \bpd{w}} \wt(P).$
\end{theorem}

We take this theorem to be the definition of the double Schubert polynomial; the single Schubert polynomial is obtained from this by setting all $y$ variables to $0$.
For example, the bumpless pipe dreams for $w = 42153$ are 
\begin{equation}\label{eq:bpd_ex}
  \begin{tikzpicture}[x=0.8em, y=0.8em]
  \draw[step=1, gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick, rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,4.5)--(1.5,4.5)--(1.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,3.5)--(.5,3.5)--(.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,2.5)--(4.5,2.5)--(4.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,1.5)--(2.5,1.5)--(2.5,1);
     \end{tikzpicture}
\hspace{2em}
  \begin{tikzpicture}[x=0.8em, y=0.8em]
    \draw[step=1, gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick, rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,4.5)--(1.5,4.5)--(1.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,3.5)--(2.5,3.5)--(2.5,2.5)--(.5,2.5)--(.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,2.5)--(4.5,2.5)--(4.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,1.5)--(2.5,1.5)--(2.5,1);
     \end{tikzpicture}
\hspace{2em}
  \begin{tikzpicture}[x=0.8em, y=0.8em]
    \draw[step=1, gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick, rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
   
     \draw[thick, rounded corners,ForestGreen] (5,3.5)--(.5,3.5)--(.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,2.5)--(4.5,2.5)--(4.5,1);
     \draw[thick, rounded corners,ForestGreen] (5,1.5)--(2.5,1.5)--(2.5,1);
  \draw[thick, rounded corners,ForestGreen] (5,4.5)--(2.5,4.5)--(2.5,2.5)--(1.5,2.5)--(1.5,1);
     \end{tikzpicture}.
\end{equation}
Hence, 
\scalebox{0.92}{$
\Schub_{42153}(\mathbf x;\mathbf y) = (x_1 - y_1)(x_1 - y_2) (x_1-y_3)(x_2-y_1)\bigg(  (x_4-y_3) + (x_3-y_1) + (x_2-y_2)  \bigg).
$}


%\subsection{Local moves on bumpless pipe dreams}

%The {\bf Rothe bumpless pipe dream} of $w$ is the (unique) bumpless pipe dream $P_w$ that has a $\begin{tikzpicture}[scale=.5][x=1.5em, y=1.5em]
%	\draw[color=black, thick](0,1)rectangle(1,2);
%	\draw[thick, rounded corners,ForestGreen] (.5,1)--(.5,1.5)--(1,1.5);
%	\end{tikzpicture}$ tile in position $(i, w(i))$ for all $i$ and contains no $\begin{tikzpicture}[scale=.5][x=1.5em, y=1.5em]
%	\draw[color=black, thick](0,1)rectangle(1,2);
%	\draw[thick, rounded corners,ForestGreen] (.5,2)--(.5,1.5)--(0,1.5);
%	\end{tikzpicture}$ tiles. 
%	It is the only bumpless pipe dream $P \in \bpd w$ satisfying $D(P) = D_w$.
%For example, the first bumpless pipe dream in \eqref{eq:bpd_ex} is the Rothe bumpless pipe dream of $42153$.

We also need to consider bumpless pipe dreams for partial permutations.  Let $w\in M_{m,n}$ be a partial permutation and $\widetilde{w}$ its completion to a permutation. We define 
${\sf BPD}(w)=\{P\mid_{m\times n}:P\in {\sf BPD}(\widetilde{w})\},$ where $P\mid_{m\times n}$ denotes the restriction of $P$ to its first $m$ rows and $n$ columns.


%\subsection{The transition formula}
(Double) Schubert polynomials satisfy a recurrence called \textbf{transition}.
Let $t_{ij}$ be the transposition $(i\; j) \in S_n$.
For $v \in S_n$ and $r \in [n]$, we define
$
I(v,r) = \{i < r: v \lessdot vt_{ir}\}$ and $\Phi(v,r) = \{ vt_{ir} : i \in I(v,r)\}.
$
An {\bf inversion} in $w \in S_n$ is a pair $(i,j)$ such that $i < j$ and $w(i) > w(j)$. {\bf Lexicographic order} on inversions of $w$ is given by $(i_1, j_1) > (i_2, j_2)$ if $i_1 > i_2$ or if $i_1 = i_2$ and $j_1 > j_2$.

\begin{theorem}[Equivariant Transition, {\cite[Proposition 4.1]{KohnertVeigneau}}]%\footnote{We believe this result was known by experts prior, but we are unaware of  any explicit earlier reference in the literature. The ordinary cohomology case appeared first in~\cite{Lascoux.Schutzenberger:tree}.}]
	\label{t:LS_tree}
	
	Let $w \in S_n$ with lexicographically largest inversion $(r,w^{-1}(s))$ and let $v \coloneqq  w t_{rw^{-1}(s)}.$
	Then $v \lessdot w$ and 
	\[
	\mathfrak{S}_w = (x_r - y_s)\mathfrak{S}_v  + \sum_{u \in \Phi(v,r)} \mathfrak{S}_u.
	\]
\end{theorem}


%This result is a straightforward consequence of the equivariant Monk's rule, which determines the equivariant cohomology of $\flags_n$.
The combinatorics of bumpless pipe dreams is compatible with transition.


\begin{lemma}
	\label{lemma:diagramtransition}
	There is a bijection  $\Psi:\bpd{v}\cup \bigcup_{u\in \Phi(v,r)} \bpd{u}\rightarrow \bpd{w}$ so that
\[D(\Psi(P))=\begin{cases}
D(P)\cup\{(r,s)\}& \text{if}\, P\in \bpd{v} \, \text{and}\\
D(P) &\text{otherwise}.
\end{cases}\]
\end{lemma}
%\begin{proof}
%This follows by restricting the bijection in \cite[Proposition~5.2]{Weigandt} to reduced bumpless pipe dreams.
%\end{proof}


Continuing our running example $w = 42153$, the lexicographically largest inversion is $(r,w^{-1}(s)) = (4,5)$, so we have $v = w t_{45} = 42135$. Since 
$
\Phi(v,4) = \{ u^{(1)} = 43125, u^{(2)} = 42315 \},
$
Lemma~\ref{lemma:diagramtransition} claims a bijection between $\bpd{w}$ and the unions of $\bpd{u^{(1)}}$, $\bpd{u^{(2)}}$, and $\bpd{v}$. Indeed, in this case, each of these three permutations $u^{(1)}, u^{(2)}, v$ is dominant and has a unique bumpless pipe dream:

\begin{equation}\label{eq:trans_ex}
 v \colon 
  \begin{tikzpicture}[x=0.8em,y=0.8em]
  \draw[step=1,gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick,rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,4.5)--(1.5,4.5)--(1.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,3.5)--(.5,3.5)--(.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,2.5)--(2.5,2.5)--(2.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,1.5)--(4.5,1.5)--(4.5,1);
     \end{tikzpicture}
\hspace{2em}
u^{(1)} \colon
  \begin{tikzpicture}[x=0.8em,y=0.8em]
    \draw[step=1,gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick,rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,4.5)--(2.5,4.5)--(2.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,3.5)--(.5,3.5)--(.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,2.5)--(1.5,2.5)--(1.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,1.5)--(4.5,1.5)--(4.5,1);
     \end{tikzpicture}
\hspace{2em}
u^{(2)} \colon
  \begin{tikzpicture}[x=0.8em,y=0.8em]
    \draw[step=1,gray, thin] (0,1) grid (5,6);
      \draw[color=black, thick](0,1)rectangle(5,6);
     \draw[thick,rounded corners,ForestGreen] (5,5.5)--(3.5,5.5)--(3.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,4.5)--(1.5,4.5)--(1.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,3.5)--(2.5,3.5)--(2.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,2.5)--(.5,2.5)--(0.5,1);
     \draw[thick,rounded corners,ForestGreen] (5,1.5)--(4.5,1.5)--(4.5,1);
     \end{tikzpicture}.
\end{equation}
The diagram of the first bumpless pipe dream of \eqref{eq:bpd_ex} consists of the diagram of the bumpless pipe dream for $v$ together with $(r,s)=(4,3)$. The diagram of the second bumpless pipe dream of \eqref{eq:bpd_ex} is that of $u^{(2)}$; the diagram of the third is that of $u^{(1)}$.

We use a diagrammatic interpretation of transition, described in~\cite[Section 2]{Knutson.Yong}.
The \textbf{maximal corner} of $w$ is the lexicographically maximal cell $(r,s)$ in $D_w$.
Amongst the $\bullet$'s in $D_w$ that are northwest of the maximal corner, we call the maximally southeast ones \textbf{pivots}.
For $(i,j)$ a pivot of $w$, the \textbf{marching operation} is a two-step procedure on $D_w$.
First remove the lines emanating from the $\bullet$ at $(i,j)$.
Next, for every cell in $D_w$ in the rectangle with corners $(i,j)$ and $(r,s)$, move that cell strictly to the northwest in the unique way such that each cell fills a position vacated either by the removed lines or by another cell.
The resulting diagram  is $D_u$ for some $u \in S_n$, and we say $w \xrightarrow{i} u$.
\begin{lemma}[Implicit in \cite{Knutson.Yong}]
\label{l:marching}
If $w \in S_n$ with maximal corner $(r,s)$ and $v = wt_{rw^{-1}(s)}$,
then the pivots of $w$ are $\{(i,w(i)): i \in I(v,r)\}$ and
$
\Phi(v,r) = \{u^{(i)}: w \xrightarrow{i} u^{(i)} \ \mbox{for}\ i \in I(v,r)\}.
$
\end{lemma}

%\subsection{Gr\"obner bases}
Recall $R=\mathbb C[Z]$.  A {\bf monomial order} is a linear ordering on monomials in $R$ such that, for any monomials ${\sf m}$, ${\sf n},$ and ${\sf p}$, we have
${\sf m}<{\sf n}$ if and only if ${\sf m p}<{\sf np}$; and
 ${\sf m}\leq{\sf mp}$.  
An {\bf (anti)diagonal} term order on R is a monomial order so that the initial term of any minor of $Z$ is the product of the entries on its main (anti)diagonal.  
Fix a monomial order on $R$.  
Given $f\in R$ its {\bf initial term} $\init{f}$ is the term whose monomial is largest with respect to the order.  
For a set of polynomials $F$, we define $\init{F}=\{\init{f}:f\in F\}$.  
If $F$ is an ideal, then $\init{F}$ is called the {\bf initial ideal} of $F$.  If $X=\Spec(R/I)$, the {\bf initial scheme} $\init{X}$ is $\Spec(R/\init{I})$.
A {\bf Gr\"obner basis} of an ideal $I$ is a subset $G$ such that $\langle \init{G} \rangle=\init{I}$.  
Every ideal $I\subseteq R$ admits a finite Gr\"obner basis;
if $G$ is a Gr\"obner basis for $I$, then $I=\langle G\rangle$.  

%A Gr\"obner basis for a diagonal (resp.\ antidiagonal) term order is called a {\bf diagonal} (resp.\ {\bf antidiagonal}) Gr\"obner basis.  A subset $G$ of an ideal $I$ is a {\bf universal} Gr\"obner basis if it is a Gr\"obner basis for $I$ with respect to all monomial orders.



%\subsection{Equivariant cohomology}

We need some basic notions of equivariant cohomology. 
Consider the torus $T \subset \GL_n(\mathbb{C})$ of invertible diagonal matrices and its Lie algebra $\mathfrak{t}$ of diagonal matrices. There is a natural left action of $T \times T$ on $\Spec R$ given by scaling rows and columns separately:
$
(t, \tau) \cdot M = t M \tau^{-1}.
$
Now, $\Spec R$ has a $(T \times T)$-equivariant cohomology ring $H_{T \times T}(\Spec R)$. Since $\Spec R$ is contractible, we have by definition that 
\[
H_{T \times T}(\Spec R) \cong H_{T \times T}({\rm pt}) \cong \mathcal{O}(\mathfrak{t} \otimes \mathfrak{t}) \cong \mathbb{Z}[x_1, \dots, x_n, y_1, \dots, y_n].
\]
Every setwise-stable subscheme $X \subseteq \Spec R$ has an equivariant class $[X]_{T \times T}$, which we may thereby identify with an integral polynomial in $2n$ variables. For $\mathcal B \subset [n] \times [n]$, let $C_ \mathcal B$ be the coordinate subspace $\Spec(R / \langle z_{ij} : (i,j) \notin  \mathcal B \rangle)$. 

We need only the following three properties of equivariant classes in $H_{T \times T}(\Spec R)$:

\noindent \textbf{Normalization:} For any coordinate subspace $C_{\mathcal{B}}$, we have
$
[C_{\mathcal B}]_{T \times T}=\prod_{(i,j)\in \mathcal B} (x_i-y_j).
$
\noindent \textbf{Additivity:}
For any $X \subseteq \Spec R$,
$
[X]_{T \times T}=\sum_{j} {\rm mult}_{X_j}(X)\,[X_j]_{T \times T},
$
where the sum is over the top-dimensional components of $X$ and ${\rm mult}_Y( X)$ denotes the multiplicity of $X$ along the reduced irreducible variety $Y$.
In particular, if $X=\bigcup_{i=1}^m X_i$, is a reduced scheme, then
$
[X]_{T \times T}=\sum_{j}\ [X_j]_{T \times T},
$
summing over components $X_j$ with $\dim X_j = \dim X$.

\noindent \textbf{Degeneration:}
If $\init{X}$ is a Gr\"obner degeneration of $X$, then 
$
[X]_{T \times T}= [\init{X}]_{T \times T}.
$


For any $X$ and any term order, $\init{X}$ is cut out of $\Spec R$ by a monomial ideal. Hence, $\init{X}$ is a (schemy) union of coordinate subspaces, and its equivariant class may be computed by Additivity and Normalization. Thus, the equivariant class of any $X \subseteq \Spec R$ may be computed from these properties, given enough information about $\init{X}$.
One of the key results of \cite{Knutson.Miller} is the following.

\begin{theorem}[{\cite[Theorem~A]{Knutson.Miller}}]
\label{t:KM}
	The matrix Schubert variety $X_w$ satisfies
	\[
	[X_w]_{T \times T}=\Schub_w(\xx;\mathbf{y}).
	\]
\end{theorem}


\section{CDG generators, skew sums, and monomial ideals}
\label{s:cdg}

For $\lambda$ an integer partition, let $Z^\lambda$ be the matrix obtained from $Z=(z_{ij})$ by specializing $z_{ij}$ to $0$ for all $(i,j)$ in the Young diagram of $\lambda$. 
The {\bf dominant part} of $D_w$ is
$
{\rm Dom}(w)=\{(i,j)\in D_w:r_w(i,j)=0\}.
$
 The cells of ${\rm Dom}(w)$ make up the Young diagram of a partition $\lambda$.
Define $\Esshat(w) \coloneqq \Ess(w) - \Dom(w)$.
For example, with $w = 42153$ we have ${\rm Dom}(w)  = \{ (1,1), (1,2), (1,3), (2,1) \} = $ the partition $(3,1)$.  Furthermore,   $\Ess'(w) = \{(4,3)\}$ and
\[
Z^{(3,1)}_{[4],[3]} = \begin{bmatrix}
	0 & 0 & 0 \\
	0 & z_{22} & z_{23} \\
	z_{31} & z_{32} & z_{33} \\
	z_{41} & z_{42} & z_{43}  
\end{bmatrix}.
\]


Let 
$
G'_w = \bigcup_{(i,j)\in \Esshat(w)} \left\{ \text{minors of size $r_w(i,j)+1$ in } Z^{{\rm \Dom}(w)}_{[i],[j]} \right\}.
$
Then $I_w$ is generated by 
$
G_w = G'_w \cup \{ z_{ij} : (i,j) \in \Dom(w) \}. 
$
We call this set $G_w$ of polynomials the {\bf CDG generators} of $I_w$ (after the authors of \cite{Conca.DeNegri.Gorla} who studied something similar). 
We are interested in when $G_w$ is a diagonal Gr\"obner basis for $I_w$; in this case, we say that $w$ and $I_w$ are {\bf CDG}. 
Note that if $w$ is CDG, then $\init{I_w}$ is reduced, since the initial terms of the polynomials in $G_w$ are all squarefree.


\begin{example}
Let $w = 42153$. Then the CDG generators of $I_w$ are
\begin{equation}\label{eq:CDG_ex}
z_{11}, z_{12}, z_{13}, z_{21},  z_{22}z_{33}z_{41} + z_{23}z_{31}z_{42} - z_{22}z_{31}z_{43} - z_{23}z_{32}z_{41}.
\end{equation}
Note that this is a smaller set than the Fulton generators from~\eqref{eq:Fulton_ex}.
\end{example}


In $S_4$, all permutations are CDG; in $S_5$, $13254$ and $21543$ are the only permutations which are not CDG.  Note that ${\rm Dom}(13254)=\emptyset$, so the CDG generators are simply the Fulton generators in this case.   

%We now observe a special class of permutations that are CDG, and indeed whose CDG generators are universal Gr\"obner. 

%\begin{proposition}
%\label{prop:universalgb}
%Fix $w\in  S_n$ so that there is a unique $(e_1,e_2)\in \Esshat(w)$.  
%Furthermore, assume that $r_w(e_1,e_2)=\min\{e_1,e_2\}-1$.  
%Then $G_w$ is a universal Gr\"obner basis for the Schubert determinantal ideal $I_w$.
%\end{proposition}
%
%
%\begin{example}
%Continuing our running example, let $w=42153$.  Since $|\Esshat(w)|=\{(4,3)\}$ and $r_w(4,3)=2$, the generators in Equation~(\ref{eq:CDG_ex}) are a universal Gr\"obner basis for $I_w$ by Proposition~\ref{prop:universalgb}.
%\end{example}

Given two diagrams $D_1 \subseteq [a] \times [b]$ and $D_2 \subseteq [c] \times [d]$, let $\square = [c]\times [b]$ and define
\[
D_1 \boxplus D_2 = %\ytableausetup{boxsize=2.9em}
\begin{ytableau}
\square  & D_2\\
 D_1
 \end{ytableau}.
\]
\begin{lemma}
\label{l:block}
For $u$ and $v$ partial permutations,
there exists $w \in S_\infty$ with $D(w) = D(u) \boxplus D(v).$
%\[
%D(w) = \ytableausetup{boxsize=2.9em}
%\begin{ytableau}
%\text{\tiny{all boxes}}  & D(v)\\
% D(u)
% \end{ytableau}.
%\]
\end{lemma}


Given partial permutations $u, v$ with $n, m$ rows respectively, the {\bf skew sum} $u \ominus v$ of $u$ and $v$ is the unique partial permutation with $n+m$ rows constructed as in Lemma~\ref{l:block}.
When $u$ and $v$ are permutations, $u \ominus v$ is the usual skew sum.
%For $u, v$ partial permutations, we can understand many  properties of $u \ominus v$ in terms of $u$ and $v$.
%Skew sums interact naturally with bumpless pipe dreams.

\begin{lemma}
	\label{l:block_bpd}
	For $u,v$ partial permutations and $w = u \ominus v$, there is a bijection from $\bpd u \times \bpd v$ to $\bpd w$ mapping the pair $(B_u,B_v)$ to $B_w$ satisfying $D(B_w) = D(B_u) \boxplus D(B_v)$.
%	\[
%D(B_w) = \ytableausetup{boxsize=3.0em}
%\begin{ytableau}
%\text{\tiny{all boxes}}  & D(B_v)\\
% D(B_u)
% \end{ytableau}.
%\]
\end{lemma}


For $z_{ij}$ an indeterminate, let 
\[\downshift_a(z_{ij}) :=\begin{cases} z_{i+a\ j} & \text{if} \enspace i+a\leq m \\
 0 &\text{otherwise}
\end{cases}
\quad \text{and} \quad 
\rightshift_b(z_{ij}) :=\begin{cases} z_{i\ j+b} & \text{if} \enspace j+b\leq n \\
 0 &\text{otherwise.}
\end{cases}
\]
Extend these operators to act indeterminate-by-indeterminate on monomials, linearly on polynomials and pointwise on sets of polynomials. 
From now on, all term orders are assumed to be diagonal, unless otherwise specified.

\begin{lemma}
\label{l:block_grobner}
	Let $u$ and $v$ be partial permutations such that $u$ has $b$ columns and $v$ has $a$ rows.
	If $F_u$ and $F_v$ are Gr\"obner bases of the Schubert determinantal ideals $I_u$ and $I_v$, then 	
$
	\downshift_a(F_u)\ \cup\ \quad {\rightshift_b(F_v)} \cup\ \{z_{ij}: 1 \leq i \leq a,\ 1 \leq j \leq b\}
$
is a Gr\"obner basis for $I_{u \ominus v}$.
\end{lemma}




\begin{corollary}
\label{c:block_cdg}
	Let $u$ and $v$ be CDG partial permutations.
	Then $u \ominus v$ is CDG.
\end{corollary}


%\section{Monomial ideal recurrences for block predominant permutations}
%\label{s:monomial}

We say that a partial permutation $w$ is {\bf predominant} if there is a partition $\lambda = (\lambda_1,\dots,\lambda_k)$ so that
$
c(w) = (\lambda_1,\dots,\lambda_k,0,\dots,0,\ell,0,\dots) = \lambda 0^h \ell,
$
for some $h, k, \ell \in \mathbb{Z}_{\geq 0}$. 
Note that we allow $h$ to be zero, so $\ell$ can immediately follow $\lambda_k$, even if $\lambda_k < \ell$.  We say a partial permutation is {\bf copredominant} if it is the transpose of a predominant permutation.
A partial permutation $w$ is {\bf block predominant} if it is a finite skew sum of predominant partial permutations. A predominant permutation is \textbf{indecomposable} if it cannot be written as the permutation associated to a skew sum of predominant partial permutations.  A predominant partial permutation is \textbf{indecomposable} if its associated permutation is indecomposable.
Note that only identity permutations are simultaneously dominant and indecomposable.
The class of block predominant permutations is closed under transition:
\begin{corollary}
\label{cor:predom_transition}
Let $\pi$ be block predominant with maximal corner $(r,s)$ and $\Phi(\pi t_{r\pi^{-1}(s)},r) = \{\tau^{(1)},\dots,\tau^{(m)}\}$.
Then each $\tau^{(i)}$ is block predominant.
\end{corollary}


%For $w$ a permutation, recall that $G_w$ is the set of CDG generators for $I_w$.
We define the monomial ideal
$
J_w = \langle \init g: g \in G_w \rangle.
$
Note that, by definition, $G_w$ is a Gr\"obner basis for $I_w$ if and only if $J_w = \init {I_w}$.
Similarly, let $J^\lambda_{ij}$ be the ideal generated by initial terms of non-zero maximal minors in the matrix $Z^\lambda_{[i],[j]}$.
By construction,
\begin{equation}
\label{eq:fake_initial}
J_w = \langle z_{ij}: (i,j) \in \Dom(w)\rangle + \sum_{(i,j) \in \Ess'(w)} J^{{\rm Dom}(w)}_{ij}.
\end{equation}


We will show that transition gives a recurrence on the monomial ideals $J_w$ for block predominant permutations.
%To do so, we first prove some technical lemmas about the ideals $J_w$.
Consulting Example~\ref{ex:transition} may help clarify this result.


%\begin{lemma}
%\label{lem:containment}
%For $w$ a block predominant permutation, $J_v \subseteq J_w$ and $J_v \subseteq J_{u^{(i)}}$ for each $i$.
%\end{lemma}
%
%
%
%\begin{lemma}
%\label{lem:good_ideal}
%
%For $w$ a block predominant permutation, we have $J_w/J_v = z_{rs} J^\lambda_{r{-1}\ s{-1}}/J_v$.
%
%\end{lemma}

%We now establish the key recurrence on the monomial ideals $J_w$.

\begin{theorem}
\label{thm:predom_transition}
If $w$ is a block predominant permutation, then
$
J_w = (J_v + \langle z_{rs} \rangle) \cap \left( \bigcap J_{u^{(i)}} \right).
$
\end{theorem}



\begin{example}
\label{ex:transition}
Let $w = 67341\ 10\ 2589$.
Transition gives
$
v= 673419258$ and $\Phi(w,6) = \{u^{(1)} = 693417258, u^{(2)} = 673914258, u^{(3)} = 673491258\}
$
 with diagrams
 \[
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\hspace{1em}
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\draw (-2,4) node  [align=left] {$D_v=$};
     \end{tikzpicture}
\hspace{1em}
   \begin{tikzpicture}[x=0.8em,y=0.8em]
      \draw[color=black, thick](0,1)rectangle(9,7);
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     \filldraw[color=black, fill=gray!30, thick](0,3)rectangle(1,4);
     \filldraw[color=black, fill=gray!30, thick](1,6)rectangle(2,7);
	\filldraw[color=black, fill=gray!30, thick](1,5)rectangle(2,6);
	\filldraw[color=black, fill=gray!30, thick](1,4)rectangle(2,5);
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     \end{tikzpicture}\]
\[
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	\filldraw[color=black, fill=gray!30, thick](3,5)rectangle(4,6);
	\filldraw[color=black, fill=gray!30, thick](4,6)rectangle(5,7);
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	\filldraw[color=black, fill=gray!30, thick](3,3)rectangle(4,4);

     \draw[thick,ForestGreen] (9,6.5)--(5.5,6.5)--(5.5,1);
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\hspace{1em}
\begin{tikzpicture}[x=0.8em,y=0.8em]
      \draw[color=black, thick](0,1)rectangle(9,7);
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	\filldraw[color=black, fill=gray!30, thick](1,5)rectangle(2,6);
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	\filldraw[color=black, fill=gray!30, thick](1,3)rectangle(2,4);
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	\filldraw[color=black, fill=gray!30, thick](3,6)rectangle(4,7);
	\filldraw[color=black, fill=gray!30, thick](3,5)rectangle(4,6);
	\filldraw[color=black, fill=gray!30, thick](4,6)rectangle(5,7);
	\filldraw[color=black, fill=gray!30, thick](4,5)rectangle(5,6);
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	\filldraw[color=black, fill=gray!30, thick](4,2)rectangle(5,3);
	\filldraw[color=black, fill=gray!30, thick](7,2)rectangle(8,3);
	\filldraw[color=black, fill=gray!30, thick](0,2)rectangle(1,3);

     \draw[thick,ForestGreen] (9,6.5)--(5.5,6.5)--(5.5,1);
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     \filldraw [black](2.5,4.5)circle(.1);
     \filldraw [black](.5,1.5)circle(.1);
 \filldraw [black](8.5,2.5)circle(.1);
     \filldraw [black](3.5,3.5)circle(.1);
\draw (-2.25,4) node  [align=left] {$D_{u^{(3)}}=$};
     \end{tikzpicture}.\]
     
The dominant component in each is $\lambda = (5^2,2^2,0^1)$ except that ${\rm Dom}(u^{(3)}) = \lambda' = (5^2,2^3)$.
The monomials coming from minors corresponding to the essential boxes $(6,2)$ and $(6,5)$ are in $J_v$, as do those corresponding to $(6,9)$ that do not involve $z_{69}$.
A lemma guarantees that   $J_v \subseteq J_w,J_{u^{(1)}},J_{u^{(2)}},J_{u^{(3)}}$.
Therefore, Theorem~\ref{thm:predom_transition} follows in this case by showing $J_w/J_v = \left(\langle z_{69}\rangle  \cap J_{u^{(1)}} \cap J_{u^{(2)}} \cap J_{u^{(3)}}\right)/J_v$.
Recall $J^\lambda_{ij}$ is the ideal generated by initial terms of non-zero maximal minors in the matrix $Z^\lambda_{[i],[j]}$.

A direct argument shows $J_w / J_v = z_{69} J^{\lambda}_{58}  / J_v$ and $J_{u^{(1)}}/J_v = J^\lambda_{28}/J_v$.
We claim
$(J_{u^{(1)}} \cap J_{u^{(2)}})/J_v = J^\lambda_{48}$.
Observe that $J^\lambda_{48} \subseteq J_{u^{(2)}}$ by Equation~\eqref{eq:fake_initial}.
The opposite containment follows since the minors coming from $(4,5) \in \Ess(u^{(2)})$ correspond to $J^\lambda_{45}$, while $(J^\lambda_{45} \cap J^\lambda_{28}) \subseteq J^\lambda_{48}$.
Next, we show $(J_{u^{(1)}} \cap J_{u^{(2)}} \cap J_{u^{(3)}})/J_v = J^\lambda_{58}/J_v$.
To show the forward containment, we study $(5,2),(5,5),(5,8) \in \Ess(u^{(3)})$ individually.
We have
$
J^\lambda_{52} \cap J^\lambda_{48}, J^{\lambda'}_{55} \cap J^\lambda_{28}, J^{\lambda'}_{58} \subseteq J^\lambda_{58}.
$
Moreover, the only monomials generating $J^\lambda_{58}$ not found in $J^{\lambda'}_{58}$ are those involving $z_{51}$ and $z_{52}$, so we see $J^\lambda_{58} \subseteq J^{\lambda'}_{58} + J^\lambda_{52} \cap J^\lambda_{48}$.
\end{example}


%Recall $R = \mathbb{C}[Z]$ where $Z = (z_{ij})_{i \in [m],j\in[n]}$.
Given a bumpless pipe dream $P$, write $\mathcal L_P=\langle z_{ij}:(i,j)\in D(P) \rangle.$ 

\begin{proposition}
\label{p:predominant}
	Suppose $w$ is a block predominant permutation.  Then 
	the CDG generators are a diagonal Gr\"obner basis for $I_w$, and
		$\displaystyle \init{I_w}=\bigcap_{P\in\bpd{w}}\mathcal L_P$.
\end{proposition}


A similar fact is true for vexillary permutations.

\begin{proposition}
\label{p:vexillary}
If $w$ is vexillary, then $w$ is CDG and 
$
 \init{I_w}=\bigcap_{P\in\bpd{w}}\mathcal L_P.
$
\end{proposition}



We say a permutation is \textbf{banner} if it is a skew sum of predominant, copredominant, and vexillary partial permutations.
The following theorem is our main result.

\begin{theorem}
\label{t:banner}
	Let $w$ be a banner permutation. Then
		$w$ is CDG, and 
		$\init {I_w} = \bigcap_{P \in \bpd w} \mathcal{L}_P.$ In particular, $\init {I_w}$ is radical.
\end{theorem}

Theorem~\ref{t:banner} follows from Propositions~\ref{p:predominant} and~\ref{p:vexillary} together with the following.

\begin{proposition}
\label{p:block_CDG}
If $w=u^{(1)}\ominus \cdots \ominus u^{(k)}$ is a skew sum of partial permutations and
 $\init{I_{u^{(i)}}}=\bigcap_{P\in \bpd{u^{(i)}}}\mathcal L_P$ for all $i\in [k]$, then $\init{I_{w}}=\bigcap_{P\in \bpd{w}}\mathcal L_P,$
and so $\init{I_w}$ is radical.
\end{proposition}



Theorem~\ref{t:banner} is a special case of Conjecture~\ref{c:main}, and provides further evidence for the general statement of the conjecture.
Theorem~\ref{t:banner} does not exhaust the set of all CDG permutations.
For example, $w=25143$ is not banner, but one can compute that its CDG generators are a diagonal Gr\"obner basis for $I_w$.
We conjecture the following characterization of CDG permutations.


\begin{conjecture}\label{conj:patterns}
Let $w$ be a permutation. The CDG generators are a diagonal Gr\"obner basis for $I_w$ if and only if $w$ avoids all eight of the patterns
$13254$, $21543$, $214635$, $215364$, $241635$, $315264$, $215634$, $4261735$.
In particular, this holds if $\Schub_w(\mathbf x)$ is a multiplicity-free sum of monomials. 
\end{conjecture}
Conjecture~\ref{conj:patterns} has now been proven by Klein \cite{Klein}.
If Conjecture~\ref{c:main} holds, then as a consequence, $\Spec R/\init{I_w}$ is reduced if and only if each $P\in\bpd{w}$ has a distinct diagram.  Data suggests that this condition is also governed by pattern containment \cite{Heck}.


\acknowledgements{We are very grateful for  helpful conversations with Patricia Klein, Allen Knutson, Jenna Raj\-chgot, David Speyer, Alex Woo, and Alexander Yong. We are also grateful to the referees.}


\printbibliography



\end{document}
