%% 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]{FPSAC2018}
\articlenumber{52}
%% 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{lipsum}



\usepackage{amsfonts,amsmath, latexsym, amssymb,epsfig}
%\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{graphicx}
\usepackage{pgf,tikz}
\usetikzlibrary{decorations.markings}
\usepackage{ytableau}
\usepackage[all]{xy}
\usepackage{wrapfig}
\usepackage{enumitem}
\newcommand{\comment}[1]{$\star${\sf\textbf{#1}}$\star$}
\renewcommand{\familydefault}{ppl}



\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{Claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem*{claim*}{Claim}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{Main Conjecture}[theorem]{Main Conjecture}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{Remark}[theorem]{Remark}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{plain}
\newtheorem{question}[theorem]{Question}
\newtheorem*{MainThm}{Main Theorem}
\newtheorem*{Fact}{Fact}
%% define your title in the usual way
\title[Prism Tableaux for Alternating Sign Matrix Varieties (extended
abstract)]{Prism Tableaux for Alternating Sign Matrix Varieties (extended
abstract)}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Anna Weigandt\thanks{\href{mailto:weigndt2@illinois.edu}{weigndt2@illinois.edu}}\addressmark{1} }%\and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Illinois at Urbana-Champaign}

%% 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{
A prism tableau is a set of reverse semistandard tableaux,  each  positioned within an ambient grid.  Prism tableaux were introduced in joint work with A.~Yong to provide a formula for the Schubert polynomials of A.~Lascoux and M.-P.~Sch\"utzenberger.  This formula directly generalizes the well known expression for Schur polynomials as a sum over semistandard tableaux.    Alternating sign matrix varieties generalize the matrix Schubert varieties of  W.~Fulton.  We use prism tableaux to give a  formula for the multidegree of an alternating sign matrix variety.  
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Schubert polynomials, Alternating sign matrices, Prism tableaux}

%% 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{52-Weigandt.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
\section{Introduction}



 An {\bf alternating sign matrix} (ASM) is a square matrix with entries in $\{-1,0,1\}$ so that 
the nonzero entries in each row and column sum to 1 and alternate in sign.
Let ${\sf ASM}(n)$ be the set of  $n\times n$ ASMs.  
The enumeration of ASMs has drawn much interest, the 
sequence for $n\geq 1$ being
\[ 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460,\ldots.\]
 There is a closed form expression for this sequence; the celebrated \emph{alternating sign matrix conjecture} of 
W.~H.~Mills--D.~P.~Robbins--H.~Rumsey \cite{mills1983alternating} asserts that
\[|{\sf ASM}(n)|=\prod_{j=0}^{n-1}\frac{(3j+1)!}{(n+j)!}.\]
The original proof  was given by D.~Zeilberger \cite{zeilberger1996proof}.  G.~Kuperberg  gave a second proof using  the six-vertex model of statistical mechanics \cite{kuperberg1996another}. See \emph{Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture}, by D.~Bressoud, for links between ASMs and hypergeometric series, plane partitions, and lattice paths \cite{bressoud1999proofs}.

  Fix $A=(a_{ij})_{i,j=1}^n\in{\sf ASM}(n)$.  The {\bf corner sum function} is
$\displaystyle r_A(i,j)=\sum_{k=1}^i\sum_{\ell =1}^ja_{k\ell}$.
 Corner sums  define a lattice structure on ${\sf ASM}(n)$; say
\begin{equation}
\label{eqn:asmposetdef}
A\leq B \enspace \text{if and only if} \enspace  r_A(i,j)\geq r_B(i,j) \enspace \text{for all} \enspace  1\leq i,j\leq n.
\end{equation}
Restricted to permutation matrices, (\ref{eqn:asmposetdef}) is the {\bf Bruhat order} on the symmetric group $\mathcal S_n$.  A.~Lascoux and M.-P.~Sch\"utzenberger showed that ${\sf ASM}(n)$ is the smallest lattice which contains  $\mathcal S_n$ as an order embedding  \cite[Lemma 5.4]{lascoux1996treillis}.  

A permutation $w$ is {\bf Grassmannian} if it has a unique descent, i.e.~a position $i$ so that $w(i)>w(i+1)$.  A permutation $u$ is {\bf biGrassmannian} if both $u$ and $u^{-1}$ are Grassmannian.   A.~Lascoux and M.-P.~Sch\"utzenberger showed that  biGrassmannian permutations  are the \emph{basic elements} of $\mathcal S_n$, and hence ${\sf ASM}(n)$.







 A {\bf partition} $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_k)$ is a weakly decreasing sequence of nonnegative integers. Each $\lambda_i$ is a {\bf part} of lambda.  The {\bf length} $\ell(\lambda)$ is the number of positive parts of $\lambda$.
Each partition $\lambda$ has an associated  (French) {\bf Young diagram} which consists of left justified boxes with $\lambda_1$ boxes in the bottom row, $\lambda_2$ in the next, and so on.    
%We will freely identify $\lambda$ with its Young diagram.

A {\bf reverse semistandard tableau} is a filling of the Young diagram of $\lambda$ with positive integers so that labels 
 weakly decrease within rows (from left to right) and 
 strictly decrease (from bottom to top) within columns.
  Write ${\tt RSSYT}(\lambda,d)$ for the set of reverse semistandard fillings of $\lambda$ which use labels from the set $[d]:=\{1,2,\ldots,d\}$.  
  


 Fix tuples  of partitions and positive integers
\begin{equation}
\label{eqn:tupledef}
\boldsymbol \lambda= (\lambda^{(1)},\ldots, \lambda^{(k)}) \enspace \text{and} \enspace  \mathbf d=(d_1,\ldots,d_k) \enspace \text{so that} \enspace  d_i\geq \ell(\lambda^{(i)}) \enspace \text{for all} \enspace  i.
\end{equation}  
%Prism tableaux were first defined in \cite{weigandt2015prism} to study Schubert polynomials. 
Let \[{\tt AllPrism}(\boldsymbol \lambda,\mathbf d)={\tt RSSYT}(\lambda^{(1)},d_1)\times \cdots \times {\tt RSSYT}(\lambda^{(k)},d_k).\]  An element of ${\tt AllPrism}(\boldsymbol \lambda,\mathbf d)$ is called a {\bf prism tableau}.   We associate to each $(\boldsymbol \lambda,\mathbf d)$ an ASM, denoted
 $A_{\boldsymbol \lambda,\mathbf d}$, which is the least upper bound of a list of Grassmannian permutations (see (\ref{def:asmassociated})).
Conversely,  for any $A\in {\sf ASM}(n)$, there exists some $(\boldsymbol \lambda,\mathbf d)$ so that $A=A_{\boldsymbol \lambda,\mathbf d}$.  See Section~\ref{section:combprism} for details.



For the discussion which follows, it is not enough to think of a prism tableau as a tuple of reverse semistandard tableaux.  Rather, we position each of the component tableaux  in the $\mathbb Z_{> 0}\times \mathbb Z_{> 0}$ grid.  We use matrix coordinates to refer boxes in  the grid; $(i,j)$ indicates the box in the $i$th row (from the top) and $j$th column (from the left) of the grid.  The $i$th {\bf antidiagonal} of $\mathbb Z_{> 0}\times \mathbb Z_{> 0}$ consists of the boxes \[\{(i,1),(i-1,2),\ldots, (1,i)\}.\]


We identify the shape of each $\lambda^{(i)}$ with 
\begin{equation}
\label{eqn:tableauposition}
\lambda^{(i)}=\{(a,b):b\leq \lambda^{(i)}_{d_i-a+1}\}\subseteq \mathbb Z_{>0}\times \mathbb Z_{>0}.
\end{equation}
The {\bf prism shape} for $(\boldsymbol \lambda,\mathbf d)$ is obtained by overlaying the $\lambda^{(i)}$'s: 
\begin{equation}
 \mathbb S(\boldsymbol \lambda,\mathbf d):=\bigcup_{i=1}^k \{(a,b):b\leq \lambda^{(i)}_{d_i-a+1}\}.\end{equation}
From this perspective, a  prism tableau  for $(\boldsymbol \lambda,\mathbf d)$ is a filling of $\mathbb S(\boldsymbol \lambda,\mathbf d)$ which assigns a label of color $i$ from  $[d_i]$ to each $(a,b)\in \lambda^{(i)}$ so that labels of color $i$
 weakly decrease along rows from left to right  and strictly decrease along columns from bottom to top.
Such fillings are  in immediate bijection with ${\tt AllPrism}(\boldsymbol \lambda,\mathbf d)$.
 


Weight $\mathcal T$ as follows:
\[{\tt wt}(\mathcal T)=\prod_{i=1}^\infty x_i^{n_i}\] where 
$n_i$ is the number of antidiagonals which contain the label $i$ (in any color).

\begin{example}
\label{example:prism}
Let $\boldsymbol \lambda=((1),(3,2),(2,1,1))$ and $\mathbf d=(2,5,6)$.  Below, we give an example of $\mathcal T\in {\tt AllPrism}(\boldsymbol \lambda,\mathbf d)$.
\ytableausetup{boxsize=.95em, aligntableaux=center}
\[
\mathcal T=\left(
\begin{ytableau}
1
\end{ytableau}
\, , \,
\begin{ytableau}
2&2\\
3&3&2
\end{ytableau}
\, , \, 
\begin{ytableau}
1\\
2\\
6&3
\end{ytableau}
 \right) 
\hspace{2em}
\longleftrightarrow
\hspace{2em}
\begin{tikzpicture}[x=1em,y=1em,baseline=4.2em]
\draw[step=1,gray!30,very thin] (0,1) grid (7,8); 
\node[] at (1.5,5){
\begin{ytableau}
\none\\
{\color{Rhodamine}1}\\
\none\\
{\color{Plum}2}{\color{blue}1}&{\color{Plum}2}\\
{\color{Plum}3}{\color{blue}2}&{\color{Plum}3}&{\color{Plum}2}\\
{\color{blue}6}&{\color{blue}3}\\
\end{ytableau}};
\end{tikzpicture}
\]
The corresponding weight monomial is ${\tt wt}(\mathcal T)=x_1^2x_2^3x_3^3x_6$.
\qed
\end{example}

Let 
${\tt deg}(\boldsymbol \lambda,\mathbf d)=\min\{{\tt deg}({\tt wt}(\mathcal T)):\mathcal T\in {\tt AllPrism}(\boldsymbol \lambda,\mathbf d)\}.
$  Say
$\mathcal T\in {\tt AllPrism}(\boldsymbol \lambda,\mathbf d)$ is {\bf minimal} if ${\tt deg}({\tt wt}(\mathcal T))={\tt deg}(\boldsymbol \lambda,\mathbf d).$
 Let $\ell_c$ be a label $\ell$ of color $c$. 
Labels $\{\ell_c,\ell_d, \ell'_e\}$
in the same antidiagonal form an {\bf unstable triple}
if $\ell<\ell'$ and replacing the $\ell_c$ with $\ell'_c$
gives a prism tableau.  For instance, in Example~\ref{example:prism}, there is an unstable triple in the fifth antidiagonal; the blue 2 may be replaced with a 3.
Write
\begin{equation}{\tt Prism}(\boldsymbol \lambda,\mathbf d)=\{\mathcal T\in {\tt AllPrism}(\boldsymbol \lambda,\mathbf d): \mathcal T \text{ is minimal and has no unstable triples}\}.
\end{equation}
Let
$\displaystyle \mathfrak A_{\boldsymbol \lambda,\mathbf d}=\sum_{\mathcal T\in {\tt Prism}(\boldsymbol \lambda,\mathbf d)}{\tt wt}(\mathcal T).
$
Call $\mathfrak A_{\boldsymbol \lambda,\mathbf d}$ an {\bf ASM polynomial}. 

If $\boldsymbol \lambda=(\lambda)$ and $\mathbf d=(d)$, the polynomial  $\mathfrak A_{\boldsymbol \lambda,\mathbf d}$ is the {\bf Schur polynomial} $s_{\lambda}(x_1,\ldots,x_d)$.  This follows immediately from the usual definition of $s_{\lambda}$ as a weighted sum over \emph{semistandard tableaux}.  The \emph{Schubert polynomials} $\{\mathfrak S_w:w\in \mathcal S_\infty\}$ of  A.~Lascoux and M.-P.~Sch\"utzenberger \cite{Lascoux.Schutzenberger} generalize Schur polynomials.  The purpose of  \cite{weigandt2018prism} was to provide a prism formula for Schubert polynomials.  We prove the following generalization.
\begin{theorem}
\label{theorem:schubertsum}
$\displaystyle \mathfrak A_{\boldsymbol \lambda,\mathbf d}=\sum_{w\in{\tt MinPerm}(A_{\boldsymbol \lambda,\mathbf d})}\mathfrak S_w$.
\end{theorem}
  Here, ${\tt MinPerm}(A)$ denotes the set permutations above $A$  which have the minimum possible  Coxeter length.  
Our proof of Theorem~\ref{theorem:schubertsum} is purely combinatorial; we give a bijection between ${\tt Prism}(\boldsymbol \lambda,\mathbf d)$ and the set of maximum dimensional facets in a union of the \emph{subword complexes} of \cite{knutson2004subword}.  The Schubert polynomial is a weighted sum over the facets of its corresponding subword complex \cite{bergeron1993rc,fomin1996yang,knutson2005grobner}.





 $\mathfrak A_{\boldsymbol \lambda,\mathbf d}$ also has a geometric interpretation; it is the \emph{multidegree} of an \emph{alternating sign matrix variety}.   
Write ${\sf Mat}(n)$ for the space of $n\times n$ matrices over an algebraically closed field $\Bbbk$.  Given $M\in {\sf Mat}(n)$, let  $M_{[i],[j]}$ be the submatrix of $M$ which consists of the first $i$ rows and $j$ columns of $M$.  
We define the {\bf alternating sign matrix variety} 
\begin{equation}
X_A:=\{M\in{\sf Mat}(n): {\rm rank}(M_{[i],[j]})\leq r_A(i,j) \enspace \text{for all} \enspace  1\leq i,j\leq n\}.
\end{equation}
If $w\in \mathcal S_n$, then $X_w$ is a {\bf matrix Schubert variety} as defined in \cite{fulton1992flags}.  

ASM varieties are stable under multiplication by the group of  invertible, diagonal matrices ${\sf T}\subset {\sf GL}(n)$.  There is a corresponding $\mathbb Z^{n}$ grading and multidegree
\[\mathcal C(X_A;\mathbf x)\in \mathbb Z[x_1,\ldots,x_n].\]  Whenever $w\in \mathcal S_n$,  we have $\mathfrak S_w=\mathcal C(X_w;\mathbf x)$.  This was shown in \cite{knutson2005grobner} and is equivalent to earlier statements in the language of equivariant cohomology  \cite{feher2003schur} and degeneracy loci \cite{fulton1992flags}.
We show $\mathfrak A_{\boldsymbol \lambda,\mathbf d}$ is the multidegree of the ASM variety $X_{A_{\boldsymbol \lambda,\mathbf d}}$.
\begin{theorem}
\label{theorem:main}
Fix $\boldsymbol \lambda$ and $\mathbf d$ as in (\ref{eqn:tupledef}).   
Then
$\displaystyle\mathcal C(X_{A_{\boldsymbol \lambda,\mathbf d}};\mathbf x)=\mathfrak A_{\boldsymbol \lambda,\mathbf d}.$
\end{theorem}
The irreducible components of $X_A$ are always matrix Schubert varieties.   Theorem~\ref{theorem:main} follows from Theorem~\ref{theorem:schubertsum} and the additivity of multidegrees.  

 We also discuss the explicit connection of prism tableaux to the Gr\"obner geometry of $X_A$.
Let $Z=(z_{ij})_{i,j=1}^n$ be the generic $n\times n$ matrix.  
Define the {\bf ASM ideal} by
\begin{equation}
\label{def:detIdeal}
I_A:=\langle  \text{minors of size} \enspace r_A(i,j)+1 \enspace \text{in} \enspace  Z_{[i],[j]}\rangle.
\end{equation}  It is immediate  that $I_A$ provides set-theoretic equations for $X_A$.  In fact, for any $A\in {\sf ASM}(n)$, we have that $I_A$ is radical.   This follows from the Frobenius splitting argument given in \cite[Section~7.2]{knutson2009frobenius}.  We make the connection to ASM varieties explicit.
\begin{proposition}[{\cite{knutson2009frobenius}}]
\label{proposition:maindegeneration}
Fix any antidiagonal term order $\prec$ on $\Bbbk[Z]$.
\begin{enumerate}
\item The essential (and hence defining) generators of $I_A$ form a Gr\"obner basis under  $\prec$.
\item $I_A$ is radical and its initial ideal is a square-free monomial ideal.
\item The Stanley--Reisner complex of ${\tt init}(X_A)$ is $\Delta(Q_{n\times n}, A)$.  
\end{enumerate}
\end{proposition}
Since ${\tt Prism}(\boldsymbol \lambda,\mathbf d)$ is in weight preserving bijection with the facets of maximum dimension in $\Delta(Q_{n\times n}, A)$, this yields a second proof of Theorem~\ref{theorem:main}.  





\section{Combinatorial Prism Models}

\label{section:combprism}



We start by presenting a generalization of Rothe diagrams to ASMs.  Plot $A\in {\sf ASM}(n)$ in the $n\times n$ grid by placing a black dot for each 1 in $A$ and a white dot for each $-1$.     Strike out hooks to the right and below each black dot which stop if they encounter the boundary of a box which contains a white dot.  The boxes which remain form the  {\bf Rothe diagram} $D(A)$.  Equivalently, $(i,j)\in D(A)$ if and only if $(i,j)$ is an \emph{inversion} of A (see  \cite{mills1983alternating} for this definition).
The {\bf essential set} $\mathcal Ess(A)$ consists of the southeast most corners of each connected component of $D(A)$.

\begin{example}
\label{example:asmDiagram}

\[ A=\left(\begin{matrix} 0 & 0 & 0 & 1\\ 0&1&0&0\\ 1&-1&1&0\\ 0 &1 & 0 & 0\end{matrix}\right)
\hspace{3em} 
D(A)=\begin{tikzpicture}[x=1.3em,y=1.3em,baseline=3.6em]
	\draw[step=1,gray!30,very thin] (0,1) grid (4,5); 
      \draw[color=black, thick](0,1)rectangle(4,5);
     \filldraw[color=black, fill=gray!30, thick](0,4)rectangle(1,5);
     \filldraw[color=black, fill=gray!30, thick](1,4)rectangle(2,5);
     \filldraw[color=black, fill=gray, thick](2,4)rectangle(3,5);
     \filldraw[color=black, fill=gray, thick](0,3)rectangle(1,4);
     \filldraw[color=black, fill=gray, thick](1,2)rectangle(2,3);
     \filldraw [black](3.5,4.5)circle(.1);
     \filldraw [black](1.5,3.5)circle(.1);
     \filldraw [black](.5,2.5)circle(.1);
     \filldraw [color=black,fill=white,thick](1.5,2.5)circle(.1);
     \filldraw [black](2.5,2.5)circle(.1);
     \filldraw [black](1.5,1.5)circle(.1);
     \draw[thick] (3.5,4.5)--(3.5,4);
     \draw[thick] (4,4.5)--(3.5,4.5);
     \draw[thick] (1.5,3.5)--(1.5,3);
     \draw[thick] (2,3.5)--(1.5,3.5);
     \draw[thick] (.5,2.5)--(.5,2);
     \draw[thick] (1,2.5)--(.5,2.5);
     \draw[thick] (2.5,2.5)--(2.5,2);
     \draw[thick] (3,2.5)--(2.5,2.5);
     \draw[thick] (1.5,1.5)--(1.5,1);
     \draw[thick] (2,1.5)--(1.5,1.5);
     \draw[thick] (3,3.5)--(2,3.5);
     \draw[thick] (4,3.5)--(3,3.5);
     \draw[thick] (4,2.5)--(3,2.5);
     \draw[thick] (3,1.5)--(2,1.5);
     \draw[thick] (4,1.5)--(3,1.5);
     \draw[thick] (3.5,4)--(3.5,3);
     \draw[thick] (3.5,3)--(3.5,2);
     \draw[thick] (.5,2)--(.5,1);
     \draw[thick] (2.5,2)--(2.5,1);
     \draw[thick] (3.5,2)--(3.5,1);
     \end{tikzpicture}\]
The boxes of the diagram of $A$ are shaded gray.  The essential boxes are dark gray. \qed
\end{example}





Notice that  $D(A)$ is similar to the ASM diagram defined by A.~Lascoux \cite{lascoux2008chern}.  However, our conventions on inversions differ; we include the set of \emph{negative inversions}.
If $w$ is a permutation matrix, $D(w)$ and $\mathcal Ess(w)$ coincide with the usual Rothe diagram and essential set, as defined in \cite{fulton1992flags}.
Any permutation is uniquely determined by the restriction of the corner sum function to its essential set \cite[Lemma 3.10]{fulton1992flags}.  The same statement holds more generally for ASMs, see Proposition~\ref{prop:bigrassList}.





BiGrassmannian permutations in $\mathcal S_n$ are naturally labeled by triples of integers $(i,j,r)$ which satisfy the following conditions:
\begin{enumerate}[label=(B\arabic*),ref=(B\arabic*)]
\item \label{item:B1} $1\leq i,j$
\item \label{item:B2} $0\leq r <\min(i,j)$
\item \label{item:B3} $i+j-r\leq n$.
\end{enumerate}
Let $I_k$ denote the $k\times k$ identity matrix.  Then we write 
\begin{equation}
\label{eqn:biGrDef}
[i,j,r]_b:=\left(\begin{array}{c|c|c|c}
I_{r}& &&\\
\hline
&&I_{i-r}&\\
\hline
&I_{j-r}&&\\
\hline
&&&I_{n-i-j+r}\\
\end{array}
\right)
\end{equation}
 for the (unique) biGrassmannian encoded by this triple.    In the case $r=\min(i,j)$, let $[i,j,r]_b$ be the identity permutation.  

 Let
${\tt biGr}(A)$ be the set of maximal biGrassmannians below $A$ in the lattice ${\sf ASM}(n)$.  Then $A=\vee{\tt biGr}(A)$.
The next proposition shows how to recover ${\tt biGr}(A)$ from $\mathcal Ess(A)$. 
\begin{proposition}
\label{prop:bigrassList}
${\tt biGr}(A)=\{[i,j,r_A(i,j)]_b:(i,j)\in \mathcal Ess(A)\}.$
\end{proposition}
Proposition~\ref{prop:bigrassList} is  discussed in \cite[Section~5]{lascoux1996treillis},  using essential points of  monotone triangles.  It appears in a  more general context in \cite[Theorem~5.1]{fortin2008macneille}.     As a  consequence,  $A$ is determined by the restriction of $r_A$ to $\mathcal Ess(A)$.  This generalizes \cite[Lemma 3.10]{fulton1992flags}.





Left justifying the Rothe diagram of a Grassmannian permutation $u$ produces the (French) Young diagram of partition $\lambda^{(u)}$.  If $d\geq \ell(\lambda)$, write $[\lambda,d]_g$ for the (unique) Grassmannian with descent at position $d$ and associated partition $\lambda$.  If $\lambda=()$, then for any $d$ we say $[\lambda,d]_g={\rm id}$.  Then given $(\boldsymbol \lambda,\mathbf d)$ as in (\ref{eqn:tupledef}), we define 
\begin{equation}
\label{def:asmassociated}
A_{\boldsymbol \lambda,\mathbf d}=\vee\{[\lambda^{(1)},d_1]_g,\ldots,[\lambda^{(k)},d_k]_g\}.
\end{equation}



We now describe two ways of taking $A\in{\sf ASM}(n)$ as a input and producing a pair $(\boldsymbol \lambda,\mathbf d)$ so that $A=A_{\boldsymbol \lambda,\mathbf d}$.  Both procedures are entirely combinatorial.  We start with BiGrassmannian prism tableaux, which were defined in \cite{weigandt2018prism}.




\begin{definition}[BiGrassmannian Prism Tableaux]
Suppose \[{\mathcal Ess}(A)=\{(i_1,j_1),(i_2,j_2),\ldots,(i_k,j_k)\}.\]  Let
\begin{equation}
\label{eqn:betapartition}
\beta^{(\ell)}=(i_\ell-r_A(i_\ell,j_\ell))\times(j_\ell-r_A(i_\ell,j_\ell)).
\end{equation}  Define
 $\boldsymbol \beta_A=(\beta^{(1)},\ldots,\beta^{(k)})$ and $\mathbf b_A=\{i_1,\ldots,i_k\}.$
Elements of ${\tt AllPrism}(\boldsymbol \beta_A,\mathbf b_A)$ are {\bf biGrassmannian} prism tableaux.
\end{definition}



We now introduce the parabolic prism model.  Our definition uses the \emph{monotone triangles} of W.~H.~Mills, D.~P.~Robbins, and H.~Rumsey \cite{mills1983alternating}.  
Given $A=(a_{i j})_{i,j=1}^n\in{\sf ASM}(n)$ let $C_A$ be the matrix of partial column sums, i.e.~$C_A(i,j)=\sum_{\ell=1}^i a_{\ell j}$. The $i$th row of $m_A$ records (in increasing order) the positions of the 1s in the $i$th row of $C_A$.  The array $m_A$ is called a {\bf monotone triangle}. 
\begin{example}
\label{example:monotonetriangle}
\[A=
\begin{pmatrix}
0& 0&1& 0\\
1&0&-1&1\\
0&0&1&0\\
0&1&0&0
\end{pmatrix}
\hspace{2em}
C_A=\begin{pmatrix}
0& 0&1& 0\\
1&0&0&1\\
1&0&1&1\\
1&1&1&1
\end{pmatrix}
\hspace{2em}
m_A=
\begin{array}{cccccccccc}
&&&&3\\
&&&1&&4\\
&&1&&3&&4\\
&1&&2&&3&&4
\end{array}
\]
\qed
\end{example}


Given $A$ and $1\leq \ell\leq n$, we define 
\begin{equation}
\lambda^{(A,\ell)}=(m_A(\ell,\ell)-\ell,m_A(\ell,\ell-1)-(\ell-1),\ldots, m_A(\ell,1)-1).
\end{equation}
Since $m_A$ strictly increases along rows, $\lambda^{(A,\ell)}$ is a partition. 


\begin{definition}[Parabolic Prism Tableaux]
 Write $\{i:(i,j)\in \mathcal Ess(A)\}=\{i_1,\ldots,i_k\}$ for the indices of essential rows of $A$.  Let \[\text{$\boldsymbol \rho_A=(\lambda^{(A,i_1)}, \lambda^{(A,i_2)}, \ldots, \lambda^{(A,i_k)})$ and $\mathbf p_A=(i_1,\ldots,i_k)$}.\]  
Elements of ${\tt AllPrism}(\boldsymbol \rho_A,\mathbf p_A)$ are {\bf parabolic} prism tableaux.
\end{definition}




\begin{proposition}
\label{prop:therightasm}
\begin{enumerate}
\item $A=A_{\boldsymbol \beta_A,\mathbf b_A}$.
\item $A=A_{\boldsymbol \rho_A,\mathbf p_A}$.
\end{enumerate}
\end{proposition}

\section{Subword complexes and prism tableaux}




Let $\mathbb P(S)$ denote the power set of $S$.  
A {\bf simplicial complex} $\Delta$ is a subset of $\mathbb P([N])$ so that whenever $f\in \Delta$ and $f'\subseteq f$, we have $f'\in \Delta$.  An element $f\in \Delta$ is called a {\bf face}.   The {\bf dimension} of $f$ is $\dim(f)=|f|-1$.   Write $\dim(\Delta)=\max\{\dim(f):f\in \Delta\}$. If $f\in \Delta$, the {\bf codimension} of $f$ is ${\rm codim}(f)=\dim(\Delta)-\dim(f)$.
The set of faces of $\Delta$ ordered by inclusion form a poset.  Let 
$
F(\Delta)$
 denote the set of {\bf facets} of $\Delta$, i.e.~the maximal faces.
 Then define 
$
F_{\tt max}(\Delta)=\{f\in\Delta: {\rm codim}(f)=0\}.
$
Necessarily, $F_{\tt max}(\Delta)\subseteq F(\Delta)$.  When this containment is an equality, $\Delta$ is called {\bf pure}.
 Given two simplicial complexes $\Delta_1,\Delta_2 \subseteq \mathbb P([N])$, we may refer without ambiguity to the intersection (or union) of $\Delta_1$ and $\Delta_2$;  it is precisely their intersection (or union) as sets.  A straightforward verification shows that $\Delta_1\cap \Delta_2$ and $\Delta_1\cup \Delta_2$ are themselves simplicial complexes.  




We now recall the definition of a subword complex, following \cite{knutson2004subword}.  Let $\Pi$ be a Coxeter group  generated by simple reflections $\Sigma$.  A {\bf word} is an ordered list $Q=(s_{1},\ldots,s_{m})$ of simple reflections in $\Sigma$.   A {\bf subword} of $Q$ is an ordered subsequence $P=(s_{i_i},\ldots, s_{i_k})$.  Subwords of $Q$ are naturally identified with faces of the simplicial complex $\mathbb P([m])$.
A word  $P$ {\bf represents} $w\in \Pi$ if $w=s_{1}\cdots s_{m}$ and $\ell(w)=m$, i.e.~the ordered product is a reduced expression for $w$. We say $Q$ {\bf contains} $w$ if $Q$ has a subword which represents $w$.  Then define the {\bf subword complex} $\Delta(Q,w)=\{Q-P:P \text{ contains } w\}$.  
We will abbreviate $\mathcal F_P:=Q-P$.   Immediately by definition, $
\mathcal F_P\subseteq \mathcal F_{P'}$ if and only if $ P\supseteq P'.
$
A well known characterization of the Bruhat order on $\mathcal S_n$ is via subwords: 
\begin{equation}
\label{eqn:bruhatword}
\text{$w\geq v$  if and only if some (and hence every) reduced word for $w$ contains $v$}.
\end{equation}  See \cite[Section 5.10]{humphreys1992reflection}.  This is equivalent to the order on $\mathcal S_n$ as defined in (\ref{eqn:asmposetdef}). See \cite[Theorem 2.1.5]{bjorner2006combinatorics} for a proof.





The {\bf Demazure algebra} of $(\Pi,\Sigma)$ over a ring $R$ is freely generated by $\{e_w:w\in \Pi\}$ with multiplication given by 
\[e_we_s=
\begin{cases}
e_{ws} & \text { if} \enspace \ell(ws)>\ell(w)\\
e_w &\text{ if} \enspace \ell(ws)<\ell(w).
\end{cases}
\]
If $Q=(s_1, \ldots ,s_k)$, the {\bf Demazure product} $\delta(Q)$ is defined by  $e_{s_1}\cdots e_{s_m}=e_{\delta(Q)}$.  The faces of $\Delta(Q,w)$ have a natural description in terms of the Demazure product.
\begin{lemma}[{\cite[Lemma 3.4]{knutson2004subword}}]
\label{lemma:dez}
$\delta(P)\geq w$ if an only if $P$ contains $w$.  
\end{lemma}
Notice
$
\Delta(Q,w)=\{\mathcal F_P:\delta(P)\geq w\}.$
This motivates the following definition.  Let $\Pi=\mathcal S_n$ and $\Sigma=\{(i,i+1):i=1,\ldots,n-1\}$ be the set of simple transpositions. Given $A\in {\sf ASM}(n)$, define 
$
\Delta(Q,A)=\{\mathcal F_P:\delta(P)\geq A\}.
$
This is itself a simplicial complex, but need not be a subword complex.
  Immediately from the definition, 
\begin{equation}
\label{eqn:subwordcont}
\text{if \enspace $A\geq B$ \enspace then \enspace $\Delta(Q,A)\subseteq \Delta(Q,B)$.}
\end{equation}
We will show that $\Delta(Q,A)$ is a union of subword complexes.  In particular, if $A\in {\sf ASM}(m)$ with $m\leq n$ each of these subword complexes correspond to permutations in $\mathcal S_m$.
Write ${\tt Min}(S)$ for the minimal elements in $S$.  Let 
$
{\tt Perm}(A):={\tt MIN}(\{w\in \mathcal S_n:w\geq A\})$
and 
\begin{equation}
\label{example:permminperm}
{\tt MinPerm}(A):=\{w\in {\tt Perm}(A):\ell(w)={\tt deg}(A)\}.
\end{equation}
\begin{proposition}
\label{prop:interesctunion}
Fix a word $Q$ and $A\in {\sf ASM}(n)$.  
\begin{enumerate}
\item $\displaystyle\Delta(Q,A)=\bigcup_{w\in {\tt Perm}(A)} \Delta(Q,w)$.
\item If $A=\vee\{A_1,\ldots,A_k\}$ then $\displaystyle\Delta(Q,A)=\bigcap_{i=1}^k \Delta(Q,A_i)$.
\item $F(\Delta(Q,A))=\{\mathcal F_P: P \text{ is a reduced expression for some } w\in{\tt Perm}(A)\}.$
\end{enumerate} 
\end{proposition}



For the rest of this section, we focus on a fixed ambient word $Q$.  Write $s_i$ for the simple transposition $(i,i+1)\in \mathcal S_{2n}$.  Define the {\bf square word} \[Q_{n\times n}=s_n \, s_{n-1} \, \cdots \, s_1 \quad s_{n+1} \,  s_{n} \, \cdots \, s_2 \quad \cdots \quad s_{2n-1} \, s_{2n-2} \, \cdots s_n.\]  
Order the boxes of the $n\times n$ grid by reading along rows from right to left, starting with the top row and working down to the bottom.  This ordering identifies each letter of $Q_{n\times n}$ with a cell in the $n\times n$ grid.  


A {\bf plus diagram} is a subset of the $n\times n$ grid.  We indicate $(i,j)$ is in the plus diagram by marking its position in the grid with a $+$.  The identification of the letters in  $Q_{n\times n}$ with the grid defines a natural bijection between subwords of $Q_{n\times n}$ and plus diagrams.  As such,
we freely identify each word with its plus diagram. 

\begin{example}
When $n=3$, we have 
$Q_{n\times n}= s_3 s_2 s_1 s_4 s_3 s_2 s_5 s_4 s_3$.
Below, we label the entries of the $3\times 3$ grid with their corresponding simple transpositions.  We also give a subword of $Q_{3\times 3}$ and  its corresponding plus diagram.  
\[\begin{array}{ccc}
s_1 &s_2 &s_3 \\
 s_2 &s_3 &s_4 \\
  s_3 &s_4 &s_5
\end{array}\hspace{3em} 
 s_3 \, -\, -\,  -\, s_3\, s_2 \,- \,-\, \,-  \hspace{3em} 
 \begin{array}{ccc}
\cdot &\cdot &+ \\
+&+&\cdot \\
  \cdot &\cdot &\cdot
\end{array}  \]
Notice that $P$ is not a reduced expression, $s_3s_3s_2=s_2$.  Therefore, it is not a facet of  $\Delta(Q_{n\times n},A)$ for any $A\in {\sf ASM}(n)$. \qed
\end{example}



For brevity, write  $\Delta_A:=\Delta(Q_{n\times n},A)$.  
Assign $\mathcal F_P$ the weight
$\displaystyle {\tt wt}(\mathcal F_P)=\prod_{i=1}^n x_i^{n_i}$  where $n_i=|\{j:(i,j)\in P\}|.$
When $w\in \mathcal S_n$, the complex $\Delta_w$ is a pure simplicial complex.  Its facets are in immediate bijection with pipe dreams (also known as RC-graphs). 
\begin{theorem}[\cite{bergeron1993rc, fomin1996yang,knutson2005grobner}]
\label{thm:schubequation}
$\displaystyle \mathfrak S_w=\sum_{\mathcal F_P\in  F(\Delta_w)}{\tt wt}(\mathcal F_P).$
\end{theorem}
For permutations, $\Delta_w$ is the Stanley--Reisner complex of a degeneration of the Schubert determinantal ideal $I_w$ \cite[Theorem~B]{knutson2005grobner}.  The same holds for $I_A$ and $\Delta_A$.
As a consequence of Theorem~\ref{thm:schubequation}, we have the following corollary.
\begin{corollary}
\label{cor:schubsumposet}
\begin{enumerate}
\item
$\displaystyle \sum_{w\in{\tt Perm}(A)} \mathfrak S_w=\sum_{\mathcal F_P\in F(\Delta_A)} {\tt wt}(\mathcal F_P).$
\item $\displaystyle \sum_{w\in{\tt MinPerm}(A)} \mathfrak S_w=\sum_{\mathcal F_P\in F_{\tt max}(\Delta_A)} {\tt wt}(\mathcal F_P).$
\end{enumerate}
\end{corollary}


There is a natural map from ${\tt AllPrism}(\boldsymbol \lambda,\mathbf d)$ to the set of plus diagrams: if $\mathcal T$ has the label $i$ in an antidiagonal, place a plus in row $i$ of this same antidiagonal.  
For instance, continuing Example~\ref{example:prism}, we have
\ytableausetup{boxsize=.95em, aligntableaux=center}
\[\mathcal T=\begin{tikzpicture}[x=1em,y=1em,baseline=4.2em]
\draw[step=1,gray!30,very thin] (0,1) grid (7,8); 
\node[] at (1.5,5){
\begin{ytableau}
\none\\
{\color{Rhodamine}1}\\
\none\\
{\color{Plum}2}{\color{blue}1}&{\color{Plum}2}\\
{\color{Plum}3}{\color{blue}2}&{\color{Plum}3}&{\color{Plum}2}\\
{\color{blue}6}&{\color{blue}3}\\
\end{ytableau}};
\end{tikzpicture} 
\hspace{2em}
 \mapsto \hspace{2em}
P_{\mathcal T}=\begin{array}{ccccccc}
\cdot &+ &\cdot &+ &\cdot &\cdot &\cdot \\
\cdot &\cdot &+&+ &\cdot &+ &\cdot \\
\cdot &\cdot &+ &+ &+ &\cdot &\cdot \\
\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\
\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\
+ &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot \\
\cdot &\cdot &\cdot &\cdot &\cdot &\cdot &\cdot 
\end{array} 
 \]Our proof of Theorem~\ref{theorem:schubertsum} involves showing that the restriction of this map to ${\tt Prism}(\boldsymbol \lambda,\mathbf d)$ is a bijection onto the maximum dimensional facets of $\Delta_{A_{\boldsymbol \lambda,\mathbf d}}$.





\section{ASM varieties and determinantal ideals}
\label{section:asmgeo}




 Recall ${\sf Mat}(n)$ is the space of $n\times n$ matrices over an algebraically closed field $\Bbbk$. 
Write ${\sf GL}(n)$ for the invertible matrices in ${\sf Mat}(n)$ and ${\sf T}$ for the torus of diagonal matrices in ${\sf GL}(n)$.  There is a natural action of ${\sf GL}(n)$, and hence ${\sf T}$, on ${\sf Mat}(n)$ by left multiplication.  A variety $X\subseteq {\sf Mat}(n)$ is {\bf ${\sf T}$ stable} if ${\sf T} X\subseteq X$.

Recall $Z=(z_{ij})_{i,j=1}^n$.  Write $\Bbbk[Z]=\Bbbk[z_{11},z_{12},\ldots, z_{nn}]$ for the coordinate ring of ${\sf Mat}(n)$.  There is a \emph{multigrading} defined by the action ${\mathsf T}$ on ${\sf Mat}(n)$.  In particular,  ${\sf T}$ stable subvarieties of ${\sf Mat}(n)$ have coordinate rings that are $\Bbbk[Z]$-graded modules and define a \emph{multidegree}.  Multidegrees are characterized by three properties, \emph{normalization}, \emph{additivity}, and \emph{degeneration} \cite[Theorem 8.44]{miller2004combinatorial}.

Fix $X\subseteq {\sf Mat}(n)$ with $X$ being ${\sf T}$ stable.  If $\Bbbk[Z]/I$ is the coordinate ring of $X$,  write $\mathcal C(X;\mathbf x):=\mathcal C(\Bbbk[Z]/I;\mathbf x)$ for its multidegree.  By additivity,
$
\mathcal C(X;\mathbf x)=\sum_{i=1}^k \mathcal C(X_i;\mathbf x)
$
where $\{X_1,\ldots,X_k\}$ are the maximal dimensional irreducible components of $X$.  Since $I$ is radical,  this is a multiplicity free sum.

 Given an $n\times n$ matrix $M$, write $M_{[i],[j]}$ for the submatrix of $M$ which consists of the first $i$ rows and $j$ columns of $M$.  
Fix $w\in \mathcal S_n$. The {\bf matrix Schubert variety} is 
\begin{equation}
\label{eqn:matrixschubert}
X_w:=\{M\in {\sf Mat}(n): {\rm rank}(M_{[i],[j]})\leq r_w(i,j) \enspace \text{for all} \enspace  1\leq i,j\leq n\}.
\end{equation}
Matrix Schubert varieties generalize classical determinantal varieties.   W.~Fulton showed that they are irreducible \cite{fulton1992flags}.   
 By \cite[Theorem~A]{knutson2005grobner}, when $w\in \mathcal S_n$,
we have $
\mathcal C(X_w;\mathbf x)=\mathfrak S_w.$



We define the {\bf alternating sign matrix variety} \[X_A:=\{M\in {\sf Mat}(n): {\rm rank}(M_{[i],[j]})\leq r_A(i,j) \enspace \text{for all} \enspace  1\leq i,j\leq n\}.\]  
Immediately by definition, if $A\leq B$ then $X_A\supseteq X_B$.
Furthermore,
$X_A$ has the following set theoretic descriptions as unions and intersections of other ASM varieties.
\begin{proposition}
\label{prop:intersectandunion}
\begin{enumerate}
\item $\displaystyle X_A=\bigcup_{w\in {\tt Perm}(A)} X_w$.
\item If $  A=\vee\{A_1,\ldots,A_k\}$, then $\displaystyle X_A=\bigcap_{i=1}^k A_i$.
\end{enumerate}
\end{proposition}






W.~Fulton showed that each $X_w$ is defined by a smaller set of {\bf essential} conditions,
\begin{equation}
\label{eqn:matrixschubertess}
X_w=\{M\in {\sf Mat}(n): {\rm rank}(M_{[i],[j]})\leq r_w(i,j) \enspace \text{for all} \enspace  (i,j)\in\mathcal Ess(w)\}.
\end{equation}
By Proposition~\ref{prop:intersectandunion}, $\displaystyle X_A=\bigcap_{u\in {\tt biGr}(A)} X_u$.  Therefore, ASM varieties are  defined by essential conditions.
The rank of \emph{any} submatrix is preserved under the ${\sf T}$ action, so $X_w$ is ${\sf T}$ stable.  
\begin{proposition}
\label{prop:cohosum}
$\displaystyle \mathcal C(X_A;\mathbf x)=\sum_{w\in {\tt MinPerm}(A)} \mathfrak S_w$.
\end{proposition}
\begin{proof}

As a consequence of Proposition~\ref{prop:intersectandunion}, the top dimensional irreducible components of $X_A$ are $\{X_w:w\in {\tt MinPerm}(A)\}$.  Then using the additivity property of multidegrees we have 
$\displaystyle
\mathcal C(X_A;\mathbf x) =\sum_{w\in{\tt MinPerm}(A)} \mathcal C(X_w;\mathbf x) 
= \sum_{w\in{\tt MinPerm}(A)}  \mathfrak S_w.  \qedhere
$
\end{proof}
Theorem~\ref{theorem:main} follows as an immediate consequence of  Proposition~\ref{prop:cohosum} and Theorem~\ref{theorem:schubertsum}.


We now turn our discussion to defining ideals for ASM varieties.   
Recall the {\bf ASM ideal} 
$
I_A:=\langle  \text{minors of size } r_A(i,j)+1 \enspace \text{in} \enspace  Z_{[i],[j]}\rangle.
$
A matrix has rank at most $r$ if and only if all of its minors of size $r+1$ vanish.  As such, $I_A$ set-theoretically cuts out $X_A$. Furthermore, $I_A$ has generators which are homogeneous for the $\mathbb Z^n$ grading on $\Bbbk[Z]$. 
There is a smaller generating set for $I_A$.  Write 
\[{\tt EssGen}(A)=\{\text{minors of size } r_A(i,j)+1 \enspace \text{in} \enspace  Z_{[i],[j]}:(i,j)\in\mathcal Ess(A)\}.\]  for {\bf essential generators} of $I_A$. 


  We could have alternatively proved Proposition~\ref{prop:cohosum} by
  taking an explicit Gr\"obner degeneration of $I_A$ with respect to an
  \emph{antidiagonal term order}.   In this case, the Stanley--Reisner complex of $I_A$ is $\Delta(Q_{n\times n},A)$.  Furthermore, if $A=A_{\boldsymbol \lambda,\mathbf d}$,  the maximum dimensional facets of $\Delta(Q_{n\times n},A)$ have a natural labeling by elements of ${\tt Prism}({\boldsymbol \lambda,\mathbf d})$.  This produces a specific connection between the Gr\"obner geometry of $X_{A}$ and prism tableaux.


\acknowledgements{I thank my advisor, Alexander Yong, for his guidance throughout this project.
I also thank Allen Knutson for suggesting this direction of research and Jessica Striker for helpful conversations about alternating sign matrices. 
  I was supported by  a UIUC Campus Research Board and by an NSF Grant.  This work was partially completed while participating in the trimester ``Combinatorics and Interactions'' at the Institut Henri Poincar\'e.  My travel support  was provided by NSF Conference Grant 1643027.  I was funded by the Ruth~V.~Shaff and Genevie~I.~Andrews Fellowship.   I used Sage and Macaulay2 during the course of my research.}


\printbibliography



\end{document}
