%% 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[submission]{FPSAC2019}
\usepackage{tikz}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{15}
\addbibresource{15.bib}

\newcommand{\KK}{{\mathbb{K}}}
\newcommand{\CC}{{\mathbb{C}}}
\newcommand{\PP}{{\mathbb{P}}}
\newcommand{\QQ}{{\mathbb{Q}}}
\newcommand{\ZZ}{{\mathbb{Z}}}
\newcommand{\FF}{{\mathbb{F}}}

\newcommand{\xx}{{\mathbf{x}}}

\newcommand{\inv}{{\mathrm{inv}}}
\newcommand{\maj}{{\mathrm{maj}}}
\newcommand{\initial}{{\mathrm{in}}}
\newcommand{\conv}{{\mathrm{conv}}}
\newcommand{\st}{{\mathrm{st}}}
\newcommand{\SYT}{{\mathrm{SYT}}}
\newcommand{\des}{{\mathrm{des}}}

\newcommand{\symm}{{\mathfrak{S}}}

\newcommand{\Fl}{{\mathcal{F \ell}}}
\newcommand{\OP}{{\mathcal{OP}}}
\newcommand{\WWW}{{\mathcal{W}}}
\newcommand{\UUU}{{\mathcal{U}}}
\newcommand{\VVV}{{\mathcal{V}}}

\newcommand{\PM}{{\mathrm{PM}}}

\newcommand{\Mat}{{\mathrm{Mat}}}

\newcommand{\Stir}{{\mathrm{Stir}}}
\newcommand{\rev}{{\mathrm{rev}}}
\newcommand{\Hilb}{{\mathrm{Hilb}}}
\newcommand{\Frob}{{\mathrm{Frob}}}
\newcommand{\grFrob}{{\mathrm{grFrob}}}

\newcommand{\Rise}{{\mathrm{Rise}}}
\newcommand{\Val}{{\mathrm{Val}}}


%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem{defn}{Definition}

\Crefname{prop}{Proposition}{Propositions}
\Crefname{lem}{Lemma}{Lemmas}

\usepackage{lipsum}

%% define your title in the usual way
\title{Spanning line configurations}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Brendan Pawlowski\thanks{\href{mailto:bpawlows@usc.edu}{bpawlows@usc.edu}}\addressmark{1} \and 
Brendon Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Southern California \\ \addressmark{2}Department of Mathematics, University of California, San Diego}

%% 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
{We define and study a variety $X_{n,k}$ which depends on two positive integers $k \leq n$. When 
$k = n$, the variety $X_{n,k}$ is homotopy equivalent to the {\em flag variety} 
$\mathcal{F \ell}(n)$ of complete flags in $\mathbb{C}^n$. We describe an affine paving of $X_{n,k}$,
present its cohomology, and describe the cellular cohomology classes in terms of Schubert polynomials.
Just as the geometry of  $\mathcal{F \ell}(n)$ is governed by the combinatorics of permutations
in $S_n$, the geometry of $X_{n,k}$ is governed by length $n$ words on the alphabet $\{1,2, \dots, k\}$ in which
each letter appears at least once.
The space $X_{n,k}$ carries a natural action of $S_n$, and we relate the induced cohomology representation
to Macdonald theory via the Delta Conjecture of Haglund, Remmel, and Wilson.}

%% 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{Fubini word, flag variety, symmetric function, coinvariant ring}

%% 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{sample.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{Introduction}

In this extended abstract we introduce and study a variety $X_{n,k}$ depending on two positive integers 
$k \leq n$. Our goal is to provide a geometric context to study the {\em Delta Conjecture} of Haglund,
Remmel, and Wilson \cite{HRW} which extends the role played by the classical flag variety $\mathcal{F \ell}(n)$
in the study of diagonal coinvariants and the {\em Shuffle Theorem} \cite{CarlssonMellit}.
We introduce our variety $X_{n,k}$ in \cref{Spanning} below; the remainder of the introduction
is devoted to connections with the Delta Conjecture and related work of Haglund, Rhoades, and
Shimozono \cite{HRS} on {\em generalized coinvariant rings}.  
We solve the problem \cite[Prob. 7.2]{HRS} of finding a flag variety 
for the Delta Conjecture.

Consider the action of the symmetric group $S_n$ on the polynomial ring $\ZZ[x_1, \dots, x_n]$ by subscript 
permutation.  The invariant subring $\ZZ[x_1, \dots, x_n]^{S_n}$ is the ring of {\em symmetric polynomials}.
Let $\ZZ[x_1, \dots, x_n]^{S_n}_+$ be the family of symmetric polynomials with vanishing constant term.
The {\em invariant ideal} $I_n \subseteq \ZZ[x_1, \dots, x_n]$ is the ideal generated by 
$\ZZ[x_1, \dots, x_n]^{S_n}_+$.  If $e_d = \sum_{1 \leq i_i < \cdots < i_d \leq n} x_{i_1} \cdots x_{i_d}$
is the degree $d$ elementary symmetric polynomial, we have
$I_n = \langle e_1, e_2, \dots, e_n \rangle$. The {\em coinvariant ring} is 
\begin{equation}
R_n := \ZZ[x_1, \dots, x_n]/I_n = \ZZ[x_1, \dots, x_n]/\langle e_1, e_2, \dots, e_n \rangle.
\end{equation}
The ring $R_n$ is a graded $\ZZ$-algebra with a graded action of $S_n$.

Let $\CC^n$ be the standard $n$-dimensional complex vector space.  A {\em (complete) flag} in 
$\CC^n$ is a maximal sequence $V_{\bullet} = (0 = V_0 \subset V_1 \subset \cdots \subset V_n = \CC^n)$
of nested subspaces of $\CC^n$ such that $\dim(V_i) = i$ for $1 \leq i \leq n$.
The {\em flag variety} $\mathcal{F \ell}(n)$ is the family of complete flags in $\CC^n$. The identification
$\mathcal{F \ell}(n) = \mathrm{GL}_n(\CC)/B$,
where $B \subseteq \mathrm{GL}_n(\CC)$ is the upper triangular subgroup, endows
$\mathcal{F \ell}(n)$ with the structure of a complex algebraic variety.
Borel proved \cite{Borel} that the (singular, integral) cohomology of $\mathcal{F \ell}(n)$ is presented
by the coinvariant ring:
\begin{equation}
H^{\bullet}(\mathcal{F \ell}(n)) = R_n.
\end{equation}
{\bf Algebraic properties of $R_n$ and  geometric properties of $\mathcal{F \ell}(n)$ are governed by 
combinatorial properties of permutations in $S_n$.}
In no small part for this reason, $R_n$ is one of the most well-studied rings and $\mathcal{F \ell}(n)$ is
one of the most well-studied varieties in algebraic combinatorics.

Consider a polynomial ring in two sets of $n$ variables $\QQ[x_1, \dots, x_n, y_1, \dots, y_n]$
over the rational field $\QQ$.  This ring carries a {\em diagonal} action of $S_n$, viz.
$w.x_i = x_{w(i)}, w.y_i := y_{w(i)}$ for $w \in S_n$ and $1 \leq i \leq n$. 
The {\em diagonal coinvariant ring} \cite{GarsiaHaiman} is the bigraded $S_n$-module
\begin{equation}
DR^{\QQ}_n := \QQ[x_1, \dots, x_n, y_1, \dots, y_n]/
\langle \QQ[x_1, \dots, x_n, y_1, \dots, y_n]^{S_n}_+ \rangle
\end{equation}
obtained by modding out by invariants with vanishing constant term.
Setting the $y$-variables equal to zero recovers (up to ground ring) the classical coinvariant ring $R_n$
which presents the cohomology of $\mathcal{F \ell}(n)$.

It is natural to ask for the bigraded $S_n$-isomorphism type of $DR_n$.  We recall the basics of 
the Frobenius map connecting $S_n$-modules and symmetric functions.

The irreducible representations
of $S_n$ are in bijective correspondence with partitions of $n$. Given a partition $\lambda \vdash n$,
let $S^{\lambda}$ be the corresponding irreducible $\symm_n$-module.
If $V$ is any finite-dimensional $S_n$-module, there exist unique multiplicities $c_{\lambda} \geq 0$ so that 
$V \cong \bigoplus_{\lambda \vdash n} c_{\lambda} S^{\lambda}$.
The {\em Frobenius image} of $V$ is the symmetric function 
$\Frob(V) = \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}$, where $s_{\lambda}$ is the Schur function.

Going further, if $V = \bigoplus_{i \geq 0} V_i$ is a {\em graded} $S_n$-module with each graded piece 
$V_i$ finite-dimensional, the {\em graded Frobenius image} is 
$\grFrob(V; q) := \sum_{i \geq 0} \Frob(V_i) \cdot q^i$.  Finally, if $V = \bigoplus_{i,j \geq 0} V_{i,j}$ 
is a {\em bigraded} $S_n$-module with each  $V_{i,j}$ finite-dimensional, the {\em bigraded Frobenius 
image} is $\grFrob(V;q,t) = \sum_{i,j \geq 0} \Frob(V_{i,j}) \cdot q^i t^j$.

Haiman \cite{Haiman} proved that the bigraded Frobenius image of $DR_n$ is given by
\begin{equation}
\grFrob(DR_n; q,t) = \nabla e_n, 
\end{equation}
where $\nabla$ is the Bergeron-Garsia nabla operator on symmetric functions and $e_n$
is the elementary symmetric function.
Finding the bigraded isomorphism type of 
$DR_n$ therefore reduces to finding a positive formula for the Schur expansion of $\nabla e_n$.  
While there is not even 
a conjecture in this direction, Carlsson and Mellit \cite{CarlssonMellit} proved the {\em Shuffle Theorem}
which gives a monomial expansion of $\nabla e_n$.

The {\em Delta Conjecture} of Haglund, Remmel, and Wilson \cite{HRW} 
predicts a generalization of the Shuffle Theorem
which depends on two positive integers $k \leq n$. It reads
\begin{equation}
\Delta'_{e_{k-1}} e_n = \Rise_{n,k}(\xx;q,t) = \Val_{n,k}(\xx;q,t).
\end{equation}
Here $\Delta'_{e_{k-1}}$ is the primed {\em delta operator} labeled by $e_{k-1}$ and $\Rise$ and $\Val$ 
are two formal power series arising from lattice path combinatorics depending on an 
infinite set of variables $\xx = (x_1, x_2, \dots )$ and two additional parameters $q,t$.
The Delta Conjecture reduces to the Shuffle Theorem when $k = n$.


Although the Delta Conjecture is open in general, it is proven when one of the parameters $q,t$ is set to zero.
Combining results of \cite{GHRY, HRS, RhoadesOSP, Wilson}, we have
\begin{equation}
\label{five-equality}
\Delta'_{e_{k-1}} e_n \mid_{t = 0} = \Rise_{n,k}(\xx;q,0) = \Rise_{n,k}(\xx;0,q) = 
\Val_{n,k}(\xx;q,0) = \Val_{n,k}(\xx;0,q).
\end{equation}
Let $C_{n,k}(\xx;q)$ be the common symmetric function of Equation~\eqref{five-equality}.


Haglund, Rhoades, and Shimozono \cite{HRS} defined an extension of the coinvariant ring which applies to the 
Delta Conjecture.  If $k \leq n$ are positive integers, let $I_{n,k} \subseteq \ZZ[x_1, \dots, x_n]$ 
be the ideal
\begin{equation}
I_{n,k} := \langle x_1^k, x_2^k, \dots, x_n^k, e_n, e_{n-1}, \dots, e_{n-k+1} \rangle
\end{equation}
and let $R_{n,k} := \ZZ[x_1, \dots, x_n]/I_{n,k}$ be the corresponding quotient.
The ring $R_{n,k}$ is a graded $S_n$-module.
If we let $R^{\QQ}_{n,k} := \QQ \otimes_{\ZZ} R_{n,k}$, we have the graded Frobenius image \cite{HRS}
\begin{equation}
\label{rnk-frobenius}
\grFrob(R_{n,k}^{\QQ};q) = (\rev_q \circ \omega) C_{n,k}(\xx;q),
\end{equation}
where $\rev_q$ reverses the coefficient sequences of polynomials in $q$ and $\omega$ is the 
symmetric function involution trading $e_n$ and $h_n$.


Equation~\eqref{rnk-frobenius} says that the generalized coinvariant
 ring $R_{n,k}$ plays the same role for the Delta Conjecture as the classical coinvariant ring $R_n$
 for the Shuffle Theorem on the level of graded $S_n$-modules. 
 Haglund, Rhoades, and Shimozono left open the problem \cite[Prob. 7.2]{HRS} of finding a
 corresponding generalization of the flag variety: a
 variety $X_{n,k}$ whose cohomology is presented by $R_{n,k}$.
We solve this problem here. 

A word $w_1 \dots w_n$ over the positive integers is {\em Fubini} (or {\em packed})
if for any $i > 1$ such that 
$i$ appears as a letter in $w_1 \dots w_n$, so does $i-1$.
Let $\WWW_{n,k}$ be the family of length $n$ Fubini words with maximum letter $k$; when 
$k = n$ we have $\WWW_{n,k} = S_n$.
{\bf The geometry of our variety $X_{n,k}$, like the algebra of the ring $R_{n,k}$,
is governed by the combinatorics of Fubini words in $\WWW_{n,k}$.}
In addition to presenting the cohomology of $X_{n,k}$, we generalize classical Schubert
calculus theorems of Ehresmann \cite{Ehresmann} and Lascoux-Sch\"utzenberger \cite{LS}
from the flag variety $\mathcal{F \ell}(n)$ to the more general spaces $X_{n,k}$.
It is the hope of the authors that this will inspire a generalization of Schubert calculus with Fubini
words as its basis.











\section{The spanning moduli space}
\label{Spanning}

Our object of study is
the following moduli space of  line configurations 
\footnote{We use `configuration' rather than `arrangement' because we are considering {\em ordered}
tuples of lines.}
which depends on two positive integers 
$k \leq n$ and a field $\FF$.

\begin{defn}
Let $k \leq n$ be positive integers and let $\FF$ be a field. We define
\begin{equation}
X_{n,k} := \{ (\ell_1, \dots, \ell_n) \,:\, \ell_i \subseteq \FF^k \text{ a $1$-dimensional subspace and }
\ell_1 + \cdots + \ell_n = \FF^k \}
\end{equation}
to be the set of all $n$-tuples of lines through the origin in $\FF^k$ whose span equals $\FF^k$.
\end{defn}

\noindent
{\bf Warning.} {\em Do not confuse $X_{n,k}$  with the Grassmannian of $k$-dimensional
subspaces of $\FF^n$. These objects have very different combinatorial and geometric properties.}

\begin{figure}
\begin{center}
\begin{tikzpicture}


\draw [gray, fill=gray!30] (-1.5,0.5) -- (2.5,0.5) -- (1.5,-0.5) -- (-2.5,-0.5) -- (-1.5,0.5);

\draw  (0,-2) -- (0,2);

\draw (-2,0) -- (2,0);

\draw (0.5,0.5) -- (-0.5,-0.5);

\draw [line width = 0.5mm, blue] (1,0.5) -- (-1,-0.5);

\draw [line width = 0.5mm, blue] (-1.4,0.5) -- (1.4,-0.5);

\draw [line width = 0.5mm, blue] (0.5,-2) -- (-0.5,2);

\draw [line width = 0.5mm, blue] (-2,0) -- (2,0);

\node at (-1.3,2) {$\ell_2 = \ell_5$};

\node at (1.2,0.8) {$\ell_1$};

\node at (2.4,0) {$\ell_4$};

\node at (1.5,-0.8) {$\ell_3$};


\end{tikzpicture}
\end{center}
\caption{A point in $X_{5,3}$.}
\label{fig:point}
\end{figure}

A point in the space $X_{5,3}$ is shown in \cref{fig:point}. We have an ordered quintuple of lines
through the origin which together have full span $\FF^3$.
We  leave $\FF$ general for now, but we  specialize to the finite field $\FF_q$
at the end of \cref{Orbit} and the complex field $\CC$ in 
\cref{Cellular,Cohomology,Stability}.

Let $\PP^{k-1}$ stand for the projective space of lines through the origin in $\FF^k$ and let 
$(\PP^{k-1})^n$ be its $n$-fold Cartesian product. The natural inclusion $X_{n,k} \subset (\PP^{k-1})^n$
realizes $X_{n,k}$ as a Zariski open subset of $(\PP^{k-1})^n$, and therefore a smooth complex manifold
when $\FF = \CC$.  

The set $X_{n,k}$ carries an action of the symmetric group $S_n$ by the rule
\begin{equation}
w.(\ell_1, \dots, \ell_n) := (\ell_{w(1)}, \dots, \ell_{w(n)})
\end{equation}
for all $w \in S_n$ and $(\ell_1, \dots, \ell_n) \in X_{n,k}$. When $\FF = \CC$, this action is continuous and
so endows the (singular, integral) cohomology ring $H^{\bullet}(X_{n,k})$ with the 
structure of a graded $S_n$-module.

We view our moduli space $X_{n,k}$ as a generalization of the flag variety. To justify this, 
observe that when $k = n$, we have a natural surjection 
\begin{equation}
X_{n,n} = GL_n/T \twoheadrightarrow GL_n/B = \Fl(n),
\end{equation}
where $T \subseteq GL_n$ is the diagonal torus. When $\FF = \CC$, this
is a homotopy equivalence 
\footnote{It is a fiber bundle over a Hausdorff base space whose fiber --
homeomorphic to the group of upper triangular matrices with $1$'s on the diagonal -- is contractible.},
so that $X_{n,n}$ agrees with $\Fl(n)$ up to homotopy and $H^{\bullet}(X_{n,n}) = H^{\bullet}(\Fl(n))$.
At the other extreme, the space $X_{n,1} = \{*\}$ is a single point.

%{\bf The geometry of $\Fl(n)$ is governed by the combinatorics of  $S_n$. }
%Ehresmann \cite{Ehresmann} proved that $\Fl(n)$ has a CW decomposition indexed by the {\em (open) Schubert cells}
%$X_w := BwB/B$, where $w$ ranges over  $S_n$.
%Borel \cite{Borel} presented
%$H^{\bullet}(\Fl(n)) = R_n$,
%where $R_n := \ZZ[x_1, \dots, x_n]/\langle e_1, \dots, e_n \rangle$ is the {\em coinvariant ring};
%here $e_d$ is the degree $d$ elementary 
%symmetric polynomial.
%Here the variable $x_i$ represents the Chern class $-c_1(V_i/V_{i-1})$.
%Lascoux and Sch\"utzenberger \cite{LS}
%showed that the class $[\overline{X_w}]$ is represented by the 
%Schubert polynomial $\symm_{w^{\vee}} \in \ZZ[x_1, \dots, x_n]$; here
%$w^{\vee}(i) = n - w(i) + 1$.

%An {\em ordered set partition} is a set partition with a total order on its blocks.  As an example,
%$(2 \, 4  \mid 1 \, 3 \, 6 \mid 5)$ is an ordered set partition of $[6]$ with $3$ blocks. Let $\OP_{n,k}$ be the 
%family of ordered set partitions of $[n]$ with $k$ blocks. We have $|\OP_{n,k}| = k! \cdot \Stir(n,k)$,
%where $\Stir(n,k)$ is the Stirling number of the second kind counting $k$-block set partitions of $[n]$.
%The set $\WWW_{n,k}$ of {\em Fubini words}  is the family 
%of length $n$ words $w_1 \dots w_n$  over the alphabet $[k] := \{1, 2, \dots, k\}$ in which every letter
%in $[k]$ appears.
%When $k = n$, the  sets 
%$\WWW_{n,n}$ and $S_n$ coincide. The set $\WWW_{n,k}$ is in bijection with the family of ordered 
%set partitions of $[n]$ into $k$ blocks.

%For positive integers $k \leq n$,
%we will see that 
%{\bf the geometry of $X_{n,k}$ is governed by the combinatorics of $\WWW_{n,k}$.}
%We prove  extensions of the results of \cite{Borel, Ehresmann, LS} to this setting.
%The space $X_{n,k}$ admits an affine paving whose cells $C_w$ are indexed by Fubini words
%$w \in \WWW_{n,k}$.
%The cohomology of $X_{n,k}$ is given by a quotient $R_{n,k}$ of $\ZZ[x_1, \dots, x_n]$ introduced  \cite{HRS}
%by Haglund, Rhoades, and Shimozono
%\footnote{more precisely, a $\ZZ$-version thereof}
%which generalizes the coinvariant ring $R_n$.
%We describe a variant $\symm_w$ of the Schubert polynomials indexed by 
%Fubini words $w \in \WWW_{n,k}$ such that $[\overline{C_w}]$ is represented by $\symm_w$.

%We relate our results to the theory of Macdonald polynomials. The {\em Delta Conjecture} \cite{HRW}
%is a still-open generalization of the {\em Shuffle Theorem}  \cite{CarlssonMellit}.
%Our results show that $X_{n,k}$ plays the same role for the Delta Conjecture as the 
%flag variety $\Fl(n)$ plays for the Shuffle Theorem.
%We also describe various stability properties enjoyed by our spaces $X_{n,k}$ as the pair
%$(n,k)$ grows in  two natural ways subject to the condition $k \leq n$: the growth
%rule $(n,k) \leadsto (n+1,k)$ and the growth rule $(n,k) \leadsto (n+1,k+1)$.















\section{The orbit set $X_{n,k}$}
\label{Orbit}

In order to understand the combinatorics of $X_{n,k}$ and the geometry of its embedding 
inside $(\PP^{k-1})^n$, we will need  matrices.  If $\Mat_{k \times n}$ is the affine
space of $k \times n$ matrices over $\FF$, we introduce the Zariski open subsets 
$\UUU_{n,k} \subseteq \VVV_{n,k}$ by 
\begin{align}
\UUU_{n,k} &:= \{ A \in \Mat_{k \times n} \,:\, \text{$A$ has no zero columns and has full rank} \}, \\
\VVV_{n,k} &:= \{ A \in \Mat_{k \times n} \,:\, \text{$A$ has no zero columns} \}. 
\end{align}

Let $T \subseteq GL_n$ be the diagonal subgroup and let $U \subseteq GL_k$ be the group of {\em lower}
triangular $k \times k$ matrices with $1$'s on the diagonal. The product group $U \times T$ acts on 
both $\UUU_{n,k}$ and $\VVV_{n,k}$ by the rule $(u,t).A := uAt$ for all $(u,t) \in U \times T$.  
We have the  orbit set identifications
$X_{n,k} = \UUU_{n,k}/T$ and $(\PP^{k-1})^n = \VVV_{n,k}/T$.

%There holds the following result of linear algebra.

\begin{prop}
\label{action-is-free}
The action of $U \times T$ on  the set $\UUU_{n,k}$ is free.
\end{prop}

What do the $U \times T$-orbits in $\UUU_{n,k}$ look like?
%To answer this question, we will need some notation.
Given any length $n$ word
$w = w_1 \dots w_n$, a position $1 \leq j \leq n$ is {\em initial} if $w_j$ is the first occurrence of its letter.
Let $\initial(w)$ be the set of initial positions, so that $\initial(2331231) = \{1,2,4\}$.
If $w \in \WWW_{n,k}$ is Fubini, the {\em pattern matrix}
$\PM(w)$ is the $k \times n$ matrix with entries in $\{0, 1, \star \}$ whose entries 
$\PM(w)_{i,j}$ (for $1 \leq i \leq k$ and $1 \leq j \leq n$) are as follows.
%The rule for constructing the $j^{th}$ column of $\PM(w)$ depends on whether $j$ is initial in $w = w_1 \dots w_n$.
\begin{itemize}
\item  We have $\PM(w)_{i,j} = 1$ if and only if $w_j = i$.
\item  Suppose $j \in \initial(w)$ is an initial position of $w$ and $w_j \neq i$.  If $w_j > i$ and there exists $j' < j$ with
$w_{j'} = i$ then $\PM(w)_{i,j} = \star$.  Otherwise $\PM(w)_{i,j} = 0$.
\item  Suppose $j \notin \initial(w)$ is not an initial position of $w$ and $w_j \neq i$.  
If the first occurrence of $i$ in $w = w_1 \dots w_n$ is before the first occurrence of 
$w_j$ in $w = w_1 \dots w_n$ then $\PM(w)_{i,j} = \star$.
Otherwise $\PM(w)_{i,j} = 0$.
\end{itemize}
In our example $w = 2331231 \in \WWW_{7,3}$ the pattern matrix is
\begin{equation*}
\PM(w) = \PM(2331231) = 
\begin{pmatrix}
0 & 0 & 0 & 1 & 0 & 0 & 1 \\
1 & \star & \star & 0 & 1 & \star & \star \\
0 & 1 & 1 & 0 & 0 & 1 & \star
\end{pmatrix}.
\end{equation*}
The {\em dimension} $\dim(w)$ of a Fubini word $w \in \WWW_{n,k}$ is the number of $\star$'s in its pattern
matrix, so that $\dim(2331231) = 5$.

A matrix $A \in \UUU_{n,k}$ {\em fits the pattern} of a Fubini word $w \in \WWW_{n,k}$ if $A$ can 
be obtained by replacing the $\star$'s in $\PM(w)$ with field elements. The following is another application
of linear algebra.

\begin{prop}
\label{orbit-representative}
For any $U \times T$-orbit $\mathcal{O}$ in $\UUU_{n,k}$, there exists a unique Fubini word $w \in \WWW_{n,k}$
and a unique matrix $A$ which fits the pattern of $w$ such that $A \in \mathcal{O}$.
\end{prop}

\cref{action-is-free,orbit-representative} yield a disjoint union decomposition of $X_{n,k}$.
Let $\widehat{C}_w \subseteq \UUU_{n,k}$ be the set of matrices which fit the pattern of a Fubini word
$w \in \WWW_{n,k}$; this is an affine space of dimension $\dim(w)$. Define $C_w \subseteq X_{n,k}$ by
\begin{equation}
C_w := \text{image of $U \widehat{C}_w$ in $X_{n,k}$}.
\end{equation}
We have
\begin{equation}
\label{disjoint-union-decomposition}
X_{n,k} = \bigsqcup_{w \in \WWW_{n,k}} C_w.
\end{equation}

There is an enumerative result over the finite field $\FF_q$.
Recall the $q$-analogs
\begin{equation}
[n]_q := 1 + q + \cdots + q^{n-1}, \quad [n]!_q := [n]_q [n-1]_q \cdots [1]_q, \quad
{n \brack k}_q := \frac{[n]!_q}{[k]!_q [n-k]!_q}.
\end{equation}
The {\em $q$-Stirling number} $\Stir_q(n,k)$ is defined recursively by $\Stir_q(0,k) = \delta_{0,k}$ and 
\begin{equation}
\Stir_q(n,k) = \Stir_q(n-1,k-1) + [k]_q \cdot \Stir_q(n-1,k).
\end{equation}
The polynomial $[k]!_q \cdot \Stir_q(n,k)$ is called the {\em Mahonian distribution} on $\WWW_{n,k}$.
Any statistic $\mathrm{stat}: \WWW_{n,k} \rightarrow \ZZ_{\geq 0}$ which satisfies
$\sum_{w \in \WWW_{n,k}} q^{\mathrm{stat}(w)} = [k]!_q \cdot \Stir_q(n,k)$ is called a {\em Mahonian statistic};
see \cite{BilleyCoskun, RemmelWilson, RhoadesOSP} for examples.

\begin{prop}
\label{dim-is-mahonian}
The dimension statistic $\dim$ is Mahonian.
\end{prop}

\cref{action-is-free,orbit-representative,dim-is-mahonian} 
combine to yield the following interpretation of the Mahonian distribution on $\WWW_{n,k}$ in terms of 
finite fields. It is our analog of the result that the number of flags in $\FF_q^n$ is $[n]!_q$.

\begin{cor}
\label{mahonian-interpretation}
Let $q$ be a prime power. Over the field $\FF_q$ with $q$ elements, there are 
$[k]!_q \cdot \Stir_q(n,k)$ orbits in the $U \times T$-set $\UUU_{n,k}$.
\end{cor}

Billey and Coskun \cite{BilleyCoskun} relate the Mahonian distribution on $\WWW_{n,k}$
to  {\em rank varieties}. The authors do not know a geometric connection between
rank varieties and $X_{n,k}$.
















\section{A cellular decomposition and the Poincar\'e series of $X_{n,k}$}
\label{Cellular}

{\bf For the rest of the extended abstract, we work over the complex field $\CC$.}
We exploit the decomposition \eqref{disjoint-union-decomposition} 
to understand the geometry of $X_{n,k}$. 

Let $X$ be a complex algebraic variety. A {\em cellular decomposition} (a.k.a. {\em affine paving}) of $X$ 
is a filtration $X_{\bullet} = (X = X_0 \supset X_1 \supset \cdots \supset X_m = \varnothing)$ of $X$,
where each $X_i$ is a closed subvariety and each difference $X_i - X_{i+1}$ is nonempty and isomorphic
(as a variety) to a disjoint union of affine spaces.
If we express $X_i - X_{i+1} = \bigsqcup_j A_{ij}$ as such a disjoint union, the $A_{ij}$ are called the 
{\em cells} of the decomposition. We say that the partition of $X$ formed by the collection of all cells $\{ A_{ij} \}$ 
{\em induces} the decomposition $X_{\bullet}$.
The following generalizes Ehresmann's CW decomposition of $\Fl(n)$. 

\begin{thm}
\label{cellular-decomposition-of-x}
The set of cells $\{ C_w \,:\, w \in \WWW_{n,k} \}$ induces a cellular decomposition of $X_{n,k}$.
\end{thm}



\cref{cellular-decomposition-of-x} determines the structure of $H^{\bullet}(X_{n,k})$ as a graded
abelian group.  Let $X_{n,k}^+ = X_{n,k} \cup \{ \infty \}$ be the one-point compactification of $X_{n,k}$.
The {\em Borel-Moore homology} $\bar{H}_{\bullet}(X_{n,k})$ is the  homology of the pair
$(X_{n,k}^+, \{\infty\})$. By
\cref{cellular-decomposition-of-x},  $\bar{H}_d(X_{n,k})$ vanishes when $d$ is odd
and is free abelian with basis $\{ [ \overline{C_w} ] \,:\, w \in \WWW_{n,k}, \, 2 \cdot \dim(w) = d \}$ when 
$d$ is even.

The reader might ask whether the cellular decomposition of \cref{cellular-decomposition-of-x}
can be replaced by the less technical notion of a CW decomposition. 
This is impossible because the space $X_{n,k}$ is not compact. Indeed,
we will show that the Hilbert series of the cohomology ring $H^{\bullet}(X_{n,k})$ 
is not always palindromic (it equals $2q^4 + 3q^2 + 1$ when $n = 3$ and $k = 2$).
Since $X_{n,k}$ is smooth, this means that $X_{n,k}$ must be noncompact.
\footnote{The authors do not know whether the one-point compactification $X_{n,k}^+$ of $X_{n,k}$ admits
a finite CW structure given by the cells $\{ C_w \,:\, w \in \WWW_{n,k} \}$ 
together with an additional $0$-cell for the added point $\infty$.}

The variety $X_{n,k}$ is irreducible. To see this, observe that the (affine) cell 
$C_w$ for $w = 123 \dots kk \dots k \in \WWW_{n,k}$ is (Zariski) dense in $X_{n,k}$.
Poincar\'e duality asserts the isomorphism of abelian groups
$\bar{H}_d(X_{n,k}) \cong H^{\dim(X_{n,k}) - d}(X_{n,k})$.


\begin{thm}
\label{poincare-series}
Let $k \leq n$ be positive integers. The cohomology ring $H^{\bullet}(X_{n,k})$ is free abelian as a graded group,
with $\ZZ$-basis given by the classes $\{ [\overline{C_w}] \,:\, w \in \WWW_{n,k} \}$.
Furthermore, the Poincar\'e polynomial of $X_{n,k}$ is given by
\begin{equation}
\sum_{d \geq 0} \mathrm{rank}(H^d(X_{n,k})) \cdot q^d =
\rev_q( [k]!_{q^2} \cdot \Stir_{q^2}(n,k) ).
\end{equation}
In particular, $\mathrm{rank}(H^{\bullet}(X_{n,k})) = |\WWW_{n,k}| = k! \cdot \Stir(n,k)$ where
$\Stir(n,k)$ is the Stirling number of the second kind.
\end{thm}

%Theorem~\ref{poincare-series} is proven using Theorem~\ref{cellular-decomposition-of-x} 
%together with Proposition~\ref{dim-is-mahonian}.













\section{The cohomology of $X_{n,k}$}
\label{Cohomology}

\cref{poincare-series} describes the structure of $H^{\bullet}(X_{n,k})$ as a graded group.
We go further and present $H^{\bullet}(X_{n,k})$ as a graded ring.
The first step is an extension of the cellular decomposition of \cref{cellular-decomposition-of-x}
from $X_{n,k}$ to the larger space $(\PP^{k-1})^n$.

Recall that the space $\VVV_{n,k}$ of $k \times n$ matrices with no zero columns carries an action
of the product group $U \times T$.  Let $w = w_1 \dots w_n$ be an arbitrary word in $[k]^n$
(which may not be Fubini).  
The  notion of `pattern matrix' may be extended to 
define $\PM(w)$ as the $k \times n$ matrix over $\{0,1,\star\}$ whose entries 
are the same as in the Fubini case, except that any index $1 \leq i \leq n$ which does not appear indexes a row of 
zeros. We refer the reader to \cite[Sec. 5]{PR} for a more precise definition.
As an example, if $k = 4$ we have
\begin{equation*}
\PM(441121) = \begin{pmatrix}
0 & 0 & 1 & 1 & \star & 1 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & \star & 0 & \star 
\end{pmatrix}.
\end{equation*}


As before, we say that a $k \times n$ matrix {\em fits the pattern} of a word $w \in [k]^n$
if it can be obtained by replacing the $\star$'s in $\PM(w)$ with complex numbers.
If $\widehat{C}_w$ is the set of matrices which fit the pattern of $w$, we define
$C_w \subseteq (\PP^{k-1})^n$ by the rule
\begin{equation}
C_w := \text{image of $U \widehat{C}_w$ in $(\PP^{k-1})^n$}.
\end{equation}
\cref{orbit-representative} extends to the $U \times T$-set $\VVV_{n,k}$ to give the disjoint
union decomposition
\begin{equation}
\label{projective-space-decomposition}
(\PP^{k-1})^n = \bigsqcup_{w \in [k]^n} C_w.
\end{equation}

The decomposition \eqref{projective-space-decomposition} of $(\PP^{k-1})^n$ extends the decomposition
\eqref{disjoint-union-decomposition} of $X_{n,k}$ set theoretically. This statement can be strengthened 
to cellular decompositions as follows.

\begin{lem}
\label{initial-lemma}
There is a cellular decomposition 
$X_{\bullet} = (X_0 \supset X_1 \supset \cdots \supset X_m)$ of $(\PP^{k-1})^n$ with cells 
$\{ C_w \,:\, w \in [k]^n \}$ such that 
$X_i = \bigsqcup_{w \in [k]^n - \WWW_{n,k}} C_w = (\PP^{k-1})^n - X_{n,k}$ for some $0 \leq i \leq m$.
\end{lem}

Let $\iota: X_{n,k} \hookrightarrow (\PP^{k-1})^n$ be the inclusion map.
\cref{initial-lemma} and the general theory of cellular decompositions imply that 
the induced map $\iota^*: H^{\bullet}((\PP^{k-1})^n) \twoheadrightarrow H^{\bullet}(X_{n,k})$ is surjective.
In fact, if $J_{n,k} \subseteq H^{\bullet}((\PP^{k-1})^n)$ is the ideal generated by the classes of cell closures
$\{ [\overline{C_w}] \,:\, w \in [k]^n - \WWW_{n,k} \}$ corresponding to non-Fubini words, then $\iota^*$ induces an 
isomorphism of graded rings
\begin{equation}
\label{quotient-cohomology-isomorphism}
H^{\bullet}(X_{n,k}) \cong H^{\bullet}((\PP^{k-1})^n)/J_{n,k}.
\end{equation}
To exploit the isomorphism \eqref{quotient-cohomology-isomorphism} and present the cohomology
of $X_{n,k}$, we need a better understanding of the classes $[\overline{C_w}]$ inside $H^{\bullet}((\PP^{k-1})^n)$.


For $1 \leq i \leq n-1$, the {\em divided difference operator} $\partial_i$ on $\ZZ[x_1, \dots, x_n]$ is given by
\begin{equation}
\partial_i: f(x_1, \dots, x_n) \mapsto
\frac{f(x_1, \dots, x_i, x_{i+1}, \dots, x_n) - f(x_1, \dots, x_{i+1}, x_i, \dots, x_n)}{x_i - x_{i+1}}.
\end{equation}
{\em Schubert polynomials} $\{ \symm_w \,:\, w \in S_n \}$ are defined recursively by 
 $\symm_{w_0} = x_1^{n-1} x_2^{n-2} \cdots x_{n-1}^1 x_n^0$ when 
$w_0 = n(n-1) \dots 1$ and 
\begin{equation}
\symm_{w_1 \dots w_{i+1} w_i \dots w_n} = \partial_i(\symm_{w_1 \dots w_i w_{i+1} \dots w_n})
\quad \text{when $w_i > w_{i+1}$}.
\end{equation}

A word  is  {\em convex} if it does not have a subword of the form
$\dots i \dots j \dots i \dots $ for $i \neq j$.
If $w = w_1 \dots w_n \in [k]^n$, the {\em convexification} $\conv(w)$ is the unique convex word with the 
same letter multiplicities as $w$ in which the initial letters appear in the same order.
We let $\sigma(w) \in S_n$ be the unique Bruhat-minimal permutation such that $\sigma(w).\conv(w) = w$.
For example, if $w = 215235 \in [5]^6$ then $\conv(w) = 221553$ so that
$\sigma(w) = 142365 \in S_6$.  

Let $w = w_1 \dots w_n \in [k]^n$ be a word with $m$ distinct letters.
The {\em standardization}
$\st(w) \in S_{n+k-m}$ is given by
replacing the letters in noninitial positions of $w$ from left to right with
$k+1, k+2, \dots, k+n-m$, and then appending the letters in $[k]$ which do not appear in $w$ 
to the end in increasing order.
For example, if $w = 215235 \in [5]^6$ (so that $m = 4$) then $\st(w) = 2156374 \in S_7$.
We extend the Schubert polynomials to words as follows.

\begin{defn}
\label{extended-schubert-definition}
Let $k \leq n$ be positive integers and let $w \in [k]^n$ be a word. Define a polynomial
$\symm_w \in \ZZ[x_1, \dots, x_n]$ by
\begin{equation}
\symm_w := \sigma(w)^{-1}.\symm_{\st(\conv(w))}.
\end{equation}
\end{defn}

%It can be shown that $\symm_{\st(\conv(w))}$ only involves the variables $x_1, \dots, x_n$, so that the polynomial
%$\symm_w$ in Definition~\ref{extended-schubert-definition} is a well-defined polynomial. 

For $1 \leq i \leq n$, let $\ell_i$ be the $i^{th}$ tautological line bundle over the projective space
product $(\PP^{k-1})^n$. By the K\"unneth Theorem, we have the presentation
\begin{equation}
\label{projective-cohomology-presentation}
H^{\bullet}((\PP^{k-1})^n) = \ZZ[x_1, \dots, x_n]/\langle x_1^k, \dots, x_n^k \rangle,
\end{equation}
where $x_i$ represents the Chern class $c_1(\ell_i^*) \in H^{2}((\PP^{k-1})^n)$ (and so
$\deg(x_i) = 2$).
This presentation interacts with the cellular decomposition of \cref{initial-lemma} as follows;
the proof uses Fulton's theory of {\em degeneracy loci} \cite{FultonFlag}.

\begin{lem}
\label{schuberts-represent}
Let $k \leq n$ and let $w \in [k]^n$ be a word. The class $[\overline{C_w}] \in H^{\bullet}((\PP^{k-1})^n)$ 
is represented by the polynomial $\symm_w$ under the
presentation \eqref{projective-cohomology-presentation}.
\end{lem}

%The polynomial ring quotient avatar of the cohomology of $X_{n,k}$ is as follows.

The connection between $X_{n,k}$, the ring $R_{n,k}$ of Haglund, Rhoades, and Shimozono, 
and the Delta Conjecture is as follows.

\begin{thm}
\label{cohomology-of-x}
Let $k \leq n$ be positive integers. The cohomology of $X_{n,k}$ may be presented as
\begin{equation}
\label{x-cohomology-presentation}
H^{\bullet}(X_{n,k}) = R_{n,k}.
\end{equation}
Under this presentation, the variable $x_i$ represents the Chern class $c_1(\ell_i^*) \in H^2(X_{n,k})$, where
$\ell_i \twoheadrightarrow X_{n,k}$ is the $i^{th}$ tautological line bundle. 
If $w \in \WWW_{n,k}$, the class $[\overline{C_w}] \in H^{\bullet}(X_{n,k})$ is represented 
by the polynomial $\symm_w$ of \cref{extended-schubert-definition}.
\end{thm}

\begin{proof} (Sketch.)
Applying \cref{initial-lemma,schuberts-represent}, we have the
presentation
\begin{equation}
\label{first-presentation}
H^{\bullet}(X_{n,k}) = \ZZ[x_1, \dots, x_n]/K_{n,k},
\end{equation}
where $K_{n,k} := \langle \symm_w \,:\, w \in [k]^n - \WWW_{n,k} \rangle + \langle x_1^k, \dots, x_n^k \rangle$.
For $1 \leq i \leq k$, let $w^i \in [k]^n$ be the unique weakly increasing word with letters $[k] - \{ i \}$ whose 
first $k-1$ letters are distinct.  For example, the word $w^3 \in [6]^7$ is $w^3 = 1245666$.
Then $w^i$ is not Fubini, so that $\symm_{w^i}$ is a generator of $K_{n,k}$. One shows that 
$\symm_{w^i} = e_{n-i+1}$, so that we have $I_{n,k} \subseteq K_{n,k}$.

The containment $I_{n,k} \subseteq K_{n,k}$ of ideals means that we have a canonical surjection of rings
\begin{equation}
\label{second-presentation}
\pi:  R_{n,k} = \ZZ[x_1, \dots, x_n]/I_{n,k}  \twoheadrightarrow  \ZZ[x_1, \dots, x_n]/K_{n,k} =
 H^{\bullet}(X_{n,k}).
\end{equation}
By \cref{poincare-series},  the target of $\pi$ is a free $\ZZ$-module of rank
$k! \cdot \Stir(n,k)$.  One shows that the domain
$R_{n,k}$ is also a free $\ZZ$-module of rank $k! \cdot \Stir(n,k)$; 
{\em Demazure characters} play a key role in this argument.

Since any surjection between $\ZZ$-modules of the same finite rank is an isomorphism, the map
$\pi$ is an isomorphism of rings and
 \eqref{x-cohomology-presentation} is proven. 
The remainder of the theorem comes from the corresponding statements about $(\PP^{k-1})^n$.
\end{proof}

Line permutation endows the rational cohomology ring $H^{\bullet}(X_{n,k};\QQ)$ with the structure
of a graded $S_n$-module which is 
concentrated in even degree. \cref{cohomology-of-x} implies that 
\begin{equation}
\grFrob(H^{\bullet}(X_{n,k}; \QQ); \sqrt{q}) = 
\grFrob(R_{n,k}^{\QQ}; q) = 
(\rev_q \circ \omega) C_{n,k}(\xx;q),
\end{equation}
justifying our assertion that $X_{n,k}$ is the flag variety for the Delta Conjecture.

Haglund, Rhoades, and Shimozono discovered extensions of various monomial bases
of $\QQ \otimes_{\ZZ} R_n$ to $\QQ \otimes_{\ZZ} R_{n,k}$. They asked \cite[Prob. 7.2]{HRS}
 for an extension of the 
Schubert basis; such an extension (valid over the integers) is given as follows.

\begin{cor}
\label{schubert-basis} 
The set $\{ \symm_w \,:\, w \in \WWW_{n,k} \}$ descends to a $\ZZ$-basis of $R_{n,k}$.
\end{cor}

The structure constants involved in the basis of \cref{schubert-basis} 
can in general be negative.









\section{Stability for $X_{n,k}$}
\label{Stability}

There are two  ways to grow a pair of integers $(n,k)$ subject to the condition $k \leq n$:
\begin{equation}
\label{growth-rules}
(n,k) \leadsto (n+1,k) \text{ and } (n,k) \leadsto (n+1,k+1).
\end{equation}
In this section we describe stability results for these two growth rules.
%In this section we describe representation theoretic and geometric stability results for these two growth rules.

Let $\lambda = (\lambda_1, \lambda_2, \dots)$ be a partition and $n > 0$. If $n \geq |\lambda| + \lambda_1$, the 
{\em padded partition} is 
$\lambda[n] := (n - |\lambda|, \lambda_1, \lambda_2, \dots ) \vdash n$.
Any partition of $n$ has the form $\lambda[n]$ for a unique partition $\lambda$, so that any
finite-dimensional $S_n$-module
$V$ has the form $V \cong \bigoplus_{\lambda} c_{\lambda} S^{\lambda[n]}$, where the direct sum is over
{\em all} partitions $\lambda$.

Let $(V_n)_{n > 0}$ be a sequence of finite-dimensional $S_n$-modules. For each $n > 0$ we can write
$V_n \cong \bigoplus_{\lambda} c_{\lambda, n} S^{\lambda[n]}$ for some unique integers $c_{\lambda,n}$.
We call the sequence $V_n$ {\em multiplicity stable} \cite{CEF} if for any partition $\lambda$, the 
sequence $c_{\lambda,n}$ is eventually constant.

\begin{thm}
\label{representation-stability}
Fix a cohomological degree $d$. Either of the module sequences
\begin{align}
\dots, \, \, H^d(X_{n-1,k}; \QQ),\, \, &H^d(X_{n,k};\QQ), \, \, H^d(X_{n+1,k};\QQ), \, \, \dots \quad  \text{or} \\
\dots, \,  \, H^d(X_{n-1,k-1}; \QQ),\,  \, &H^d(X_{n,k};\QQ),\, \, H^d(X_{n+1,k+1};\QQ), \, \, \dots
\end{align}
is multiplicity stable.
\end{thm}

\begin{proof} (Sketch.)
Both of these module sequences are identically zero when $d$ is odd, so assume $d = 2m$ is even.

Let $\SYT(n)$ be the family of standard Young tableaux with $n$ boxes. Given a tableau
$T \in \SYT(n)$, let $\des(T)$ be the number of descents in $T$ and let $\maj(T)$ be the major index of $T$.
Work of Haglund, Rhoades, and Shimozono \cite{HRS} yields the tableau formula
\begin{equation}
\grFrob( H^{\bullet}(X_{n,k}; \QQ); \sqrt{q}) = 
\sum_{T \in \SYT(n)} q^{\maj(T)} {n - \des(T) - 1 \brack n-k}_q s_{\mathrm{shape}(T)}
\end{equation}
for fixed $k \leq n$.  A tableau $T$ only contributes to this sum when $\des(T) < k$.
Since $m = d/2$ is fixed, the representation stability asserted in the theorem follows from the 
standard combinatorial interpretation of the $q$-binomial
${n - \des(T) - 1 \brack n-k}_q$ in terms of partitions inside a box of size
$(n-k) \times (k - \des(T) - 1)$.
\end{proof}

We also mention that there exist growth rules for Fubini words
$\WWW_{n,k} \rightarrow \WWW_{n+1,k}$ and $\WWW_{n,k} \rightarrow \WWW_{n+1,k+1}$
which give rise to stability results for the word Schubert polynomials $\symm_w$. 
The space constraints of this extended abstract preclude us from expanding on this, but
see \cite{PR} for more
information.

%Let $w \in \WWW_{n,k}$ be a Fubini word. We have two larger Fubini words
%$w \circledast 1 \in \WWW_{n+1,k}$ and
%$1 \times w \in \WWW_{n+1,k+1}$
%corresponding to the growth rules \eqref{growth-rules}.  Explicitly, 
%the word $w \circledast 1$ is obtained by appending the last initial letter of $w$ to the end of $w$ while the
%word $1 \times w$ is obtained by increasing every letter in $w$ by $1$ and then prepending a $1$ to the 
%beginning.  For example, we have
%\begin{equation*}
%3313424 \circledast 1 = 33134242 \quad \text{and} \quad
%1 \times 3313424 = 14424535.
%\end{equation*}

%\begin{thm}
%Let $k \leq n$ be positive integers, let $w \in \WWW_{n,k}$ be a Fubini word.  We have
%\begin{equation}
%\symm_{w \circledast 1} = \symm_w \quad \text{and} \quad
%\widetilde{\symm_{1 \times w}} \mid_{x_{n+1} = 0} = \widetilde{\symm_w},
%\end{equation}
%where if $f = f(x_1, \dots, x_n)$ is any polynomial in an ordered variable set,
%$\widetilde{f} := f(x_n, \dots, x_1)$ is obtained through variable reversal.
%\end{thm}

%\begin{proof}
%(Sketch.) This result can be proven in two ways. It is possible to argue combinatorially using  
% Definition~\ref{extended-schubert-definition}.
%On the other hand, we have closed embeddings
%\begin{equation}
%\varphi: X_{n,k} \hookrightarrow X_{n+1,k} \quad \text{and} \quad
%\psi: X_{n,k} \hookrightarrow X_{n+1,k+1}
%\end{equation}
%given by $\varphi: (\ell_1, \dots, \ell_n) \mapsto (\ell_1, \dots, \ell_n, {\bf 1})$ and
%$\psi: (\ell_1, \dots, \ell_n) \mapsto ({\bf 1}, i(\ell_1), \dots, i(\ell_n))$.
%Here ${\bf 1}$ is the line of constant vectors and $i: \CC^k \hookrightarrow \CC^{k+1}$ is the inclusion
%$(z_1, \dots, z_n)^T \mapsto (0, z_1, \dots, z_n)^T$.
%Via these embeddings  $C_{w \circledast 1} \cap X_{n,k} = C_w$ and
%$C_{1 \times w} \cap X_{n,k} = C_w$. Now apply Theorem~\ref{cohomology-of-x}.
%\end{proof}





\acknowledgements{The authors are grateful to Sara Billey, Thomas Lam, and 
Vic Reiner for helpful conversations. B. Rhoades was partially supported by NSF
Grant DMS-1500838. The authors thank two anonymous referees
for their careful reading of the manuscript and valuable suggestions which improved
our exposition.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way

\sloppy
\printbibliography


%\bibliographystyle{plain}
%\bibliography{sample}

% \begin{thebibliography}{99}
 
 
% \bibitem{Artin}  E. Artin.  {\it Galois Theory}, Second edition.  Notre Dame Math Lectures, no. 2.
% Notre Dame: University of Notre Dame, 1944.
 
% \bibitem{BC}  S. Billey and I. Coskun.  Singularities of generalized Richardson varieties.
% {\it Comm. Alg.}, {\bf 40 (4)} (2012), 1466--1495.
 
% \bibitem{Borel}  A. Borel.  Sur la cohomologie des espaces fibr\'es principaux et des espaces
% homog\`enes de groupes de Lie compacts.  {\it Ann. of Math.}, {\bf 57} (1953), 115--207.
 
% \bibitem{CarlssonMellit}  E. Carlsson and A. Mellit.  A proof of the shuffle conjecture.  
% {\it J. Amer. Math. Soc.}, {\bf 31} (2018), 661--697.
 
% \bibitem{Ehresmann} C. Ehresmann. Sur la topologie de certains espaces homog\`enes.
% {\it Ann. Math.}, {\bf 35} (1934), 396--443.
 
% \bibitem{FultonIntersection}  W. Fulton.  {\it Intersection Theory.}  Springer-Verlag, New York, 1984.

%\bibitem{FultonFlag}  W. Fulton.  Flags, Schubert polynomials, degeneracy loci, and determinantal formulas.
%{\it Duke Math. J.}, {\bf 65} (1992), 381--420.
 
% \bibitem{Garsia}  A. M. Garsia.  Combinatorial methods in the theory of Cohen-Macaulay rings.
% {\it Adv. Math.}, {\bf 38} (1980), 229--266.
 
% \bibitem{GHRY}  A. M. Garsia, J. Haglund, J. Remmel, and M. Yoo.
% A proof of the Delta Conjecture when $q = 0$.
% Preprint, 2017.
% {\tt arXiv:1710.07078}.
 
% \bibitem{GarsiaStanton}  A. M. Garsia and D. Stanton.  Group actions on Stanley-Reisner rings
% and invariants of permutation groups.  {\it Adv. Math.}, {\bf 51 (2)} (1984), 107--201.
 
 
% \bibitem{HRW}  J. Haglund, J. Remmel, and A. T. Wilson.  The Delta Conjecture. 
% {\it Trans. Amer. Math. Soc.}, {\bf 370} (2018), 4029--4057.
 
%\bibitem{HRS}  J. Haglund, B. Rhoades, and M. Shimozono.  
%Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture.
%{\it Adv. Math.}, {\bf 329} (2018), 851--915.

%\bibitem{HRSSchur}  J. Haglund, B. Rhoades, and M. Shimozono.  
%Hall-Littlewood expansions associated to delta operators at $t = 0$.
%Submitted, 2018. {\tt arXiv:1801.08017}.

%\bibitem{LS} A. Lascoux and M.-P. Sch\"utzenberger. Polyn\^omes de Schubert.
%{\it Comptes Rendus de l'Acad\'emie des Sciences, S\'erie I},
%{\bf 294 (13)}, 447--450.



%\bibitem{RemmelWilson} J. Remmel and A. T. Wilson.  An extension of MacMahon's Equidistribution
%Theorem to ordered set partitions.  {\it J. Combin. Theory Ser. A}, {\bf 134} (2015), 242--277.

%\bibitem{RhoadesOSP}  B. Rhoades.  Ordered set partition statistics and the Delta Conjecture.
%{\it J. Combin. Theory Ser. A}, {\bf 154} (2018), 172--217.

%\bibitem{Stanley}  R. P. Stanley.  Invariants of finite groups and their applications to combinatorics.
%{\it Bull. Amer. Math. Soc.}, {\bf 1} (1979), 475--511.

%\bibitem{Stein} E. Steingr\'imsson.  Statistics on ordered partitions of sets.
%Preprint, 2017. {\tt arXiv:0605670}.

%\bibitem{Wilson} A. T. Wilson.  An extension of MacMahon's Equidistribution Theorem to ordered 
%multiset partitions.  {\it Electron. J. Combin.}, {\bf 23 (1)} (2016), P1.5.



%\end{thebibliography}

\end{document}
