%% 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]{FPSAC2020}
\articlenumber{57}
\addbibresource{fpsac.bib}
%% 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}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem*{fconj}{Fields Conjecture}
\newtheorem*{zconj}{Zabrocki Conjecture}
\newtheorem*{leit}{Leitmotif}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Defintion}
\newtheorem{prob}[thm]{Problem}
\newcommand{\aaa}{\mathbf{a}}
\newcommand{\xxx}{\mathbf{x}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\OP}{\mathcal{OP}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\Gr}{\mathrm{Gr}}
\newcommand{\WW}{\mathbb{W}}
\newcommand{\grFrob}{\mathrm{grFrob}}
\newcommand{\Frob}{\mathrm{Frob}}
\newcommand{\Rise}{\mathrm{Rise}}
\newcommand{\Val}{\mathrm{Val}}
\newcommand{\Fl}{\mathcal{F \ell}}
\newcommand{\maj}{\mathrm{maj}}
\newcommand{\des}{\mathrm{des}}
\newcommand{\shape}{\mathrm{shape}}
\newcommand{\Stir}{\mathrm{Stir}}
\newcommand{\Hilb}{\mathrm{Hilb}}
\newcommand{\Park}{\mathrm{Park}}
\newcommand{\Conf}{\mathrm{Conf}}
\newcommand{\symm}{\mathfrak{S}}
\newcommand{\one}{\mathbb{1}}
\newcommand{\FI}{\mathsf{FI}}
\newcommand{\coFI}{\mathsf{coFI}}
\newcommand{\Vect}{\mathsf{Vect}}
\usepackage{tikz, mathtools, young}
%% define your title in the usual way
\title{Spanning configurations and matroidal representation stability}
%% 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},
Eric Ramos\thanks{\href{mailto:eramos@uoregon.edu}{eramos@uoregon.edu}. Partially supported by NSF grant DMS--1704811.}\addressmark{2}, \and
Brendon Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}.
Partially supported by NSF grant DMS--1500838.}\addressmark{3} }
%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Southern California, Los Angeles, CA,
USA \\
\addressmark{2}Department of Mathematics, University of Oregon, Eugene, OR, USA \\
\addressmark{3}Department of Mathematics, University of California, San Diego, La Jolla, CA,
USA}
%% 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{Let $V_1, V_2, \dots $ be a sequence of vector spaces where $V_n$ carries an action of $\symm_n$
for each $n$. {\em Representation stability} describes when the
sequence $V_n$ has a limit. An important source of stability arises when $V_n$ is the
$d^{th}$ homology group (for fixed $d$) of the configuration space of $n$ distinct points in some topological
space $X$. We replace these configuration spaces with the variety $X_{n,k}$ of {\em spanning configurations}
of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^k$ with $\ell_1 + \cdots + \ell_n = \CC^k$
as vector spaces.
That is, we replace the configuration space condition of {\em distinctness} with the matroidal
condition of {\em spanning}.
We study stability phenomena for the homology groups $H_d(X_{n,k})$ as the
parameter $(n,k)$ grows. We also study stability phenomena for a family of multigraded
modules related to the Delta Conjecture.}
%% 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{symmetric group module, representation stability, subspace configuration}
%% 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
%% 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 and main result}
\label{Introduction}
Suppose that for each $n \geq 1$, we have a representation $V_n$ of the symmetric group $\symm_n$.
\footnote{We only consider finite-dimensional representations over $\QQ$.}
What does it mean for the sequence $V_1, V_2, V_3, \dots$ to converge?
Representation stability is an answer to this question.
We regard $\symm_n$ as the subgroup of permutations in $\symm_{n+1}$ which fix $n+1$, so
any $\symm_{n+1}$-module is also an $\symm_n$-module.
\begin{defn}
\label{representation-stability-defn} Let
$(V_n)_{n \geq 1}$ be a sequence of $\symm_n$-modules and let $f_n: V_n \rightarrow V_{n+1}$
be a sequence of linear maps. $V_n$ is
{\em (uniformly) representation stable} with respect to $f_n$ if
for $n \gg 0$
\begin{itemize}
\item the map $f_n$ is injective,
\item we have $f_n(w \cdot v) = w \cdot f_n(v)$ for all $w \in \symm_n$ and all $v \in V_n$,
\item the $\symm_{n+1}$-module generated by the image $f_n(V_n) \subseteq V_{n+1}$ is all of $V_{n+1}$, and
\item the transposition $(n+1,n+2) \in \symm_{n+2}$ acts trivially on the image of the composition
$(f_{n+1} \circ f_n)(V_n) \subseteq V_{n+2}$.
\end{itemize}
\end{defn}
Let $n \geq 0$.
A {\em partition} $\lambda$ of $n$ is a weakly decreasing sequence
$\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots)$
of positive integers with $\lambda_1 + \lambda_2 + \cdots = n$. We use $\lambda \vdash n$ to mean that $\lambda$
is a partition of $n$ and write $|\lambda| = n$ for the sum of the parts of $\lambda$.
There is a one-to-one correspondence between partitions $\lambda \vdash n$ and irreducible representations of
$\symm_n$; given $\lambda \vdash n$, let $S^{\lambda}$ be the corresponding $\symm_n$-irreducible.
If $\mu$ is any partition and $n \geq |\mu| + \mu_1$, the {\em padded partition} $\mu[n] \vdash n$ is given by
$\mu[n] := (n - |\mu|, \mu_1, \mu_2, \dots )$.
Any partition $\lambda \vdash n$ may be written uniquely as $\lambda = \mu[n]$ for some partition $\mu$:
if $\lambda = (\lambda_1, \lambda_2, \lambda_3, \dots )$ we have
$\mu = (\lambda_2, \lambda_3, \dots )$.
Let $(V_n)_{\geq 1}$ be a sequence of $\symm_n$-modules. Decomposing $V_n$ into irreducibles yields multiplicities
$m_{\mu,n} \geq 0$ such that
$V_n \cong \bigoplus_{\mu} m_{\mu,n} S^{\mu[n]},$
where the direct sum is over all partitions $\mu$.
\cref{representation-stability-defn} has the following combinatorial interpretation.
\begin{thm} (Church-Ellenberg-Farb \cite{CEF})
Let $(V_n)_{n \geq 1}$ be a sequence of $\symm_n$-modules and define
the multiplicities $m_{\mu,n}$ as above. The following are equivalent.
\begin{enumerate}
\item The sequence $(V_n)_{n \geq 1}$ is representation stable with respect to some maps
$f_n: V_n \rightarrow V_{n+1}$.
\item There exists $N$ such that for any partition $\mu$ we have
$m_{\mu,n} = m_{\mu,N}$ for all $n \geq N$.
\end{enumerate}
\end{thm}
A famous geometric instance of representation stability comes from configuration spaces.
Let $X$ be a topological space and $n \geq 0$. The {\em configuration space}
$\Conf_n X$ is the set of all $n$-tuples $(x_1, \dots, x_n)$ of distinct points in $X$.
The set $\Conf_n X$ is topologized via its inclusion into the $n$-fold product $X \times \cdots \times X$.
A point in $\Conf_3 X$ where $X$ is the torus is shown on the left of \cref{three-points}.
Let $H_{\bullet}(\Conf_n X))$ be the homology of $\Conf_n X$ (singular
with rational coefficients). For any $d \geq 0$, the symmetric group
$\symm_n$ acts continuously on $\Conf_n X$ by point permutation and so endows the vector space
$H_d(\Conf_n X)$ with the structure of an $\symm_n$-module.
Many theorems in representation stability state that if $X$ is a `nice' space and $d > 0$, the
sequence $(H_d(\Conf_n X))_{n \geq 1}$ is representation stable (for example, see \cite{Church}).
\begin{figure}
\centering
\begin{tikzpicture}[scale = 0.6]
\draw (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[xscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[rotate=180] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[yscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw (-2,.2) .. controls (-1.5,-0.3) and (-1,-0.5) .. (0,-.5) .. controls (1,-0.5) and (1.5,-0.3) .. (2,0.2);
\draw (-1.75,0) .. controls (-1.5,0.3) and (-1,0.5) .. (0,.5) .. controls (1,0.5) and (1.5,0.3) .. (1.75,0);
\draw [gray, fill=gray!30] (6.5,0.5) -- (10.5,0.5) -- (9.5,-0.5) -- (5.5,-0.5) -- (6.5,0.5);
\node at (0,-1.2) {$ \bullet \, x_1$};
\node at (-2,0.6) {$\bullet \, x_2$};
\node at (2.5,0.7) {$\bullet \, x_3$};
%%%%%%%%
\draw (8,-2) -- (8,2);
\draw (6,0) -- (10,0);
\draw (8.5,0.5) -- (7.5,-0.5);
\draw [line width = 0.5mm, blue] (9,0.5) -- (7,-0.5);
\draw [line width = 0.5mm, blue] (6.6,0.5) -- (9.4,-0.5);
\draw [line width = 0.5mm, blue] (8.5,-2) -- (7.5,2);
\draw [line width = 0.5mm, blue] (6,0) -- (10,0);
\node at (6.2,2) {$\ell_2 = \ell_5$};
\node at (9.4,1) {$\ell_1$};
\node at (10.6,0) {$\ell_4$};
\node at (9.7,-1) {$\ell_3$};
%%%%%%
\draw [gray, fill=red!20] (13.5,0.5) -- (17.5,0.5) -- (16.5,-0.5) -- (12.5,-0.5) -- (13.5,0.5);
\draw [gray, fill=red!20] (15.5,2.5) -- (14.5,1.5) -- (14.5,-2.5) -- (15.5,-1.5) -- (15.5,2.5);
% \draw [red!40, fill=red!40] (12.5,0.5) -- (12.5,-0.5) -- (11.5,-0.5) -- (11.5, 0.5) -- (12.5,0.5);
\draw [gray] (15.5, -0.5) -- (14.5,-0.5);
\draw [red!20] (15.5, -0.5) -- (15.5,0);
\draw [red!20] (15.5, 0) -- (15.5,0.5);
\draw (15,-2) -- (15,2);
\draw (13,0) -- (17,0);
\draw (15.5,0.5) -- (14.5,-0.5);
%\draw [line width = 0.5mm, blue] (13,2) -- (11,-2);
\node at (17.6,-0.4) {$W_2$};
\node at (17, 2.3) {$W_1 = W_3$};
\end{tikzpicture}
\caption{A point configuration, a line configuration, and a 2-plane configuration.}
\label{three-points}
\end{figure}
In this paper we prove a new geometric family of {\bf matroidal representation stability} results where the
configuration space condition of {\bf distinctness} is replaced by the matroidal
condition of {\bf spanning}.
The key example is as follows. Given positive integers $k \leq n$, Pawlowski and
Rhoades \cite{PR} introduced the following space of spanning line configurations:
\begin{equation}
X_{n,k} := \{ (\ell_1, \dots, \ell_n) \,:\, \ell_i \subseteq \CC^k \text{ a $1$-dimensional subspace and }
\ell_1 + \cdots + \ell_n = \CC^k \}.
\end{equation}
A point in $X_{5,3}$ is shown in the middle of \cref{three-points}.
When $k = n$, the space $X_{n,k}$ is homotopy equivalent to the variety
$\mathcal{F \ell}_n$ of complete flags in $\CC^n$.
The group $\symm_n$ acts on $X_{n,k}$ by
$w.(\ell_1, \dots, \ell_n) := (\ell_{w(1)}, \dots, \ell_{w(n)})$ for all $w \in \symm_n$ and
$(\ell_1, \dots, \ell_n) \in X_{n,k}$.
This induces an action of $\symm_n$ on the homology group $H_d(X_{n,k})$ for each $d \geq 0$.
There are two natural ways to grow a pair $(n,k)$ subject to the condition $k \leq n$:
\begin{equation*}
(n,k) \leadsto (n+1,k) \quad \quad \text{and} \quad \quad (n,k) \leadsto (n+1,k+1).
\end{equation*}
Both of these growth rules leads to a stability result.
The following matroidal stability theorem will be proved in {\bf \cref{Geometric}}.
\begin{thm}
\label{line-stability-theorem}
Fix $d \geq 0$. The following sequences of modules are representation stable with respect to
some linear maps $f_n$:
\begin{enumerate}
\item $(H_d(X_{n,k}))_{n \geq 1}$ for $k \geq 0$ fixed, and
\item $(H_d(X_{n,n-m}))_{n \geq 1}$ for $m \geq 0$ fixed.
\end{enumerate}
Here we adopt the convention $X_{n,k} = \varnothing$ for $n < k$
or $k < 0$ so that $H_{d}(X_{n,k}) = 0$ in this case.
\end{thm}
Pawlowski and Rhoades \cite{PR} presented the rational cohomology of $X_{n,k}$ as
\begin{equation}
\label{cohomology-presentation}
H^{\bullet}(X_{n,k}) = \QQ[x_1, \dots, x_n] / \langle x_1^k, \dots, x_n^k, e_n ,e_{n-1}, \dots, e_{n-k+1} \rangle.
\end{equation}
Here $e_d$ is the degree $d$ elementary symmetric polynomial and $x_i$ represents the Chern class
$c_1(\ell_i^*) \in H^2(X_{n,k})$ of the line bundle $\ell_i^* \twoheadrightarrow X_{n,k}$.
By the Universal Coefficient Theorem, this also gives the homology of $X_{n,k}$.
The graded $\symm_n$-isomorphism type of the quotient
appearing in \eqref{cohomology-presentation} was calculated by
Haglund, Rhoades, and Shimozono \cite{HRS} in terms of statistics on standard Young tableaux.
The presentation \eqref{cohomology-presentation} of $H^{\bullet}(X_{n,k})$ in \cite{PR} and the
calculation of the graded $\symm_n$-module structure
of $H^{\bullet}(X_{n,k})$ in \cite{HRS} involved a substantial amount
of combinatorics, algebra, and geometry.
One might think that the proof of the stability result of \cref{line-stability-theorem}
would rely on these or other similarly difficult arguments,
but representation stability exhibits the following leitmotif.
\begin{leit}
It is often easier to show that a sequence $(V_n)_{n \geq 1}$ of $\symm_n$-modules is representation
stable then it is to calculate the $\symm_n$-isomorphism types of the $V_n$.
\end{leit}
Indeed, in {\bf \cref{Geometric}} we prove \cref{line-stability-theorem} using
only a geometric property of $X_{n,k}$ coming from linear algebra (a realization as a terminal part of
a nonstandard affine paving of the $n$-fold projective space product $\PP^{k-1} \times \cdots \times \PP^{k-1}$
discovered in \cite{PR})
and not relying on any explicit presentation of the (co)homology of $X_{n,k}$.
We will also illustrate our leitmotif for modules $V_n$ whose isomorphism types are unknown.
\begin{itemize}
\item In \cref{Higher} we generalize \cref{line-stability-theorem} to spanning configurations
of higher-dimensional subspaces; see the right of \cref{three-points} for a spanning configuration of three
$2$-places in $\CC^3$. The cohomology rings of these moduli spaces
were presented by Rhoades \cite{Rhoades}, but their graded $\symm_n$-module decomposition is
unknown. Our proof relies only on a nonstandard affine paving of a product of Grassmannians.
\item The space $X_{n,k}$ was introduced in \cite{PR} to give geometric context to the
Haglund-Remmel--Wilson {\em Delta Conjecture} \cite{HRW} in symmetric function theory.
Zabrocki \cite{Zabrocki}
and Rhoades-Wilson \cite{RW} defined multigraded $\symm_n$-modules and conjectured that their isomorphism
types are
given by
the Delta Conjecture.
In \cref{Coinvariant} we give stability results for a family of multigraded $\symm_n$-modules
including those studied in \cite{Zabrocki, RW}. There is not even a conjecture for the
multigraded $\symm_n$-isomorphism types of these modules.
\end{itemize}
\section{Background}
\label{Background}
Let $\Lambda$ be the ring of symmetric functions in the infinite variable set $\xxx = (x_1, x_2, \dots )$
over the ground field $\QQ(q,t)$. If $V$ is any $\symm_n$-module, there are unique multiplicities
$c_{\lambda}$ so that $V \cong \bigoplus_{\lambda \vdash n} c_{\lambda} S^{\lambda}$.
The {\em Frobenius image} $\Frob(V) \in \Lambda$ is the symmetric function
$\Frob(V) := \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}$, where $s_{\lambda}$ is the
Schur function.
We will consider (multi)graded $\symm_n$-modules. Suppose $V = \bigoplus_{i \geq 0} V_i$
is a graded $\symm_n$-module. The {\em graded Frobenius image}
of $V$ is $\grFrob(V;q) := \sum_{i \geq 0} q^i \cdot \Frob(V_i)$.
More generally, if $V = \bigoplus_{i,j} V_{i,j}$ (or $V = \bigoplus_{i,j,k} V_{i,j,k}$) is a doubly
(resp. triply) graded $\symm_n$-module, the multigraded Frobenius image is
$\grFrob(V;q,t) := \sum_{i,j} q^i t^j \cdot \Frob(V_{i,j})$
(resp. $\grFrob(V;q,t,z) := \sum_{i,j,k} q^i t^j z^k \cdot \Frob(V_{i,j,k})$).
For any symmetric function $F \in \Lambda$, the {\em (primed) delta operator}
$\Delta'_F: \Lambda \rightarrow \Lambda$ is defined as follows.
For any partition $\mu$, let $\widetilde{H}_{\mu}(\xxx;q,t)$ be the modified Macdonald symmetric function.
The set $\{ \widetilde{H}_{\mu}(\xxx;q,t) \,:\, \mu$ a partition$\}$ is a basis of $\Lambda$.
The operator $\Delta'_F$ is the Macdonald eigenoperator given by
\begin{equation}
\Delta'_F: \widetilde{H}_{\mu}(\xxx;q,t) \mapsto
F( \dots, q^{i-1} t^{j-1}, \dots ) \times \widetilde{H}_{\mu}(\xxx;q,t),
\end{equation}
where $(i,j)$ ranges over all cells $\neq (1,1)$ in the Young diagram of $\mu$.
For example, if $\mu = (3,2)$ we fill the cells of $\mu$ as follows
\begin{center}
\begin{young}
$ \cdot $ & $q$ & $q^2$ \cr
$t$ & $qt$
\end{young}
\end{center}
so that $\Delta'_F: \widetilde{H}_{(3,2)}(\xxx;q,t) \mapsto F(q,q^2,t,qt) \times \widetilde{H}_{(3,2)}(\xxx;q,t)$.
The {\em Delta Conjecture} of Haglund, Remmel, and Wilson \cite{HRW} predicts the monomial expansion
of $\Delta'_{e_{k-1}} e_n$ for $k \leq n$. It reads
\begin{equation}
\Delta'_{e_{k-1}} e_n = \Rise_{n,k}(\xxx;q,t) = \Val_{n,k}(\xxx;q,t),
\end{equation}
where $\Rise$ and $\Val$ are certain formal power series defined using lattice path combinatorics;
see \cite{HRW} for more details.
We now review category-theoretic material related to representation stability.
The notion of an $\FI$-module will allow us to prove that a sequence $(V_n)_{n \geq 1}$
is representation stable by embedding it in another sequence $(W_n)_{n \geq 1}$ for which
representation stability is known.
For $n \geq 1$, write $[n] := \{1, 2, \dots, n\}$
Let $\FI$ be the category consisting of
\begin{itemize}
\item the single object $[n]$ for each positive integer $n$, and
\item morphisms given by injective maps $f: [n] \hookrightarrow [m]$.
\end{itemize}
Let $\Vect$ be the category of finite-dimensional $\QQ$-vector spaces with morphisms given by arbitrary linear
maps.
An {\em $\FI$-module} is a covariant functor $V: \FI \rightarrow \Vect$. We write $V(n)$ for the
image of the object $[n]$ in $\FI$ under $V$. More explicitly, an $\FI$-module consists of a
finite-dimensional $\QQ$-vector space $V(n)$ for each $n \geq 1$ and a linear map
$V(f): V(n) \rightarrow V(m)$ associated to any injection $f: [n] \hookrightarrow [m]$ such that
\begin{itemize}
\item if $\mathrm{id}_{[n]}: [n] \rightarrow [n]$ is the identity, then
$V(\mathrm{id}_{[n]}): V(n) \rightarrow V(n)$ is the identity, and
\item if $f: [n] \hookrightarrow [m]$ and $g: [m] \hookrightarrow [p]$, then
$V(g \circ f) = V(g) \circ V(f)$.
\end{itemize}
If $V$ is an $\FI$-module, the vector space $V(n)$ is naturally a $\symm_n$-module for each $n \geq 1$.
As an example of an $\FI$-module, fix $d \geq 0$ and let
$\QQ[x_1, \dots, x_n]_d$ be the space of polynomials in $x_1, \dots, x_n$ which
are homogeneous of degree $d$. The assignment
$[n] \mapsto \QQ[x_1, \dots, x_n]_d$ is an $\FI$-module where
$f: [n] \hookrightarrow [m]$ is sent to the map
$\QQ[x_1, \dots, x_n]_d \rightarrow \QQ[x_1, \dots, x_m]_d$ defined on variables by
$x_i \mapsto x_{f(i)}$.
If $V, W: \FI \rightarrow \Vect$ are $\FI$-modules, we say that $W$ is a {\em submodule} of $V$
if $W(n) \subseteq V(n)$ for all $n$ and for any injection $f: [n] \hookrightarrow [m]$ the following diagram commutes
\begin{center}
\begin{tikzpicture}[scale = 0.8]
\node at (0,2) (A) {$V(n)$};
\node at (4,2) (B) {$V(m)$};
\node at (4,0) (D) {$W(m)$};
\node at (0,0) (C) {$W(n)$};
\draw[->] (A) -- (B);
\draw[->] (C) -- (D);
\draw[->] (C) -- (A);
\draw[->] (D) -- (B);
\node at (2,2.4) {\begin{footnotesize}$V(f)$\end{footnotesize}};
\node at (2,-0.4) {\begin{footnotesize}$W(f)$\end{footnotesize}};
\end{tikzpicture}
\end{center}
where the vertical arrows are inclusions. If $W$ is a submodule of $V$, we have a
quotient $\FI$-module $V/W: [n] \mapsto V(n)/W(n)$.
An $\FI$-module $V$ is {\em finitely-generated} if there is a finite subset $S \subseteq \bigsqcup_{n \geq 1} V(n)$
such that no proper submodule $W \subsetneq V$ contains every element of $S$.
The $\FI$-module $\QQ[x_1, \dots, x_n]_d$ described above is finitely-generated.
In fact, it is generated by the set of monomials
\begin{equation*}
S = \bigsqcup_{n \leq d}
\{ x_1^{a_1} \cdots x_n^{a_n} \,:\, a_1 + \cdots + a_n = d \} \subseteq
\bigsqcup_{n \leq d} \QQ[x_1, \dots, x_n]_d.
\end{equation*}
We also define the category $\coFI$ to be the opposite category to $\FI$. That is, the objects of $\coFI$
are the same as those in $\FI$ but the arrows are reversed.
A $\coFI$-module is a covariant functor $V: \coFI \rightarrow \Vect$.
Submodules, quotient modules, and finite generation are defined as in the setting of $\FI$-modules.
We state two key results about $\FI$ and $\coFI$.
\begin{thm}
\label{fi-is-noetherian}
\
\begin{enumerate}
\item (Snowden \cite{Snowden})
Any submodule or quotient module of a finitely-generated $\FI$-module or $\coFI$-module is finitely-generated.
\item (Church-Ellenberg-Farb \cite{CEF})
If $V$ is a finitely-generated $\FI$-module or $\coFI$-module then the sequence $(V(n))_{n \geq 1}$ of
$\symm_n$-modules exhibits representation stability with respect to the maps
$V([n] \hookrightarrow [n+1]): V(n) \rightarrow V(n+1)$
induced by containment for $\FI$ or the duals of the maps
$V([n] \hookrightarrow [n+1]): V(n+1) \rightarrow V(n)$ for $\coFI$.
\end{enumerate}
\end{thm}
\section{Geometric proof of \cref{line-stability-theorem}}
\label{Geometric}
To use the category $\FI$ to prove \cref{line-stability-theorem},
we need the geometric notion of an affine paving.
Let $X$ be a complex algebraic variety.
A chain
\begin{equation}
\varnothing = Z_0 \subseteq Z_1 \subseteq \cdots \subseteq Z_m = X
\end{equation}
of Zariski closed subsets is an {\em affine paving} of $X$ if each difference $Z_i - Z_{i-1}$ is isomorphic
to a disjoint union $\bigsqcup_j A_{ij}$ of affine spaces (of possibly different dimensions).
For example,
let $\PP^{k-1}$ denote the $(k-1)$-dimensional complex projective space of lines through the origin
in $\CC^k$. The variety $\PP^{k-1}$ admits the following affine paving (in projective coordinates)
\begin{equation}
\label{basic-affine-paving}
\varnothing \subset [ \star : 0 : \cdots : 0] \subset [\star : \star : \cdots : 0] \subset \cdots \subset
[ \star : \star : \cdots : \star ] = \PP^{k-1}.
\end{equation}
We need only one fact about affine pavings.
Suppose $X$ is a variety and $U \subseteq X$. The inclusion $\iota: U \rightarrow X$
induces a map on homology
\begin{equation}
\iota_*: H_{\bullet}(U) \longrightarrow H_{\bullet}(X).
\end{equation}
Although the nature of the map $\iota_*$ is generally inscrutable:
\begin{quote}
Suppose $\varnothing = Z_0 \subseteq Z_1 \subseteq \cdots \subseteq Z_m = X$ is an affine paving
and $U = X - Z_i$ for some $i$. Then the induced map $\iota_*$ on homology is injective.
\end{quote}
\begin{proof}
We prove \cref{line-stability-theorem} (2); the proof of
\cref{line-stability-theorem} (1) is similar, but easier.
The strategy is to give the homology groups in question the structure
of an $\FI$-module which embeds inside a finitely-generated $\FI$-module, and then apply
\cref{fi-is-noetherian}.
We start by describing our embedding.
The $n$-fold product $(\PP^{k-1})^n$ consists of all $n$-tuples $(\ell_1, \dots, \ell_n)$ of
$1$-dimensional subspaces of $\CC^k$.
We have an inclusion $\iota: X_{n,k} \hookrightarrow (\PP^{k-1})^n$.
While one can take products of the subvarieties in \eqref{basic-affine-paving} to get
a product paving of $(\PP^{k-1})^n$, this paving interacts poorly with the inclusion $\iota$.
Pawlowski and Rhoades \cite{PR} exhibit a {\em different} affine paving
$\varnothing = Z_0 \subseteq Z_1 \subseteq \cdots \subseteq (\PP^{k-1})^n$ with
$X_{n,k} = (\PP^{k-1})^n - Z_i$ for some $i$.
Therefore, the map
\begin{equation}
\label{iota-is-injective}
\iota_*: H_{d}(X_{n,k}) \hookrightarrow H_{d}( (\PP^{k-1})^n)
\end{equation}
is an injection for all $k$ and $n$.
Given $f: [n] \hookrightarrow [p]$ define
$\nu_f: (\PP^{n-m-1})^n \rightarrow (\PP^{p-m-1})^p$ by
$\nu_f: (\ell_1, \dots, \ell_n) \mapsto (\ell'_1, \dots, \ell'_p)$ where the $\ell_j'$ are defined as follows.
Write the complement of the image of $f$ as
\begin{equation*}
\{1, 2, \dots, p \} - \{f(1), f(2), \dots, f(n) \} := \{ c_1 < c_2 < \cdots < c_{p-n} \}.
\end{equation*}
Now set
\begin{equation}
\ell_j' := \begin{cases}
\ell_i & \text{if $f(i) = j$,} \\
\langle e_{n+i} \rangle& \text{if $c_i = j$.}
\end{cases}
\end{equation}
In the first branch we consider $\ell_i$ as a line in $\CC^{p-m}$ by embedding
$\CC^{n-m} \hookrightarrow \CC^{p-m}$ into the first $n-m$ coordinates and in the second branch
$\langle e_{n+i} \rangle \subseteq \CC^{p-m}$ is the line spanned by the $(n+i)^{th}$ standard basis vector.
We have $\ell_1' + \cdots + \ell_p' = \CC^{p-m}$ whenever $\ell_1 + \cdots + \ell_n = \CC^{n-m}$
so that $\nu_f(X_{n,n-m}) \subseteq X_{p,p-m}$.
If $f: [n] \hookrightarrow [p]$ and $g: [p] \hookrightarrow [r]$ are two injections, we do {\bf not} necessarily
have the equality of maps $\nu_{g \circ f} = \nu_g \circ \nu_f$.
For example, suppose $f: [2] \hookrightarrow [4]$ and $g: [4] \hookrightarrow [6]$ are given by
\begin{equation*}
f(1) = 3, \, f(2) = 1 \quad \text{and} \quad g(1) = 2, \, g(2) = 6, \, g(3) = 5, \, g(4) = 3.
\end{equation*}
Then
\begin{equation*}
(\ell_1, \ell_2) \xmapsto{\nu_f} ( \ell_2, \langle e_3 \rangle, \ell_1, \langle e_4 \rangle )
\xmapsto{\nu_g} ( \langle e_5 \rangle , \ell_2 , \langle e_4 \rangle , \langle e_6 \rangle , \ell_1 , \langle e_3 \rangle )
\end{equation*}
whereas
\begin{equation*}
(\ell_1, \ell_2) \xmapsto{\nu_{g \circ f}} ( \langle e_3 \rangle, \ell_2, \langle e_4 \rangle, \langle e_5 \rangle, \ell_1,
\langle e_6 \rangle).
\end{equation*}
Despite the inequality $\nu_{g \circ f} \neq \nu_g \circ \nu_f$, we have the following
{\bf Claim:} {\em We have a homotopy of maps $\nu_{g \circ f} \simeq \nu_g \circ \nu_f$.}
To prove the Claim, consider the translation action of $GL_{r-m}(\CC)$ on
$(\PP^{r-m-1})^r$ given by $A \cdot (\ell_1, \dots, \ell_r) := (A \ell_1, \dots, A \ell_r)$.
For fixed injections $f: [n] \hookrightarrow [p]$ and $g: [p] \hookrightarrow [r]$, there exists a matrix
$P \in GL_{r-m}(\CC)$ such that
\begin{equation}
P \cdot \nu_{g \circ f}(\ell_1, \dots, \ell_n) = (\nu_g \circ \nu_f)(\ell_1, \dots, \ell_n)
\end{equation}
for all
$(\ell_1, \dots, \ell_n) \in (\PP^{n-m-1})^n$. The matrix $P$ simply permutes the last $r-n$ standard basis vectors
in a fashion depending on $f$ and $g$.
Since $GL_{r-m}(\CC)$ is path-connected, there is a continuous map
$\gamma: [0,1] \rightarrow GL_{r-m}(\CC)$ with $\gamma(0) = I$ (the identity matrix) and $\gamma(1) = P$.
The requisite homotopy equivalence $[0,1] \times (\PP^{n-m-1})^n \rightarrow (\PP^{r-m-1})^r$ is given by
$t \times (\ell_1, \dots, \ell_n) \mapsto \gamma(t) \cdot \nu_{g \circ f}(\ell_1, \dots, \ell_n)$. This proves the Claim.
Our Claim implies $(\nu_{g \circ f})_* = (\nu_g)_* \circ (\nu_f)_*$ as functions on
$H_d((\PP^{n-m-1})^n)$ so
the assignment $[n] \mapsto H_d((\PP^{n-m-1})^n)$ is an $\FI$-module.
The injectivity of the map $\iota_*$ in \eqref{iota-is-injective} means that
$[n] \mapsto H_d(X_{n,n-m})$ is a submodule.
The $\FI$-module $[n] \mapsto H_d((\PP^{n-m-1})^n)$ is finitely-generated, so
of \cref{line-stability-theorem} (2) follows from \cref{fi-is-noetherian}.
\end{proof}
\section{Higher dimensional subspaces}
\label{Higher}
In this section we extend \cref{line-stability-theorem} from lines to higher-dimensional subspaces.
Let $\Gr(r,k)$ be the Grassmannian of $r$-dimensional subspaces
of $\CC^k$ and consider the $n$-fold product
$\Gr(r,k)^n = \Gr(r,k) \times \cdots \times \Gr(r,k)$ of this Grassmannian with itself.
We have the variety
\begin{equation}
X_{r,n,k} := \{ (V_1, \dots, V_n) \in \Gr(r,k)^n \,:\, V_1 + \cdots + V_n = \CC^k \}
\end{equation}
of spanning subspace configurations.
The cohomology of $X_{r,n,k}$ may be presented as follows.
\begin{thm}
\label{higher-cohomology-presentation}
(Rhoades \cite{Rhoades})
Let $N = r \cdot n$ and consider a list $\xxx_N := (x_1, \dots, x_N)$ of $N$ variables.
For $1 \leq i \leq n$ denote the $i^{th}$ batch of $r$ variables by
$\xxx_N^{(i)} := (x_{(r-1)i+1}, x_{(r-1)i+2}, \dots, x_{ri})$.
We have
\begin{equation}
H^{\bullet}(X_{r,n,k}) = (\QQ[x_1, x_2, \dots, x_N]/I)^{\symm_r \times \cdots \times \symm_r}
\end{equation}
where the $n$-fold symmetric group product $\symm_r \times \cdots \times \symm_r$ permutes
variables within batches, the superscript indicates taking invariants, and $I \subseteq \QQ[x_1, \dots, x_N]$
is generated by
\begin{itemize}
\item the top $k$ elementary symmetric polynomials $e_N, e_{N-1}, \dots, e_{N-k+1}$ in the full variable set $\xxx_N$
and
\item for $1 \leq i \leq n$ the complete homogeneous symmetric polynomials
$h_k, h_{k-1}, \dots, h_{k-r+1}$ in the variable set $\xxx_N^{(i)}$.
\end{itemize}
The variables in $\xxx_N^{(i)}$ represent the Chern roots of the vector bundle $V_i^* \twoheadrightarrow X_{r,n,k}$.
\end{thm}
The action of $\symm_n$ on the cohomology $H^{\bullet}(X_{r,n,k})$ corresponds
under the presentation of \cref{higher-cohomology-presentation} to permuting the variable
batches $\xxx_N^{(1)}, \dots, \xxx_N^{(n)}$.
As an ungraded $\symm_n$-module, it follows from \cite{Rhoades} that
$H^{\bullet}(X_{r,n,k})$ is isomorphic to the column-permuting action of $\symm_n$ on the
set of 0,1-matrices of dimension $k \times n$ which have all column sums equal to $r$ and no zero rows.
When $r = 2, n = 4,$ and $k = 3$ one such matrix is
\begin{equation*}
\begin{pmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 1 & 1 \\
1 & 1 & 0 & 1
\end{pmatrix}
\end{equation*}
The following stability result concerns the graded structure of $H^{\bullet}(X_{r,n,k})$.
\begin{thm}
\label{higher-stability-theorem}
Fix a homological degree $d \geq 0$ and a subspace dimension $r$.
The following sequences of modules are representation
stable with respect to some sequence of maps $f_n$.
\begin{enumerate}
\item $(H_d(X_{r,n,k}))_{n \geq 0}$ for $k \geq 0$ fixed, and
\item $(H_d(X_{r,n,n-m}))_{n \geq 0}$ for $m \geq 0$ fixed.
\end{enumerate}
\end{thm}
The proof of \cref{higher-stability-theorem} is similar to that of \cref{line-stability-theorem}.
One exhibits a nonstandard affine paving of the product $\Gr(r,k)^n$ of Grassmannians which has
$X_{r,n,k}$ as a terminal portion. We omit the details in this extended abstract, but remark that the following
problem remains open, despite the explicit presentation in
\cref{higher-cohomology-presentation}.
\begin{prob}
\label{higher-isomorphism-type}
Calculate the graded $\symm_n$-isomorphism type of $H_{\bullet}(X_{r,n,k})$.
\end{prob}
This illustrates our introductory leitmotif: it can be easier to prove that a sequence of
modules exhibits representation stability than it is to calculate their isomorphism types.
\section{Modules from Coinvariants and Vandermondes}
\label{Coinvariant}
In this section we consider a family of multigraded $\symm_n$-modules which arise as subspaces
or quotients of rings generated by two $n$-column matrices of variables: one matrix of commuting variables
and one matrix of anticommuting variables.
We obtain stability results for modules considered by Orellana-Zabrocki \cite{OZ}, Zabrocki \cite{Zabrocki}, and
Rhoades-Wilson \cite{RW} which have (sometimes conjectural) ties to the Delta Conjecture \cite{HRW}.
Most of the modules we consider in this section do not have known decompositions into irreducibles.
In spite of this (and in keeping with our leitmotif), it is possible to show that they enjoy stability properties.
For $n, m, p \geq 0$, consider an $m \times n$ matrix $(x^{(i)}_j)_{1 \leq i \leq m, 1 \leq j \leq n}$ of (commuting)
variables
and a $p \times n$ matrix $(\theta^{(i)}_j)_{1 \leq i \leq p, 1 \leq j \leq n}$ of (anticommuting) variables.
Let $S(n,m,p)$ be the unital associative $\QQ$-algebra generated by these $mn + pn$ variables
subject to the relations
\begin{equation*}
x^{(i)}_j x^{(i')}_{j'} = x^{(i')}_{j'} x^{(i)}_j \quad \quad
x^{(i)}_j \theta^{(i')}_{j'} = \theta^{(i')}_{j'} x^{(i)}_j \quad \quad
\theta^{(i)}_j \theta^{(i')}_{j'} = - \theta^{(i')}_{j'} \theta^{(i)}_j
\end{equation*}
The algebra $S(n,m,p)$ has a multigrading obtained by considering each row of the two
variable matrices separately.
For $\alpha = (\alpha_1, \dots, \alpha_m) \in (\ZZ_{\geq 0})^m$ and
$\beta = (\beta_1, \dots, \beta_p) \in (\ZZ_{\geq 0})^p$ write $S(n,m,p)_{\alpha,\beta}$ for the piece
of $S(n,m,p)$ of homogeneous multidegree $(\alpha,\beta)$.
The group $\symm_n$ acts on $S(n,m,p)$ by the rule
$w.x^{(i)}_j := x^{(i)}_{w(j)}$ and $w.\theta^{(i)}_j := \theta^{(i)}_{w(j)}$. Orellana and Zabrocki
\cite{OZ} gave a combinatorial interpretation of the $\symm_n$-isomorphism type of $S(n,m,p)_{\alpha,\beta}$.
We consider this object as $n$ varies.
%In keeping with the leitmotif, stability is easier to prove than the results in \cite{OZ}.
\begin{prop}
\label{s-module-stability}
Let $m, p \geq 0$ and let $\alpha \in (\ZZ_{\geq 0})^m$ and $\beta \in (\ZZ_{\geq 0})^p$ be multidegrees.
The sequence $(S(n,m,p)_{\alpha,\beta})_{n \geq 1}$ is representation stable with respect to the inclusion
maps
\begin{equation*}
f_n: S(n,m,p)_{\alpha,\beta} \hookrightarrow S(n+1,m,p)_{\alpha,\beta}.
\end{equation*}
\end{prop}
%\begin{proof}
%For fixed degrees $d, e \geq 0$, we have $\coFI$-modules $[n] \mapsto \QQ[x_1, \dots, x_n]_d$ and
%$[n] \mapsto \QQ[\theta_1, \dots, \theta_n]_e$, where an injection
%$f: [n] \hookrightarrow [r]$ is sent to maps defined by
%\begin{equation}
%x_j \mapsto \begin{cases}
%x_i & \text{if $f(i) = j$,} \\
%0 & \text{if $j$ is not in the image of $f$,}
%\end{cases} \quad
%\theta_j \mapsto \begin{cases}
%\theta_i & \text{if $f(i) = j$,} \\
%0 & \text{if $j$ is not in the image of $f$.}
%\end{cases}
%\end{equation}
%One checks that these $\coFI$-modules are finitely-generated,
%observes that finite-generation is inherited by tensor
%products, and applies \cref{fi-is-noetherian}.
%\end{proof}
Let $S(n,m,p)^{\symm_n}_+ \subseteq S(n,m,p)$ be the space of $\symm_n$-invariants
with vanishing constant
term and let $I(n,m,p) \subseteq S(n,m,p)$ be the ideal generated by $S(n,m,p)^{\symm_n}_+$.
We consider the quotient
\begin{equation}
R(n,m,p) := S(n,m,p) / I(n,m,p).
\end{equation}
Write $R(n,m,p)_{\alpha,\beta}$ for the piece of homogeneous multidegree $(\alpha,\beta)$.
The $\symm_n$-modules $R(n,m,p)$ have received significant attention in algebraic combinatorics.
$R(n,1,0)$ is the classical coinvariant ring attached to the symmetric group which presents the cohomology
of the flag variety $\Fl_n$ (or the space $X_{n,n}$).
$R(n,2,0)$ is the {\em diagonal coinvariant ring} studied by Garsia and Haiman \cite{GH}.
The trigraded $\symm_n$-module
$R(n,2,1)$ was studied by Zabrocki \cite{Zabrocki}
in the context of the Delta Conjecture.
Zabrocki conjectured that
\begin{equation}
\label{zabrocki-conjecture}
\grFrob( R(n,2,1); q, t, z) = \sum_{k = 1}^n z^{n-k} \cdot \Delta'_{e_{k-1}} e_n.
\end{equation}
\begin{thm}
\label{r-module-stability}
Let $m, p \geq 0$ and let $\alpha \in (\ZZ_{\geq 0})^m$ and $\beta \in (\ZZ_{\geq 0})^p$ be multidegrees.
The sequence $(R(n,m,p)_{\alpha,\beta})_{n \geq 1}$ is representation stable with respect to some
sequence of maps $f_n$.
\end{thm}
\begin{proof}
The assignment $[n] \mapsto I(n,m,p)_{\alpha,\beta}$ is a submodule of the $\coFI$-module
$[n] \mapsto S(n,m,p)_{\alpha,\beta}$.
Now apply \cref{s-module-stability} and \cref{fi-is-noetherian}.
\end{proof}
Another Delta Conjecture model
was proposed by Rhoades and Wilson \cite{RW}. Assuming $m, p \geq 1$,
the {\em superspace Vandermonde} $\delta_{n,k} \in S(n,m,p)$ is the element
\begin{equation}
\delta_{n,k} := \varepsilon_n \cdot (x_1^{k-1} \cdots x_{n-k}^{k-1} \,
x_{n-k+1}^{k-1} x_{n-k+2}^{k-2} \cdots x_{n-1}^1 \, x_n^0 \cdot
\theta_1 \cdots \theta_{n-k}),
\end{equation}
where the $x$'s and $\theta$'s are drawn from the `first' commuting and anticommuting tensor factors
and $\varepsilon_n = \sum_{w \in \symm_n} \mathrm{sign}(w) \cdot w \in \QQ[\symm_n]$
is the antisymmetrizing group algebra element.
To describe the representations in \cite{RW} we need polarization operators.
Let $y_1, \dots, y_n$ and $z_1, \dots, z_n$ be two rows of commuting generators of
$S(n,m,p)$ (renamed $y$ and $z$ for clarity). For $j \geq 1$ the {\em commuting polarization operator}
from $y$ to $z$ of order $j$ is the operator on $S(n,m,p)$ defined by
\begin{equation}
\rho_{y \rightarrow z}^{(j)} := z_1 (\partial/\partial y_1)^j + \cdots + z_n (\partial/\partial y_n)^j.
\end{equation}
This operator lowers $y$-degree by $j$ and raises $z$-degree by $1$.
To define an anticommuting version of the $\rho_{y \rightarrow z}^{(j)}$ we need to
differentiate with respect to anticommuting variables.
Let $\xi_1, \dots, \xi_n$ and $\tau_1, \dots, \tau_n$ be two rows of anticommuting generators
of $S(n,m,p)$ (renamed $\xi$ and $\tau$ for clarity).
For $1 \leq i \leq n$, the operator $\partial/\partial \xi_i$ acts on $S(n,m,p)$ by commuting with multiplication by
any commuting variable and the rule
\begin{equation}
\partial / \partial \xi_i: \zeta_{j_1} \cdots \zeta_{j_r} \mapsto \begin{cases}
(-1)^{s-1} \zeta_{j_1} \cdots \widehat{\xi_{j_s}} \cdots \zeta_{j_r} & \text{if $\zeta_{j_s} = \xi_{i}$,} \\
0 & \text{if $\xi_i$ does not appear in $\zeta_{j_1}, \dots, \zeta_{j_r}$,}
\end{cases}
\end{equation}
where $\zeta_{j_1}, \dots, \zeta_{j_r}$ are distinct anticommuting variables and $\widehat{\cdot}$ means omission.
The {\em anticommuting polarization operator} from $\xi$ to $\tau$ is the operator on $S(n,m,p)$ defined by
\begin{equation}
\rho_{\xi \rightarrow \tau} := \tau_1 (\partial/\partial \xi_1) + \cdots + \tau_n (\partial/\partial \xi_n).
\end{equation}
This operator lowers $\xi$-degree by $1$ and raises $\tau$-degree by $1$.
Let $V(n,k,m,p)$ be the smallest linear subspace of $S(n,m,p)$ which contains the superspace
Vandermonde $\delta_{n,k}$ and
is closed under all commuting partial derivatives $\partial/\partial x_i$ as well as all
possible polarization operators. Considering commuting multidegree alone,
$V(n,k,2,1)$ is a bigraded $\symm_n$-module.
Rhoades and Wilson conjectured \cite{RW} that
\begin{equation}
\label{rw-conjecture}
\grFrob( V(n,k,2,1); q, t) = \Delta'_{e_{k-1}} e_n.
\end{equation}
This is similar in form to Zabrocki's conjecture \eqref{zabrocki-conjecture}.
Unlike \eqref{zabrocki-conjecture}, \eqref{rw-conjecture} is proven at $t = 0$.
\cref{r-module-stability} extends to the setting of \eqref{rw-conjecture}.
\begin{thm}
\label{v-module-stability}
Let $m, p, k \geq 0$ and let $\alpha \in (\ZZ_{\geq 0})^m$ and $\beta \in (\ZZ_{\geq 0})^p$ be multidegrees.
The sequence $(V(n,k,m,p)_{\alpha,\beta})_{n \geq 0}$ is representation stable with respect to some sequence
of maps $f_n$.
\end{thm}
\begin{proof}
The identity
\begin{equation}
(\partial/\partial x_1)(\partial/\partial x_2) \cdots (\partial/\partial x_n) \delta_{n+1,k} = \delta_{n,k}
\end{equation}
implies that
the map $S(n,m,p) \hookrightarrow S(n+1,m,p)$ sends
$V(n,k,m,p)$ into $V(n+1,k,m,p)$.
Now apply \cref{s-module-stability} and \cref{fi-is-noetherian}.
\end{proof}
We close with a very difficult problem which serves as a final illustration of our introductory leitmotif.
\begin{prob}
Find the Schur expansion of any of the following symmetric functions:
\begin{equation}
\Delta'_{e_{k-1}} e_n, \quad \grFrob(R(n,m,p);q,t,z), \quad \grFrob(V(n,k,m,p);q,t).
\end{equation}
In the case $k = n$, the first of these is $\nabla e_n$, where $\nabla$ is the Bergeron-Garsia nabla operator.
\end{prob}
\acknowledgements{The authors are grateful to Steven Sam and Ben Young for helpful conversations.}
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way
\printbibliography
\end{document}