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\documentclass[finalversion]{FPSAC2020}
\articlenumber{56}
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\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}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Defintion}



\newcommand{\aaa}{\mathbf{a}}
\newcommand{\xxx}{\mathbf{x}}

\newcommand{\OP}{\mathcal{OP}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\VV}{\mathbb{V}}
\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{\ann}{\mathrm{ann}}


%% define your title in the usual way
\title{Vandermondes, Superspace, and  Delta Conjecture modules}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Brendon Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}. 
Partially supported by NSF grant DMS--1500838.}\addressmark{1} \and 
Andrew Timothy Wilson\thanks{\href{mailto:andwils2@pdx.edu}{andwils2@pdx.edu}.}
\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California - San Diego, La Jolla, CA, 
92093, USA \\ 
\addressmark{2}Department of Mathematics and Statistics, Portland State University, Portland, OR, 97201, USA}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{Superspace is an algebra $\Omega_n$ with $n$ commuting generators $x_1, \dots, x_n$ 
and $n$ anticommuting generators $\theta_1, \dots, \theta_n$. We present an extension
$\delta_{n,k}$ of the Vandermonde
determinant  to $\Omega_n$ which depends on  positive
integers $k \leq n$. 
We use superspace Vandermondes to build  representations of the symmetric 
group $S_n$. In particular, we construct a doubly graded $S_n$-module 
$\VV_{n,k}$ whose bigraded Frobenius image $\grFrob(\VV_{n,k}; q, t)$
conjecturally equals the symmetric function $\Delta'_{e_{k-1}} e_n$ appearing in the
Haglund-Remmel-Wilson
Delta Conjecture. We prove the specialization 
of our conjecture at $t = 0$. 
We use a  differentiation action of $\Omega_n$ on itself to build bigraded quotients $\WW_{n,k}$ of $\Omega_n$
which extend the Delta Conjecture coinvariant rings $R_{n,k}$ defined by Haglund-Rhoades-Shimozono
and studied geometrically by Pawlowski-Rhoades. Despite the fact that the Hilbert polynomials of the $R_{n,k}$
are not palindromic, we show that  $\WW_{n,k}$ exhibits a superspace version of Poincar\'e Duality.}

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% \resume{\lipsum[2]}

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\keywords{Vandermonde, superspace, $S_n$-module}

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\begin{document}

\maketitle
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\section{Introduction}

The symmetric group $S_n$ acts on the polynomial ring $\QQ[x_1, \dots, x_n]$ by subscript permutation.
Polynomials in the invariant subring 
\begin{equation}
\QQ[x_1, \dots, x_n]^{S_n} := \{ f \in \QQ[x_1, \dots, x_n] \,:\, w.f = f \text{ for all $w \in S_n$} \}
\end{equation}
are called {\em symmetric polynomials}.
The $\QQ$-algebra $\QQ[x_1, \dots, x_n]^{S_n}$ is generated by the $n$ elementary
symmetric polynomials $e_1, e_2, \dots, e_n$. 

Let $\QQ[x_1, \dots, x_n]^{S_n}_+$ be the space of symmetric polynomials with vanishing constant term.
The {\em invariant ideal} $I_n \subseteq \QQ[x_1, \dots, x_n]$ is given by
\begin{equation}
I_n := \langle \QQ[x_1, \dots, x_n]^{S_n}_+ \rangle = \langle e_1, e_2, \dots, e_n \rangle,
\end{equation}
and the {\em coinvariant ring} is the corresponding quotient
\begin{equation}
R_n := \QQ[x_1, \dots, x_n]/I_n.
\end{equation}
The quotient $R_n$ is simultaneously a graded ring and a graded $S_n$-module.
The module $R_n$ is among the most important in algebraic combinatorics, with representation
theory tied to permutation combinatorics and a geometric realization as the cohomology of the flag variety 
\cite{Borel, Chevalley}.

The symmetric group $S_n$ acts diagonally on the polynomial ring
$\QQ[x_1, \dots, x_n, y_1, \dots,\linebreak[0] y_n]$ in $2n$ variables, viz.
$w.x_i = x_{w(i)}$ and $w.y_i := y_{w(i)}$ for all $w \in S_n$ and $1 \leq i \leq n$.
Garsia and Haiman \cite{GH} initiated the study of the the {\em diagonal coinvariant ring} 
$DR_n$ defined by modding out by those $S_n$-invariants with vanishing constant term:
\begin{equation}
DR_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}
Considering $x$-degree and $y$-degree separately, the ring $DR_n$ is a doubly graded $S_n$-module
which specializes to $R_n$ when the $y$-variables are set to zero.

Haiman proved  \cite{Haiman} that as ungraded $S_n$-modules we have 
$DR_n \cong \QQ[\Park_n] \otimes \mathrm{sign}$ where $\Park_n$ is the permutation action of $S_n$ on
size $n$ parking functions and $\mathrm{sign}$ is the 1-dimensional sign representation of $S_n$.
Haiman also proved more refined results on the bigraded $S_n$-module structure of $DR_n$; to state these 
we recall some facts about $S_n$-modules.


The irreducible representations of $S_n$ over $\QQ$ are indexed by partitions of $n$; if $\lambda \vdash n$
is a partition, let $S^{\lambda}$ be the corresponding $S_n$-irreducible.
If $V$ is any finite-dimensional $S_n$-module, there exist unique multiplicities $c_{\lambda} \geq 0$ so that
$V \cong \bigoplus_{\lambda \vdash n} c_{\lambda} S^{\lambda}$.
Let $\Lambda$ denote the ring of symmetric functions over the ground field $\QQ(q,t)$ in the infinite
variable set $\xxx = (x_1, x_2, \dots )$.
The {\em Frobenius image} of $V$ is the symmetric function $\Frob(V) \in \Lambda$ given by
$\Frob(V) := \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}$, where $s_{\lambda}$ is the Schur function.

In this extended abstract we will consider (multi)graded $S_n$-modules.
If $V = \bigoplus_{i \geq 0} V_i$ is a graded $S_n$-module with each graded piece $V_i$ 
finite-dimensional, the {\em graded Frobenius image} of $V$ is
$\grFrob(V;q) := \sum_{i \geq 0} q^i \cdot \Frob(V_i)$.  Even more generally, if $V = \bigoplus_{i,j \geq 0} V_{i,j}$
or $V = \bigoplus_{i,j,k \geq 0} V_{i,j,k}$ is a doubly or triply graded $S_n$-module, we have the associated
bigraded and trigraded Frobenius images
\begin{equation*}
\grFrob(V; q, t) := \sum_{i,j \geq 0} q^i t^j \cdot \Frob(V_{i,j}) \quad \text{or} \quad
\grFrob(V; q, t, z) := \sum_{i,j,k \geq 0} q^i t^j z^k \cdot \Frob(V_{i,j,k}),
\end{equation*}
respectively.

Haiman proved \cite{Haiman} 
that $\grFrob(DR_n; q, t) = \nabla e_n$, where $e_n$ is the degree $n$ elementary symmetric
function and $\nabla$ is the Bergeron-Garsia {\em nabla operator}.
Therefore, describing the bigraded $S_n$-isomorphism type of $DR_n$ is equivalent to finding the 
Schur expansion of $\nabla e_n$, but there is not even a conjecture in this direction.
The monomial expansion of $\nabla e_n$ is given by the {\em Shuffle Theorem} \cite{CM}.

The {\em Delta Conjecture} is a conjectural extension of the Shuffle Theorem due to Haglund, Remmel, and Wilson
\cite{HRW}.
It depends on two positive integers $k \leq n$ and reads
\begin{equation}
\Delta'_{e_{k-1}} e_n = \Rise_{n,k}(\xxx;q,t) = \Val_{n,k}(\xxx;q,t).
\end{equation}
Here $\Delta'_{e_{k-1}}$ is a certain symmetric function operator and
 $\Rise$ and $\Val$ are  formal power series defined using the combinatorics of lattice paths; see \cite{HRW} for 
 details.
When $k = n$, the Delta Conjecture reduces to the Shuffle Theorem. 

The Delta Conjecture is open
as of this writing, but combining the work of \cite{GHRY, HRS, Rhoades, Wilson} 
it is known at $q = 0$. More precisely, we have
\begin{equation}
\label{c-common-function}
\Delta'_{e_{k-1}} e_n \mid_{t = 0} = \Rise_{n,k}(\xxx;q,0) = \Rise_{n,k}(\xxx;0,q) =
\Val_{n,k}(\xxx;q,0) = \Val_{n,k}(\xxx;0,q).
\end{equation}


In this paper we define a doubly graded $S_n$-module $\VV_{n,k}$ for any positive integers
$k \leq n$ and conjecture 
that $\grFrob(\VV_{n,k}; q, t) = \Delta'_{e_{k-1}} e_n$ (see \cref{bold-v-conjecture}).
That is, we conjecture that $\VV_{n,k}$ is a module for the Delta Conjecture.
We prove this conjecture at $t = 0$.
In order to describe $\VV_{n,k}$, we introduce new combinatorial objects called {\em superspace Vandermondes}.

 
 {\em Superspace}
 of rank $n$ is the unital associative $\QQ$-algebra $\Omega_n$ 
 generated by  $2n$ symbols $x_1, \dots, x_n, \theta_1, \dots, \theta_n$ subject to the relations
 \begin{equation*}
 x_i x_j = x_j x_i  \quad \quad x_i \theta_j = \theta_j x_i \quad \quad \theta_i \theta_j = - \theta_j \theta_i
 \end{equation*}
 for all $1 \leq i, j \leq n$.\footnote{The `super' in superspace comes from supersymmetry in physics:  the $x$-variables 
index bosons and the $\theta$-variables index fermions.
Extending coefficients to the reals, $\Omega_n$ is the ring of polynomial-valued
differential forms on Euclidean $n$-space -- this is why we write  $\Omega$.}
 Setting the $\theta$-variables to zero recovers the classical polynomial ring $\QQ[x_1, \dots, x_n]$.
By considering $x$-degree and $\theta$-degree separately, $\Omega_n$ is a doubly graded algebra.
The ring $\Omega_n$ carries a diagonal action of $S_n$ given by
$w.x_i := x_{w(i)}$ and $w.\theta_i := \theta_{w(i)}$ for  $w \in S_n$ and $1 \leq i \leq n$.




\begin{defn}
Let $k \leq n$ be positive integers. The {\em superspace Vandermonde} $\delta_{n,k}$ is the following 
element of $\Omega_n$:
\begin{equation}
\delta_{n,k} := \varepsilon_n . (x_1^{k-1} x_2^{k-1} \cdots  x_{n-k+1}^{k-1} x_{n-k+2}^{k-2} \cdots x_{n-1}^1 x_n^0
\theta_1 \theta_2 \cdots \theta_{n-k}).
\end{equation}
Here $\varepsilon_n := \sum_{w \in S_n} \mathrm{sign}(w) \cdot w \in \QQ[S_n]$ is the antisymmetrizing element 
in the symmetric group algebra.
\end{defn}

For example, when $n = 3$ and $k = 2$ we have
\begin{equation*}
\delta_{3,2} = \varepsilon_3.(x_1 x_2 \theta_1) = x_1 x_2 \theta_1 - x_1 x_2 \theta_2 - x_1 x_3 \theta_1
+ x_1 x_3 \theta_3 + x_2 x_3 \theta_2 - x_2 x_3 \theta_3.
\end{equation*}
The superpolynomial
$\delta_{n,k}$ is always a nonzero element of $\Omega_n$, thanks to the $\theta$-variables.
When $k = n$, the superspace Vandermonde $\delta_{n,k}$ reduces to the classical
Vandermonde determinant
$\varepsilon_n.(x_1^{n-1} x_2^{n-2} \cdots x_{n-1}^1 x_n^0)$.

The $\delta_{n,k}$ are seeds we use to grow modules.  By starting with $\delta_{n,k}$ and closing under
various differential operators and linearity we will construct:
\begin{itemize}
\item A singly graded subspace $V_{n,k}$ of  $\Omega_n$ which 
satisfies $\grFrob(V_{n,k}; q) = \Delta'_{e_{k-1}} e_n \mid_{t = 0}$ (see {\bf \cref{Delta}}).
\item A doubly graded extension $\VV_{n,k}$ of $V_{n,k}$ with
$\grFrob(\VV_{n,k}; q, t)$  conjecturally  given by $\Delta'_{e_{k-1}} e_n$ (see {\bf \cref{Delta}}).
\item A doubly graded $S_n$-stable
quotient $\WW_{n,k}$ of $\Omega_n$ which extends $V_{n,k}$
and exhibits a number of symmetries including a superspace variant of Poincar\'e Duality
(see {\bf \cref{Poincare}}).
$\WW_{n,k}$ extends the cohomology of the space of 
spanning line configurations studied by Pawlowski and Rhoades \cite{PR}.
\end{itemize}

This paper is not the first to propose connections between the Delta Conjecture 
and superspace.
The Fields Institute Combinatorics Group in general, and Mike Zabrocki in particular,  conjectured \cite{Zabrocki}
that representation-theoretic models for the Delta Conjecture can be obtained by looking at 
coinvariant-type quotients defined using superspace $\Omega_n$ and an extension
$\Omega_n[y_1, \dots, y_n]$ of superspace involving $n$ new commuting variables
$y_1, \dots, y_n$.  We discuss the connection between our work and their conjectures in {\bf \cref{Fields}}.
In a nutshell, we are able to prove that our proposed Delta model $\VV_{n,k}$ is valid at $t = 0$, but the 
corresponding case of their conjecture remains open.



















\section{The $S_n$-modules $V_{n,k}$ and $\VV_{n,k}$ and the Delta Conjecture}
\label{Delta}

For $1 \leq i \leq n$, the partial derivative operator $\partial/\partial x_i$ acts naturally 
on the polynomial ring $\QQ[x_1, \dots, x_n]$.  Superspace admits the tensor product decomposition
\begin{equation}
\Omega_n = \QQ[x_1, \dots, x_n] \otimes \wedge \{ \theta_1, \dots, \theta_n \}
\end{equation}
where $\wedge \{\theta_1, \dots, \theta_n\}$ is the exterior algebra on the generators $\theta_1, \dots, \theta_n$.
The operator $\partial/\partial x_i$ therefore acts on $\Omega_n$ by acting on the first tensor factor.

Our first new  $S_n$-module is defined as follows.  
Starting with the superspace Vandermonde $\delta_{n,k}$, we close under the operators 
$\partial/\partial x_1, \dots, \partial/\partial x_n$ and linearity.

\begin{defn}
Let $k \leq n$ be positive integers. The vector space $V_{n,k}$ is the smallest $\QQ$-linear subspace
of $\Omega_n$ which
\begin{itemize}
\item contains the superspace Vandermonde $\delta_{n,k}$, and
\item is closed under the $n$ partial derivatives $\partial/\partial x_1, \dots, \partial/\partial x_n$.
\end{itemize}
\end{defn}

The subspace $V_{n,k} \subseteq \Omega_n$ is closed under the action of $S_n$.
Furthermore, $V_{n,k}$ a doubly graded subspace of $\Omega_n$. If we ignore the $\theta$-grading
(which is constant of degree $n-k$) and focus on the $x$-grading, we see that $V_{n,k}$ is a singly-graded
$S_n$-module.

To describe the Schur expansion of $\grFrob(V_{n,k}; q)$, we need some notation.
Let $T$ be a standard Young tableau with $n$ boxes. A number $1 \leq i \leq n-1$ is a {\em descent} of $T$
if $i$ appears in a row above $i+1$. 
The {\em descent number} $\des(T)$ is the number of descents and the {\em major index} $\maj(T)$ is the sum
of the descents in $T$.  We write $\shape(T) \vdash n$ for the partition of $n$ obtained by erasing the numbers 
in $T$.  We also use the standard $q$-numbers, $q$-factorials, and $q$-binomials:
\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  following theorem, and other results in this extended abstract, are proven in \cite{RW}.


\begin{thm}
\label{v-isomorphism-type}
Let $k \leq n$ be positive integers. The graded Frobenius image of $V_{n,k}$ is given by
either of the expressions
\begin{align}
\grFrob(V_{n,k}; q) &= 
\sum_{T \in \mathrm{SYT}(n)} q^{\maj(T) + {n-k \choose 2} - (n-k) \cdot \des(T)}
{\des(T) \brack n-k}_q s_{\shape(T)} \\
&=
\Delta'_{e_{k-1}} e_n \mid_{t = 0}
\end{align}
where the sum is over all standard Young tableaux $T$ with $n$ boxes.
\end{thm}

Equation~\eqref{c-common-function} allows us to replace the $\Delta'_{e_{k-1}} e_n \mid_{t = 0}$
in \cref{v-isomorphism-type} with 
any of the symmetric functions $\Rise_{n,k}(\xxx;q,0), \Rise_{n,k}(\xxx;0,q), \Val_{n,k}(\xxx;q,0),$ or
$\Val_{n,k}(\xxx;0,q)$.
Thanks to \cref{v-isomorphism-type}, it is easy to describe the ungraded $S_n$-isomorphism
type of $V_{n,k}$.

\begin{cor}
\label{v-ungraded-isomorphism-type}
Let $k \leq n$ be positive integers and consider the permutation action of $S_n$ on the family 
$\OP_{n,k}$ of $k$-block ordered set partitions $(B_1, B_2, \dots, B_k)$ of $\{1, 2, \dots, n\}$.
As ungraded $S_n$-modules we have
\begin{equation}
V_{n,k} \cong \QQ[\OP_{n,k}] \otimes \mathrm{sign}
\end{equation}
where $\mathrm{sign}$ is the 1-dimensional sign representation of $S_n$.
\end{cor}

The {\em (signless) Stirling number of the second kind} $\Stir(n,k)$ counts
(unordered) $k$-block set partitions of 
$\{1, 2, \dots, n\}$.
\cref{v-ungraded-isomorphism-type} implies $\dim V_{n,k} = k! \cdot \Stir(n,k)$.
The graded dimension of $V_{n,k}$ is given by a suitable $q$-analog of this formula.


Recall that the {\em Hilbert series} of a graded vector space $V = \bigoplus_{i \geq 0} V_i$ is the formal 
power series $\Hilb(V; q) := \sum_{i \geq 0} q^i \cdot \dim V_i$.
The {\em $q$-Stirling number} $\Stir_q(n,k)$ is defined by the recursion
\begin{equation}
\Stir_q(n,k) = \Stir_q(n-1,k-1) + [k]_q \cdot \Stir_q(n-1,k)
\end{equation}
together with the initial conditions $\Stir_q(0,0) = 1$ and $\Stir_q(0,k) = 0$ for any $k > 0$.


\begin{cor}
\label{v-hilbert-series}
The Hilbert series of $V_{n,k}$
is $\Hilb(V_{n,k}; q) = [k]!_q \cdot \Stir_q(n,k)$.
\end{cor}

In order to describe our proposed model for the Delta Conjecture, we need more variables.
Let $y_1, \dots, y_n$ be $n$ new commuting variables and consider the extension 
$\Omega_n[y_1, \dots, y_n]$ of superspace defined formally by the tensor product
\begin{equation}  \Omega_n[y_1, \dots, y_n] := 
\QQ[x_1, \dots, x_n] \otimes \QQ[y_1, \dots, y_n] \otimes \wedge \{\theta_1, \dots, \theta_n \}.
\end{equation}
This is a {\em triply} graded $S_n$-module with action
$w.x_i := x_{w(i)}, w.y_i := y_{w(i)}, w.\theta_i := \theta_{w(i)}$.
This ring admits an action of partial derivatives $\partial/ \partial x_i$ and $\partial/\partial y_i$ in both
the $x$-variables and $y$-variables.


\begin{defn}
For $k \leq n$, let $\VV_{n,k}$ be the smallest $\QQ$-linear subspace of $\Omega_n[y_1, \dots, y_n]$ 
which
\begin{itemize}
\item contains the superspace Vandermonde $\delta_{n,k}$ (in the $x$-variables and $\theta$-variables alone),
\item is closed under the {\em polarization operator}
$\sum_{s = 1}^n y_s (\partial/\partial x_s)^j$ for each $j \geq 1$, and
\item is closed under the $2n$ partial derivatives 
$\partial/\partial x_1, \dots, \partial/\partial x_n, \partial/\partial y_1, \dots, \partial/\partial y_n$.
\end{itemize}
\end{defn}

The $S_n$-module $\VV_{n,k}$ is concentrated in $\theta$-degree $n-k$.  By considering $x$-degree and
$y$-degree, 
the space $\VV_{n,k}$ attains the structure of a doubly graded $S_n$-module.


\begin{conj}
\label{bold-v-conjecture}
Let $k \leq n$ be positive integers. The doubly graded Frobenius image of $\VV_{n,k}$ is given by
\begin{equation}
\grFrob(\VV_{n,k};q,t) = \Delta'_{e_{k-1}} e_n.
\end{equation}
\end{conj}

\cref{bold-v-conjecture} is true at $t = 0$ by \cref{v-isomorphism-type}.
\cref{bold-v-conjecture} is true when $k = n$ by the work of Haiman \cite{Haiman}.
\cref{bold-v-conjecture} has been checked on computer for $n \leq 4$ (and at $n = 5$
on the level of bigraded Hilbert series).
Since every increase $n \rightarrow n+1$ adds two new commuting variables and one
new anticommuting variable, studying \cref{bold-v-conjecture} involves considerable computational
challenges as $n$ grows.




\section{The Fields and Zabrocki Conjectures}
\label{Fields}

In this section we describe alternative conjectural representation-theoretic models for the Delta Conjecture
arising from quotients of $\Omega_n$ and $\Omega_n[y_1, \dots, y_n]$.
Recall that the symmetric group $S_n$ acts diagonally on superspace $\Omega_n$.  
Solomon proved \cite{Solomon} that 
the ring 
$(\Omega_n)^{S_n} \subseteq \Omega_n$ of $S_n$-invariants is a free $\QQ[x_1, \dots, x_n]^{S_n}$-module
on the generating set $\{ de_{i_1} \cdots de_{i_r} \,:\, 1 \leq i_1 < \cdots < i_r \leq n \}$ where
$d := \sum_{j = 1}^n \theta_j \cdot (\partial/\partial x_j)$ is the total derivative operator.


Let 
$\langle (\Omega_n)^{S_n}_+ \rangle \subseteq \Omega_n$ be the two-sided ideal of $\Omega_n$ generated by
$S_n$-invariants with vanishing constant term.
By considering $x$-degree and $\theta$-degree, the quotient 
$\Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle$
is a doubly graded $S_n$-module.
We view this quotient as a `superspace coinvariant ring'.
The following conjecture about its doubly graded Frobenius image was made by the Combinatorics Group
at the Fields Institute.

\begin{fconj} (see \cite{Zabrocki})
Let $n$ be a positive integer.  The doubly graded Frobenius image of 
$\Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle$ 
is given by
\begin{equation}
\grFrob( \Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle; q, z) = 
\sum_{k = 1}^n z^{n-k} \cdot \Delta'_{e_{k-1}} e_n \mid_{t = 0},
\end{equation}
where $q$ tracks $x$-degree and $z$ tracks $\theta$-degree.
\end{fconj}

If the Fields Conjecture is true, the bigraded Hilbert series of $\Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle$  would
be given by
\begin{equation}
\Hilb(\Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle ; q, z) = \sum_{k = 1}^n
z^{n-k} \cdot [k]!_q  \cdot \Stir_q(n,k)
\end{equation}
where $q$ tracks $x$-degree and $z$ tracks $\theta$-degree.
The Fields Combinatorics Group proved (personal communication) the inequality
\begin{equation}
\Hilb(\Omega_n / \langle (\Omega_n)^{S_n}_+ \rangle ; q, z) \geq \sum_{k = 1}^n
z^{n-k} \cdot [k]!_q  \cdot \Stir_q(n,k)
\end{equation}
where $f(q,z) \geq g(q,z)$ means that the difference $f(q,z) - g(q,z)$ is a polynomial in $q,z$ with nonnegative 
coefficients. 


Recall that the {\em alternating subspace} of an $S_n$-module $V$ is given by 
\begin{equation*}
\{v \in V \,:\, w.v = \mathrm{sign}(w) \cdot v \text{ for all $w \in S_n$} \}.
\end{equation*}
Let $A_n$ be the alternating subspace of 
$\Omega_n/\langle (\Omega_n)^{S_n}_+ \rangle$.
The alternant space $A_n$ is a doubly graded vector space.
The Fields Conjecture would imply that
\begin{equation}
\label{swanson-wallach-theorem}
\Hilb(A_n; q, z) = 
\sum_{k = 1}^n z^{n-k} \cdot q^{{k \choose 2}} \cdot {n-1 \brack k-1}_q.
\end{equation}
Equation~\eqref{swanson-wallach-theorem} has been verified by Swanson and Wallach \cite{SW}, 
giving further evidence
for the Fields Conjecture.

If the Fields Conjecture is true, we would have an isomorphism of ungraded $S_n$-modules
$\Omega_n/\langle (\Omega_n)^{S_n}_+ \rangle \cong \bigoplus_{k = 1}^n (\QQ[\OP_{n,k}] \otimes \mathrm{sign})$. 
At present, it is unknown whether either of these $S_n$-modules injects into the other.

The symmetric functions appearing in the Fields Conjecture and \cref{v-isomorphism-type}
are closely related.  We propose the following `bridge conjecture' whose truth would yield the Fields 
Conjecture.
Let $\varphi$ be the composite linear map
\begin{equation}
\varphi: V_{n,1} \oplus \cdots \oplus V_{n,n} \hookrightarrow \Omega_n \twoheadrightarrow
 \Omega_n/\langle (\Omega_n)^{S_n}_+ \rangle 
\end{equation}
obtained by including the direct sum $V_{n,1} \oplus \cdots \oplus V_{n,n}$ into superspace
and then projecting onto the superspace coinvariant ring.

\begin{conj}
\label{phi-is-bijective}
The linear map $\varphi$ is bijective.
\end{conj}

Mike Zabrocki studied the triply diagonal action of $S_n$ on the ring $\Omega_n[y_1, \dots, y_n]$
and the associated space $\Omega_n[y_1, \dots, y_n]^{S_n}_+$ of $S_n$-invariants with vanishing constant term.
He checked the following conjecture by computer for $n \leq 6$.

\begin{zconj} (\cite{Zabrocki})
Let $n$ be a positive integer. We have
\begin{equation}
\grFrob( \Omega_n[y_1, \dots, y_n] / \langle \Omega_n[y_1, \dots, y_n]^{S_n}_+ \rangle; q, t, z)
= \sum_{k = 1}^n z^{n-k} \cdot \Delta'_{e_{k-1}} e_n
\end{equation}
where $q$ tracks $x$-degree, $t$ tracks $y$-degree, and $z$ tracks $\theta$-degree.
\end{zconj}

The Zabrocki Conjecture is related to \cref{bold-v-conjecture} in the same way as the 
Fields Conjecture is related to \cref{v-isomorphism-type}.
Since \cref{v-isomorphism-type} is proven whereas the Fields Conjecture remains open,
superspace Vandermondes might prove an easier road to Delta Conjecture modules than
quotient rings.














\section{The ring  $\WW_{n,k}$ and Super Poincar\'e Duality}
\label{Poincare}


So far we have built $S_n$-modules $V_{n,k}$ and $\VV_{n,k}$ by starting with the superspace
Vandermonde $\delta_{n,k}$ and closing under partial derivatives in the commuting variables $x_i, y_i$
(and potentially polarization operators).
The modules $V_{n,k}$ and $\VV_{n,k}$ have the defect of not being closed under multiplication and not 
admitting a natural ring structure.
In this section we build a new bigraded $S_n$-module $\WW_{n,k}$ from $\delta_{n,k}$. The module 
$\WW_{n,k}$ is naturally a bigraded quotient of $\Omega_n$. 
The module $\WW_{n,k}$ turns out to extend both $V_{n,k}$ and the cohomology ring
$H^{\bullet}(X_{n,k}; \QQ)$ of a variety $X_{n,k}$ of line configurations studied by Pawlowski and 
Rhoades. In order to define $\WW_{n,k}$, we need operators $\partial/\partial \theta_i$  on $\Omega_n$
which differentiate with respect to anticommuting variables.

For $1 \leq i \leq n$, let $\partial/\partial \theta_i: \Omega_n \rightarrow \Omega_n$ be the 
$\QQ[x_1, \dots, x_n]$-module endomorphism characterized by
\begin{equation}
\partial/\partial \theta_i: \theta_{j_1} \cdots \theta_{j_r} \mapsto
\begin{cases}
(-1)^{s-1} \theta_{j_1} \cdots \widehat{\theta_{j_s}} \cdots \theta_{j_r} & \text{if $j_s = i$} \\
0 & \text{if $i \neq j_1, \dots, j_r$}
\end{cases}
\end{equation}
where $1 \leq j_1, \dots, j_r \leq n$ are distinct indices and $\widehat{\cdot}$ means omission.
The sign $(-1)^{s-1}$ is necessary to ensure that $\partial/\partial \theta_i$ is well-defined.

\begin{defn}
For positive integers $k \leq n$, let $\WW_{n,k}$ be the smallest linear subspace of $\Omega_n$
which
\begin{itemize}
\item contains the superspace Vandermonde $\delta_{n,k}$, and
\item is closed under the $2n$ operators 
$\partial/\partial x_1, \dots, \partial/\partial x_n, \partial/\partial \theta_1, \dots, \partial/\partial \theta_n$.
\end{itemize}
\end{defn}

The vector space $\WW_{n,k}$ is a bigraded $S_n$-module. We use an action of superspace on itself
to show that $\WW_{n,k}$ is naturally a bigraded quotient of $\Omega_n$.

The operators $\partial/\partial \theta_i$ and $\partial/\partial x_i$ on $\Omega_n$ satisfy the relations
\begin{footnotesize}
\begin{equation*}
(\partial/\partial x_i) (\partial/\partial x_j) = (\partial/\partial x_j) (\partial/\partial x_i) \quad
(\partial/\partial x_i) (\partial/\partial \theta_j) = (\partial/\partial \theta_j) (\partial/\partial x_i)  \quad
(\partial/\partial \theta_i) (\partial/\partial \theta_j) = - (\partial/\partial \theta_j) (\partial/\partial \theta_i)
\end{equation*}
\end{footnotesize}
for all $1 \leq i, j \leq n$.  These are the defining relations of  $\Omega_n$, so for any 
superpolynomial $f = f(x_1, \dots, x_n, \theta_1, \dots, \theta_n)$ we have an unambiguous operator
$\partial f$
on $\Omega_n$ obtained by replacing each $x_i$ in $f$ with $\partial/\partial x_i$ and each $\theta_i$ in $f$
by $\partial/\partial \theta_i$.  This gives rise to an action $\odot: \Omega_n \times \Omega_n \rightarrow \Omega_n$
of superspace on itself by the rule
\begin{equation}
f \odot g := \partial f (g).
\end{equation}

\begin{prop}
\label{w-is-a-ring}
Let $\ann (\delta_{n,k}) := \{ f \in \Omega_n \,:\, f \odot \delta_{n,k} = 0 \}$ be the annihilator in 
$\Omega_n$ of the superspace Vandermonde $\delta_{n,k}$.  Then  $\ann (\delta_{n,k})$ is a two-sided
ideal in $\Omega_n$ which is $S_n$-stable and bigraded. 
The canonical composition 
\begin{equation}
\WW_{n,k} \hookrightarrow \Omega_n \twoheadrightarrow \Omega_n / \ann( \delta_{n,k})
\end{equation}
is an isomorphism of bigraded $S_n$-modules.
\end{prop}

Thanks to \cref{w-is-a-ring}, there is a natural multiplication operation on $\WW_{n,k}$, so that
the anticommuting differentiation operators $\partial/\partial \theta_i$ give rise to a ring structure which 
$V_{n,k}$ and $\VV_{n,k}$ lack.

What do the bigraded $S_n$-modules $\WW_{n,k}$ look like?
We display $\grFrob(\WW_{4,2}; q, z)$ in matrix format, with rows labeling $\theta$-degree
and columns labeling $x$-degree.
\begin{equation}
\label{frob-example}
\grFrob(\WW_{4,2}; q, z) = 
\begin{pmatrix}
s_4 & s_4 + s_{31} & s_4 + s_{31} + s_{22} & s_{31} \\
s_{31} & 2 s_{31} + s_{22} + s_{211} & s_{31} + s_{22} + 2 s_{211} & s_{211} \\
s_{211} & s_{22} + s_{211} + s_{1111} & s_{211} + s_{1111} & s_{1111}
\end{pmatrix}
\end{equation}
The matrices $\grFrob(\WW_{n,k}; q, z)$ enjoy the following properties. Let $U_n = S^{(n-1,1)}$ 
be the $(n-1)$-dimensional
reflection representation of $S_n$.

\begin{thm}
\label{w-basic-facts}
There hold the following facts about the bigraded $S_n$-module $\WW_{n,k}$.
\begin{enumerate}
\item {\em (Special $k$)} We have $\WW_{n,n} \cong R_n$ (coinvariant ring)
and $\WW_{n,1} \cong \wedge U_n$ (exterior algebra).
\item {\em (Bottom $x$-degree)} The $x$-degree $0$ piece of $\WW_{n,k}$ is isomorphic to
$\bigoplus_{j = 0}^{n-k} \wedge^j U_n$.
\item {\em (Top $x$-degree)} The top $x$-degree of 
$\WW_{n,k}$ is ${k \choose 2} + (n-k) \cdot (k-1)$; this piece of $\WW_{n,k}$
is isomorphic to 
$\bigoplus_{j = 0}^{n-k} \wedge^j U_n \otimes \mathrm{sign}$.
\item {\em (Top $\theta$-degree)} The top ($= n-k$) $\theta$-degree piece of $\WW_{n,k}$ is isomorphic to $V_{n,k}$.
\item {\em (Bottom $\theta$-degree)}  Let $I_{n,k} \subseteq \QQ[x_1, \dots, x_n]$ be 
$I_{n,k} := \langle x_1^k, \dots, x_n^k, e_n, e_{n-1}, \dots, e_{n-k+1} \rangle$
 and let 
$R_{n,k} := \QQ[x_1, \dots, x_n]/I_{n,k}.$
The $\theta$-degree $0$ piece of $\WW_{n,k}$ is isomorphic to $R_{n,k}$.
\end{enumerate}
\end{thm}

The quotient rings $R_{n,k}$  in Item 5 of \cref{w-basic-facts} were introduced
by Haglund, Rhoades, and Shimozono \cite{HRS}.  They proved that 
\begin{equation}
\label{r-isomorphism-type}
\grFrob(R_{n,k}; q) = (\mathrm{rev}_q \circ \omega) \Delta'_{e_{k-1}} e_n \mid_{t = 0},
\end{equation}
where $\omega$ is the symmetric function involution which trades $s_{\lambda}$ and $s_{\lambda'}$ 
and $\mathrm{rev}_q$ reverses the coefficient sequences of polynomials in $q$.
The ring $R_{n,k}$ was the first model for a coinvariant ring attached to the Delta Conjecture.

The rings $R_{n,k}$ have a geometric interpretation.
A {\em line} in the $k$-dimensional complex vector space $\CC^k$ is a $1$-dimensional linear subspace.
Pawlowski and Rhoades defined \cite{PR} the variety $X_{n,k}$ of spanning configurations of $n$ lines in $\CC^k$:
\begin{equation}
X_{n,k} := \{ (\ell_1, \dots, \ell_n) \,:\, \ell_i \subseteq \CC^k \text{ a line and } \ell_1 + \cdots + \ell_n = \CC^k \}.
\end{equation}

The space $X_{n,k}$ and its cohomology ring $H^{\bullet}(X_{n,k}; \QQ)$ admit $S_n$-actions 
by line permutation. Pawlowski and Rhoades presented \cite{PR} the cohomology  $H^{\bullet}(X_{n,k}; \QQ)$  as 
\begin{equation}
H^{\bullet}(X_{n,k}; \QQ) = \QQ[x_1, \dots, x_n]/I_{n,k}  = R_{n,k}.
\end{equation}
We may therefore interpret the $\theta$-degree 0 piece of $\WW_{n,k}$ as the cohomology of $X_{n,k}$.

The `twist' $(\mathrm{rev}_q \circ \omega)$ involved in Equation~\eqref{r-isomorphism-type}
can be visualized in the matrix representing $\grFrob(\WW_{4,2}; q, z)$ in \eqref{frob-example}.
Namely, the top row can be obtained from the bottom row by reversal together with applying the 
operator $\omega$.
The  reader may notice that the middle row of $\grFrob(\WW_{4,2}; q, z)$ is  invariant
under reversal followed by $\omega$.  This observation generalizes as follows.

\begin{thm}
\label{w-symmetry}
The matrix representing $\grFrob(\WW_{n,k}; q, z)$ is  invariant under $180^{\circ}$ rotation
followed by the application of $\omega$ to each entry.
\end{thm}



Recall that a sequence of numbers $(a_0, a_1, \dots, a_d)$ is  {\em palindromic} if $a_i = a_{d-i}$ for all $i$
and {\em unimodal} if $a_0 \leq a_1 \leq \cdots \leq a_r \geq a_{r+1} \geq \cdots \geq a_d$ for some $r$.
A famous example of a polynomial in $\QQ[q]$ with a palindromic and unimodal coefficient sequence is the
$q$-factorial $[n]!_q = (1+q)(1+q+q^2) \cdots (1+q+ \cdots + q^{n-1})$.  While these facts about $[n]!_q$ 
follow from showing that if $f(q)$ and $g(q)$ have palindromic unimodal coefficient sequences, so does
$f(q) \cdot g(q)$, there is a deeper derivation coming from geometry.

A finite-dimensional graded $\QQ$-algebra $A = \bigoplus_{i = 0}^d A_i$ is a {\em Poincar\'e Duality Algebra}
if $A_d \cong \QQ$ is $1$-dimensional and if for all $0 \leq i \leq d$ the  map
$A_i \otimes A_{d-i} \rightarrow A_d \cong \QQ$ is a perfect pairing.
This forces $\dim A_i = \dim A_{d-i}$.

Let $\Fl_n$ be the variety of complete flags in $\CC^n$. Borel proved \cite{Borel} that the cohomology of $\Fl_n$ has 
presentation $H^{\bullet}(\Fl_n; \QQ) = R_n$ given by the coinvariant ring.  Since $\Fl_n$ is a 
compact complex manifold, the ring $H^{\bullet}(\Fl_n; \QQ)$ is a 
Poincar\'e Duality Algebra and the palindromicity
of its Hilbert polynomial $[n]!_q$ follows.

The complex variety $X_{n,k}$ is smooth, but usually not compact. Indeed, the cohomology ring 
$H^{\bullet}(X_{n,k}; \QQ) = R_{n,k}$ does not usually have a palindromic Hilbert series, e.g.
 $\Hilb(R_{3,2};q) = 1 + 3q + 2q^2$.
However, the extension $\WW_{n,k} \supseteq R_{n,k}$ exhibits a superspace version of Poincar\'e Duality.

Let $A = \bigoplus_{i = 0}^d \bigoplus_{j = 0}^e A_{i,j}$ be a finite-dimensional bigraded $\QQ$-algebra.
We say that $A$ is a {\em Super Poincar\'e Duality Algebra} if $A_{d,e} \cong \QQ$  and 
$A_{i,j} \otimes A_{d-i,e-j} \rightarrow A_{d,e}$ is a perfect pairing for all $0 \leq i \leq d$ and $0 \leq j \leq e$.

\begin{thm}
\label{super-theorem}
The bigraded algebra $\WW_{n,k}$ is a Super Poincar\'e Duality Algebra.
\end{thm}

Does \cref{super-theorem} have geometric meaning?
Is there a `superspace version' of cohomology which yields $\WW_{n,k}$ when applied to $X_{n,k}$?

The unimodality of $[n]!_q$ also has geometric meaning.  A Poincar\'e Duality Algebra
 $A = \bigoplus_{i = 0}^d A_i$ satisfies the {\em Hard Lefschetz Property} if there exists an element 
 $\ell \in A_1$ (called a {\em Lefschetz element}) such that for any $i \leq d/2$ the map
 $A_i \xrightarrow{ \times \ell^{d-2i} } A_{d-i}$ of multiplication by $\ell^{d-2i}$ is bijective.

Since $\Fl_n$ is a compact complex manifold and $H^{\bullet}(\Fl_n; \QQ) = R_n$,
the ring $R_n$ satisfies the 
Hard Lefschetz Property. Maneo, Numata, and Wachi proved  \cite{MNW} that 
a linear form $\ell = c_1 x_1 + \cdots + c_n x_n$ is a Lefschetz element if and only if  
$c_1, \dots, c_n \in \QQ$ are distinct.

As a closing example, we display the bigraded Hilbert series $\Hilb(\WW_{4,2}; q, z)$ as a matrix
where rows index $\theta$-degree and columns index $x$-degree.
\begin{equation}
\label{hilb-example}
\Hilb(\WW_{4,2}; q, z) = \begin{pmatrix}
1 & 4 & 6 & 3 \\
3 & 11 & 11 & 3 \\
3 & 6 & 4 & 1
\end{pmatrix}
\end{equation}
Either \cref{w-symmetry} or \cref{super-theorem} imply that the matrix
$\Hilb(\WW_{n,k}; q, z)$ is always invariant under $180^{\circ}$ rotation.

\begin{conj}
\label{super-conj}
Each row and column in the matrix representing $\Hilb(\WW_{n,k}; q, z)$ is unimodal.
\end{conj}

\cref{super-conj} would be best proven by showing that $\WW_{n,k}$ satisfies an
as-yet-undefined `Super Hard Lefschetz Property'.



\acknowledgements{The authors are grateful to Josh Swanson and Mike Zabrocki for helpful conversations.}

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