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\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
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\newtheorem{rmk}[thm]{Remark}
\newtheorem{conj}[thm]{Conjecture}

\newcommand{\CC}{{\mathbb {C}}}
\newcommand{\FDR}{{\mathrm {FDR}}}
\newcommand{\PP}{{\mathbb {P}}}
\newcommand{\Frob}{{\mathrm {Frob}}}
\newcommand{\sign}{{\mathrm {sign}}}
\newcommand{\GL}{{\mathrm {GL}}}
\newcommand{\ZZ}{{\mathbb {Z}}}
\newcommand{\cont}{{\mathrm {cont}}}
\newcommand{\ch}{{\mathrm {ch}}}
\newcommand{\Nar}{{\mathrm {Nar}}}
\newcommand{\Cat}{{\mathrm {Cat}}}
\newcommand{\gr}{{\mathrm {gr}}}
\newcommand{\EEE}{{\mathcal {E}}}
\newcommand{\GGG}{{\mathcal {G}}}
\newcommand{\grFrob}{{\mathrm {grFrob}}}
\newcommand{\wt}{{\mathrm {wt}}}
\newcommand{\asc}{{\mathrm {asc}}}
\newcommand{\codinv}{{\mathrm {codinv}}}



%\usepackage{lipsum}

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\title{Fermionic diagonal coinvariants and exterior Lefschetz elements}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Jongwon Kim\thanks{\href{mailto:jk1093@sas.upenn.edu}{jk1093@sas.upenn.edu}}\addressmark{1} \and Brendon Rhoades
\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}. B. Rhoades was partially supported by NSF Grants DMS-1500838 and 
DMS-1953781.}\addressmark{2}}

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\address{\addressmark{1}Department of Mathematics, University of Pennsylvania \\ 
\addressmark{2}Department of Mathematics, University of California, San Diego}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{Let $W$ be a complex reflection group acting irreducibly on its reflection representation $V$.
The group $W$ acts diagonally on the exterior algebra $\wedge (V \oplus V^*)$ over the direct sum of $V$
with its dual space $V^*$. We study the \emph{$W$-fermionic diagonal coinvariant ring}
$FDR_W$ obtained by quotienting $\wedge (V \oplus V^*)$ by the ideal generated by $W$-invariants with 
vanishing constant term.}

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\keywords{reflection group, fermions, diagonal coinvariants}

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\begin{document}

\maketitle
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\section{The $W$-Fermionic Diagonal Coinvariant Ring}

Let $X_n = (x_1, \dots, x_n)$ and $Y_n = (y_1, \dots, y_n)$ be two lists of $n$ variables and let 
$\CC[X_n, Y_n]$ be the polynomial ring in these $2n$ variables over the complex numbers.
The symmetric group acts diagonally on $\CC[X_n, Y_n]$, viz.
\begin{equation*}
w \cdot x_i := x_{w(i)} \quad \quad w \cdot y_i := y_{w(i)} \quad \quad 
\text{for all $w \in S_n$ and $1 \leq i \leq n$.}
\end{equation*}
Let $\CC[X_n, Y_n]^{S_n} := \{ f \in \CC[X_n, Y_n] \,:\, w \cdot f = f \text{ for all $w \in S_n$} \}$ be the 
corresponding invariant subring and let 
$\CC[X_n, Y_n]^{S_n}_+$ be the space of $S_n$-invariants with vanishing constant term.
The {\em diagonal coinvariant ring} is the quotient
\begin{equation}
DR_n := \CC[X_n, Y_n] / \langle \CC[X_n, Y_n]^{S_n}_+ \rangle.
\end{equation}
This is a doubly graded $S_n$-module, with one grading coming from $X_n$ and another coming from $Y_n$.
Haiman \cite{Haiman} proved that $DR_n$ carries (up to sign twist) the permutation action of $S_n$ on size $n$
parking functions and gave an expression for the bigraded $S_n$-isomorphism type of $DR_n$ in terms 
of the $\nabla$ operator on symmetric functions.  

Over the last couple years, many researchers in algebraic combinatorics 
\cite{Bergeron, BRT, Griffin, HRS, OZ, RW, Zabrocki} have considered variations  
on the diagonal coinvariant ring $DR_n$. Some of these modules 
\cite{Bergeron, BRT, OZ, RW, Zabrocki} have involved the use of not only commuting variables, 
but also anticommuting variables.
This was motivated in part by the {\em superspace} ring 
$$\Omega_n = \CC[x_1, \dots, x_n] \otimes \wedge \{ \theta_1, \dots, \theta_n \}$$
given by the tensor product of a rank $n$ polynomial ring with a rank $n$ exterior algebra. 
The ring $\Omega_n$ has a long history in physics; the variables $x_i$ are called {\em bosonic}
(with a power $x_i^2$ corresponding to two indsintinguishable bosons in state $i$) and the 
variables $\theta_i$ are called {\em fermionic}
(with the relation $\theta_i^2 = 0$ corresponding to the {\em Pauli Exclusion Principle}:
two fermions cannot occupy the same state at the same time).
The `super' in superspace comes from `supersymmetry' between bosons and fermions;
in {\bf Section~\ref{Bergeron}} we recall a beautiful conjecture of F. Bergeron \cite{Bergeron}
on supersymmetry in $S_n$-coinvariant theory and explain
(Remark~\ref{supersymmetry-remark})
how it relates to our work.

In this extended abstract we define and study a variant on $DR_n$ in which all variables are fermionic
(i.e. anticommuting). Our construction extends simply and uniformly 
from the symmetric group $S_n$ to any complex reflection group $W$.
Before defining our ring, we recall the relevant reflection group terminology.

Let $V = \CC^n$ be an $n$-dimensional complex vector space. An element
$t \in \GL(V)$ is called a {\em reflection} if it has finite order and if the fixed space
$V^t := \{ v \in V \,:\, t \cdot v = v \}$ has codimension one in $V$.
A {\em complex reflection group} is a finite subgroup $W \subseteq \GL(V)$ which is generated by reflections.
The group $W$ acts naturally on $V$; 
we call $W$  {\em irreducible} if $V$ is an
irreducible $W$-module. If $W$ is irreducible, we say that $W$ has {\em rank} $n = \dim V$ and call
$V$ the {\em reflection representation} of $W$.

Let $W$ be a complex reflection group of rank $n$ acting irreducibly on its reflection representation $V$.
We may construct several $W$-modules from $V$, including
\begin{itemize}
\item the $n$-dimensional dual space $V^*$ of all linear functionals $V \rightarrow \CC$,
\item the $2n$-dimensional direct sum $V \oplus V^*$ of $V$ with its dual space, and finally
\item the $2^{2n}$-dimensional exterior algebra $\wedge (V \oplus V^*)$ over $V \oplus V^*$.
\end{itemize}
This last space $\wedge (V \oplus V^*)$ may be regarded as a bigraded $W$-module by placing $V$ in
bidegree $(1,0)$ and $V^*$ in bidegree $(0,1)$. The $(i,j)$-graded piece $\wedge (V \oplus V^*)_{i,j}$
is then given by $\wedge^i V \otimes \wedge^j V^*$.

It will sometimes be convenient to coordinatize our spaces.
Let $\Theta_n = (\theta_1, \dots, \theta_n)$ be a basis of $V$ and let 
$\Xi_n = (\xi_1, \dots, \xi_n)$ be the corresponding 
dual basis of $V^*$ characterized by $\xi_i(\theta_j) = \delta_{i,j}$ (Kronecker delta). Write
$$
\wedge \{ \Theta_n, \Xi_n \} := \wedge \{ \theta_1, \dots, \theta_n, \xi_1, \dots, \xi_n \}
$$
for the exterior algebra (over $\CC$) generated by these $2n$ symbols. We have a natural identification
of bigraded $W$-modules $\wedge \{ \Theta_n, \Xi_n \} \cong \wedge (V \oplus V^*)$.
The following quotient ring is our object of study.

\begin{defn}
Let $W$ be an irreducible complex reflection group acting on its reflection representation $V = \CC^n$.
Let $\wedge (V \oplus V^*)^W_+ \subseteq \wedge (V \oplus V^*)$ be the subspace of $W$-invariants
with vanishing constant term and let  $\langle \wedge (V \oplus V^*)^W_+  \rangle$ be the ideal generated
by this subspace.  The {\em $W$-fermionic diagonal coinvariant ring} is the quotient
\begin{equation}
FDR_W := \wedge (V \oplus V^*) / \langle \wedge (V \oplus V^*)^W_+  \rangle.
\end{equation}
\end{defn}

The ring $FDR_W$ is a bigraded $W$-module, with one grading coming from $V$ and another from $V^*$. We 
denote its bigraded decomposition as
\begin{equation}
FDR_W = \bigoplus_{i,j = 0}^n (FDR_W)_{i,j}.
\end{equation}
In our coordinate model, we may write 
\begin{equation}
FDR_W = \wedge \{ \Theta_n, \Xi_n \} / \langle \wedge \{ \Theta_n, \Xi_n \}^W_+ \rangle.
\end{equation}











\section{Bigraded Structure of $FDR_W$}


As with any representation, the first and most basic question one can ask about $FDR_W$ 
is its vector space dimension and the dimension of its bigraded pieces $(FDR_W)_{i,j}$.
Here we get a combinatorial surprise: an appearance of the {\em Catalan}
and {\em Narayana numbers}:
\begin{equation}
\Cat(n) := \frac{1}{n+1} {2n \choose n} \quad \quad 
\Nar(n,k) := \frac{1}{n} {n \choose k} {n \choose k-1}.
\end{equation}
%The Narayana numbers refine the Catalan numbers in that $\sum_{k = 0}^n \Nar(n,k) = \Cat(n)$.


\begin{thm}
\label{vector-space-structure}
Let $W$ be a rank $n$ irreducible reflection group and let 
$FDR_W = \bigoplus_{i,j = 0}^n (FDR_W)_{i,j}$ 
be the fermionic 
diagonal coinvariant ring. 
\begin{enumerate}
\item We have $\dim FDR_W = {2n+1 \choose n}$.
\item The bigraded component $(FDR_W)_{i,j}$ is zero unless $i + j \leq n$.
\item When $i+j \leq n$ we have
$\dim (FDR_W)_{i,j} = {n \choose i} {n \choose j} - {n \choose i-1} {n \choose j-1}$.
\item \label{item:FDR-Nar}For any $0 \leq k \leq n$ we have $\dim (FDR_W)_{k,n-k} = \Nar(n,k)$, the Narayana number.
\item  We have $\sum_{k = 0}^n \dim (FDR_W)_{k,n-k} = \Cat(n)$, the Catalan number.
\end{enumerate}
\end{thm}

Theorem~\ref{vector-space-structure} says that the ring $FDR_W$ lives in the `triangle' of bidegrees
defined by $i, j \geq 0$ and $i + j \leq n$.  The `extreme' bidegree components
$i + j = n$ give a Catalan-dimensional $W$-module with a natural direct sum decomposition into 
Narayana-dim\-en\-sion\-al submodules.
The natural $\mathrm{GL}_2$-action on the $2 \times n$ matrix of variables
$\left(\begin{smallmatrix} \theta_1 & \cdots & \theta_n \\ \xi_1 & \cdots & \xi_n \end{smallmatrix}\right)$ induces actions
on $\wedge \{ \Theta_n, \Xi_n \}$, its quotient $FDR_W$, and on its Catalan-dimensional submodule
$\bigoplus_{i + j = n} (FDR_W)_{i,j}$.
The representation theory of $\mathrm{GL}_2$ combines with 
point~(\ref{item:FDR-Nar}.) of Theorem~\ref{vector-space-structure} to explain algebraically
the palindromicity and unimodality of the sequence
$( \Nar(n,k) )_{k = 1}^n$ of Narayana numbers.


\begin{rmk}
While there exist Catalan numbers $\Cat(W)$ and Narayana numbers $\Nar(W,k)$ defined
for any reflection group $W$ in terms of invariant degrees, only their type A versions show
up in Theorem~\ref{vector-space-structure}. In contrast to the bosonic case, results in fermionic diagonal
coinvariant theory tend to depend on only the rank of the group $W$, rather than $W$ itself.
\end{rmk}

Our next result enhances Theorem~\ref{vector-space-structure} by describing the bigraded 
$W$-isomorphism type of $FDR_W$.  We state our answer in terms of the {\em Grothendieck ring}
of $W$. This is the $\mathbb{Z}$-algebra generated by isomorphism classes $[U]$ of $W$-modules
subject to relations $[U] = [U'] + [U'']$ for any short exact sequence
$0 \rightarrow U' \rightarrow U \rightarrow U'' \rightarrow 0$. Multiplication in the Grothendieck ring
corresponds to tensor product, viz. $[U] \cdot [U'] := [U \otimes U']$ where $W$ acts diagonally
on $U \otimes U'$.


\begin{thm}
\label{w-module-structure}
Let $W$ be a rank $n$ irreducible reflection group. For any $i + j \leq n$, in the Grothendieck
ring of $W$ there holds the equation
\begin{equation}
[ (FDR_W)_{i,j} ] = [ \wedge^i V ] \cdot [\wedge^j V^* ] - [ \wedge^{i-1} V] \cdot [ \wedge^{j-1} V^* ]
\end{equation}
where we interpret $\wedge^{-1} V = \wedge^{-1} V^* = 0$.
\end{thm}

Our next result gives a basis of $FDR_W$. To do this, we use
our coordinate model $FDR_W = \wedge \{ \Theta_n, \Xi_n \} / \langle \wedge \{ \Theta_n, \Xi_n \}^W_+ \rangle$
and interpret exterior monomials in $\wedge \{ \Theta_n, \Xi_n \}$ in terms of certain lattice paths.

We denote by $\Pi(n)$ the family of $n$-step lattice paths in $\ZZ^2$ which 
\begin{itemize}
\item
start at the origin $(0,0)$ and
\item
consist of up-steps $(1,1)$, down-steps $(1,-1)$, and horizontal steps $(1,0)$, where
\item
every horizontal step has one of the decorations $\theta$ or $\xi$.
\end{itemize}
We define $\Pi(n)_{\geq 0} \subseteq \Pi(n)$ to be the subfamily of paths which remain weakly
above the $x$-axis.

Paths in $\Pi(n)$ encode monomials in $\wedge \{ \Theta_n, \Xi_n \}$. To see how,
 let $\sigma = (\sigma_1,  \dots, \sigma_n) \in \Pi(n)$ be a path with steps $\sigma_1, \dots, \sigma_n$.
 For $1 \leq i \leq n$, we assign the step $\sigma_i$ the following weight:
 \begin{equation}
 \wt(\sigma_i) = \begin{cases}
 1 & \text{if $\sigma_i$ is an up-step,} \cr
 \theta_i & \text{if $\sigma_i$ is a horizontal step decorated $\theta$,} \cr
 \xi_i & \text{if $\sigma_i$ is a horizontal step decorated $\xi$,} \cr
 \theta_i \xi_i & \text{ if $\sigma_i$ is a down-step.}
 \end{cases}
 \end{equation}
 The path $\sigma$ itself is weighted by the product of its step weights:
 \begin{equation}
 \wt(\sigma) = \wt(\sigma_1) \cdots \wt(\sigma_n).
 \end{equation}
The family $\{ \wt(\sigma) \,:\, \sigma \in \Pi(n) \}$ of all path weights is precisely the family 
 of monomials in $\wedge \{ \Theta_n, \Xi_n \}$ (up to sign). By considering paths which do not 
 sink below the $x$-axis, we get a monomial basis of $FDR_W$.
 
 \begin{thm}
 \label{w-basis-theorem}
 Let $W$ be a rank $n$ irreducible complex reflection group.
 The set 
 \begin{equation}
 \{ \wt(\sigma) \,:\, \sigma \in \Pi(n)_{\geq 0} \} 
 \end{equation}
 descends to a monomial basis of $FDR_W$.
 \end{thm}
 
 In fact, the basis of Theorem~\ref{w-basis-theorem} is a {\em standard monomial basis}
 in the sense of (exterior) Gr\"obner theory.
 We close this section by remarking that the conclusions
 Theorems~\ref{vector-space-structure}, \ref{w-module-structure}, and \ref{w-basis-theorem} hold
 in greater generality than their hypotheses.
 
 \begin{rmk}
 The results of this section apply not just to a complex reflection group $W \subseteq \GL(V)$,
  but to any subgroup $G \subseteq \GL(V)$ for which the modules
  $\wedge^0 V, \wedge^1 V, \dots, \wedge^{\dim V} V$ are inequivalent 
  $G$-irreducibles. One example of such a group $G$ is the general linear group $\GL(V)$ itself.
 \end{rmk}

















\section{The case of the symmetric group}
\label{Symmetric}

In this section we specialize to the case $W = S_n$ of the symmetric group. In this setting, it is traditional
to consider the $n$-dimensional permutation representation $U = \CC^n$ rather than its 
$(n-1)$-dimensional irreducible reflection submodule $V$. In the coinvariant context, this has only minor effects.
Indeed, we have $U = V \oplus U^{S_n}$ and $U^* = V^* \oplus (U^*)^{S_n}$ so that 
\begin{multline*}
\wedge (U \oplus U^*) \cong \wedge [ (V \oplus U^{S_n}) \oplus (V^* \oplus (U^*)^{S_n}) ] \\
\cong \wedge [ (V \oplus V^*) \oplus (U^{S_n} \oplus (U^*)^{S_n}] 
\cong [ \wedge (V \oplus V^*) ] \otimes [ \wedge (U^{S_n} \oplus (U^*)^{S_n}) ]
\end{multline*}
and modding out by the ideals generated by $S_n$-invariants with vanishing constant term gives
\begin{equation}
\wedge (U \oplus U^*) / \langle \wedge (U \oplus U^*)^{S_n}_+ \rangle \cong 
\wedge (V \oplus V^*) / \langle \wedge (V \oplus V^*)^{S_n}_+ \rangle.
\end{equation}
We may therefore harmlessly adopt the convenient realization of the $S_n$-structure of
\begin{equation}
FDR_n := FDR_{S_n} \cong \wedge \{ \Theta_n, \Xi_n \} / \langle \wedge \{ \Theta_n, \Xi_n \}^{S_n}_+ \rangle
\end{equation}
 by subscript permutation.

To state Theorems~\ref{vector-space-structure} and
\ref{w-module-structure} 
for the $S_n$-module
 $FDR_n$, we recall some ideas from symmetric group representation theory.
Let $\Lambda = \bigoplus_{n \geq 0} \Lambda_n$ be the ring of symmetric functions
in an infinite variable set $X = (x_1, x_2, \dots )$ with
$\Lambda_n$ denoting its $n^{th}$ graded piece.
Bases of $\Lambda_n$ are indexed by partitions $\lambda \vdash n$. Given a partition $\lambda$, we let 
$s_{\lambda}(X)$ be the associated {\em Schur function}.

Irreducible representations of $S_n$ are indexed by partitions of $n$; if $\lambda \vdash n$ is a partition
we let $V^{\lambda}$ be the corresponding $S_n$-irreducible. Any finite-dimensional $S_n$-module
$V$ has a unique decomposition $V = \bigoplus_{\lambda \vdash n} c_{\lambda} V^{\lambda}$
for some multiplicities $c_{\lambda} \geq 0$. The {\em Frobenius image} of $V$
is the symmetric function $\Frob(V) := \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}$
obtained by replacing the irreducible $V^{\lambda}$ with the Schur function $s_{\lambda}$.

As an example of what the $FDR_n$ modules look like, the Frobenius images of the bigraded pieces of
$FDR_4$ are displayed in matrix format below, where the rows correspond to increasing $\theta$-degree
and the columns correspond to increasing $\xi$-degree.
\begin{equation}
\label{4-example}
FDR_4 \leftrightarrow
\begin{pmatrix}
s_4 & s_{31} & s_{211} & s_{1111} \\
s_{31} & s_{211} + s_{22} + s_{31} & s_{1111} + s_{211} + s_{22} \\
s_{211} & s_{1111} + s_{211} + s_{22}  \\
s_{1111}
\end{pmatrix}
\end{equation}
The symmetry of \eqref{4-example} across the main diagonal reflects the symmetry between the 
$\Theta_n$ and $\Xi_n$ in the definition of $FDR_n$.

The natural external product on the graded ring $\Lambda = \bigoplus_{n \geq 0} \Lambda_n$ 
corresponds to induction product of symmetric group modules. Far more mysterious is the 
{\em Kronecker product} $*$ defined on each graded piece $\Lambda_n$ by 
$s_{\lambda} * s_{\mu} = \sum_{\gamma \vdash n} g_{\lambda,\mu,\gamma} \cdot s_{\gamma}$
where the {\em Kroencker coefficient} $g_{\lambda,\mu,\gamma} \geq 0$ is given by
$V^{\lambda} \otimes V^{\mu} \cong_{S_n} 
\bigoplus_{\gamma \vdash n} g_{\lambda,\mu,\gamma} \cdot V^{\gamma}$.
No general combinatorial formula is known for the numbers $g_{\lambda,\mu,\gamma}$.


\begin{thm}
\label{sn-structure-theorem}
The following facts hold concerning the bigraded $S_n$-module \[FDR_n = \bigoplus_{i,j = 0}^n (FDR_n)_{i,j}.\]
\begin{enumerate}
\item  We have $\dim  FDR_n = {2n - 1 \choose n}$.
\item  We have $(FDR_n)_{i,j} = 0$ unless $i + j \leq n-1$.
\item  When $i + j \leq n-1$ we have
\begin{equation}
\Frob(FDR_n)_{i,j} = s_{(n-i,1^i)} * s_{(n-j,1^j)} - s_{(n-i+1,1^{i-1})} * s_{(n-j+1,1^{j-1})}
\end{equation}
where we interpret the second term to be zero when $i = 0$ or $j = 0$.
\item For any $0 \leq k \leq n-1$ we have $\dim (FDR)_{k,n-k-1} = \Nar(n-1,k)$.
\item We have $\sum_{k = 0}^{n-1} \dim (FDR)_{k,n-k-1} = \Cat(n-1)$.
\end{enumerate}
\end{thm}



Theorem~\ref{sn-structure-theorem} (1) proves a conjecture of Mike Zabrocki \cite{ZabrockiCH}.
The Kronecker products of hook-shaped Schur functions appearing in Theorem~\ref{sn-structure-theorem} (2)
may be evaluated using work of Rosas \cite{Rosas}. 
The discrepancies between Theorem~\ref{sn-structure-theorem} and 
Theorems~\ref{vector-space-structure} and \ref{w-module-structure} come from the reducibility of the 
permutation representation of $S_n$.

A glance at the matrix \eqref{4-example} shows some patterns. There is a unique copy of the 
trivial representation $s_4$ in bidegree $(0,0)$ and a single copy of the sign representation
$s_{1111}$ in the extreme bidegrees $(i,j)$ with $i+j = 3$.  These patterns, along with an expression
for the bigraded multiplicities of hook representations, generalize as follows.
Let $\langle - , - \rangle$ be the {\em Hall inner product} on $\Lambda_n$ obtained by declaring the 
Schur basis $\{ s_{\lambda} \,:\, \lambda \vdash n \}$ to be orthonormal.
We also define the {\em $q,t$-number} $[m]_{q,t} := q^{m-1} + q^{m-2} t + \cdots + q t^{m-2} + t^{m-1}$.

\begin{thm}
\label{multiplicities}
For any partition $\lambda \vdash n$, let 
$c_{\lambda}(q,t) := \sum_{i,j \geq 0} \langle \Frob (FDR_n)_{i,j}, s_{\lambda} \rangle \cdot q^i t^j$ be the bigraded
multiplicity of $V^{\lambda}$ in $FDR_n$.  
\begin{enumerate}
\item We have $c_{\lambda}(q,t) = 0$ unless $\lambda_3 < 3$.
\item We have $c_{(n)}(q,t) = 1$.
\item We have $c_{(1^n)}(q,t) = [n]_{q,t}$.
\item  We have $c_{(n-k,1^k)}(q,t) = [k+1]_{q,t} + (qt) \cdot [k]_{q,t}$ for any $0 < k < n-1$.
\end{enumerate}
\end{thm}

%The space $FDR_n$ admits a similar lattice path basis to $FDR_W$ involving modified edge
%weights. For $1 \leq i \leq n$, define $\xi_i' := \xi_1 + \cdots + \xi_{i-1} + 2 \xi_i + \xi_{i+1} + \cdots + \xi_n$.
%Given any lattice path $\sigma \in \Pi(n)$, define a {\em primed weight}
%$\wt'(\sigma) \in \wedge \{ \Theta_n, \Xi_n \}$ in the same way as $\wt(\sigma)$, except
%we declare the weight of a $\xi$-decorated horizontal step $\sigma_i$ to be $\xi'_i$ instead of 
%$\xi_i$ and the weight of a down-step $\sigma_i$ to be $\theta_i \xi'_i$ instead of $\theta_i \xi_i$.

%\begin{thm}
%\label{sn-basis}
%The set $\{ \wt'(\sigma) \,:\, \sigma \in \Pi(n)_{> 0} \}$ descends to a basis of $FDR_n$.
%\end{thm}



\section{F. Bergeron's Combinatorial Supersymmetry Conjecture}
\label{Bergeron}

The diagonal coinvariants $DR_n$ (two sets of bosonic variables and no fermionic variables)
and the ring $FDR_n$ (no bosonic variables and two sets of fermionic variables) motivate the 
study of a multigraded coinvariant $S_n$-quotient involving $r$ sets of bosonic variables and $s$
sets of fermionic variables.
To formalize this, for $n, r, s \geq 0$ let
$X_{r \times n}$ be an $r \times n$ matrix of bosonic variables and $\Theta_{s \times n}$ be an
$s \times n$ matrix of fermionic variables and let
\begin{equation}
S(n;r,s) := \CC[X_{r \times n}] \otimes \wedge \{ \Theta_{s \times n} \}
\end{equation}
be the tensor product of the polynomial ring over $X_{r \times n}$ with the exterior algebra 
over $\Theta_{s \times n}$.

The symmetric group $S_n$ acts in a `multidiagonal' way on $S(n;r,s)$ by simultaneously permuting 
the columns of $X_{r \times n}$ and $\Theta_{s \times n}$. Let $I(n;r,s) \subseteq S(n;r,s)$ be the ideal
generated by the $S_n$-invariants with vanishing constant term and consider the 
{\em boson-fermion coinvariant ring}
\begin{equation}
R(n;r,s) := S(n;r,s)/I(n;r,s).
\end{equation}
The ring $R(n;r,s)$ is an $S_n$-module which carries an $r$-fold grading in the bosonic variable sets and 
an $s$-fold grading in the fermionic variable sets.
Explicit generators for the defining ideal $I(n;r,s)$ of $R(n;r,s)$ were found by Orellana and Zabrocki \cite{OZ}.
Special cases of the ring $R(n;r,s)$  are as follows.
\begin{itemize}
\item When $r = s = 0$, the ring $R(n;0,0)$ is the ground field $\CC$.
\item When $r = 1$ and $s = 0$, the ring $R(n;1,0)$ is the classical coinvariant algebra
$\CC[X_n]/\langle \CC[X_n]^{S_n}_+ \rangle$ of the symmetric group. This $S_n$-module carries
the regular representation $\CC[S_n]$ and presents the cohomology of the variety 
$\mathcal{F \ell}_n$ of complete flags in $\CC^n$.
\item When $r = 2$ and $s = 0$, the ring $R(n;2,0)$ is the diagonal coinvariant ring $DR_n$.
\item When $r = 0$ and $s = 1$, it is not difficult to see that 
$R(n;0,1) = \wedge \{ \Theta_n \} / \langle \wedge \{ \Theta_n \}^{S_n}_+ \rangle$ has dimension $2^{n-1}$.
\item When $r = s = 1$, the ring $R(n;1,1)$ is the {\em superspace coinvariant ring}
$\Omega_n/\langle (\Omega_n)^{S_n}_+ \rangle$ studied in \cite{RW, Zabrocki}. It is conjectured
that the dimension of this quotient counts ordered set partitions of $\{1, \dots, n \}$.
\item When $r = 2$ and $s = 1$, Zabrocki conjectured \cite{Zabrocki} that $R(n;2,1)$ gives a 
representation-theoretic model for the symmetric function $\Delta'_{e_{k-1}} e_n$ as $k$ ranges
from $1$ to $n$, where $\Delta'$ is a {\em delta operator}; see \cite{Zabrocki} for details.
\item When $r = 0$ and $s = 2$, the ring $R(n;0,2) = FDR_n$ is the quotient ring studied in this 
extended abstract.
\end{itemize}

F. Bergeron proposed \cite{Bergeron} the following unified approach to studying $R(n;r,s)$ as $r$ and $s$ vary.
The general linear group $\GL_r$ acts on the matrix $X_{r \times n}$ of bosonic variables by left multiplication. 
Similarly, the matrix $\Theta_{s \times n}$ of fermionic variables carries an action of $\GL_s$.
These actions combine to give an action of $\GGG := \GL_r \times \GL_s \times S_n$ on the ring
$S(n;r,s)$ and its quotient $R(n;r,s)$.
We describe how to record the $\GGG$-character of $R(n;r,s)$ as a formal power series.
For the remainder of this section, we assume knowledge of the power series operation of 
{\em plethysm}.


Let $W$ be a $\GGG$-module.
To capture the isomorphism type of $W$ in a polynomial, we need three alphabets of variables: an 
$r$-letter alphabet $Q_r = (q_1, \dots, q_r)$ tracking $\GL_r$, an
$s$-letter alphabet $Z_s = (z_1, \dots, z_s)$ tracking $\GL_s$, and an infinite
alphabet $X  = (x_1, x_2, \dots )$ tracking $S_n$.
For any partition $\lambda \vdash n$, we let $W^{\lambda}$ be the $\lambda$-isotypic component of $W$
as an $S_n$-module;
the space $W^{\lambda}$ carries a $\GL_r \times \GL_s$-action.
We define the {\em $\GGG$-character} $\ch_{\GGG}(W)$ to be the formal power series
\begin{equation}
\ch_{\GGG}(W)[Q_r + Z_s; X] :=
\sum_{\lambda \vdash n}  
\mathrm{trace}_{W^{\lambda}} 
(\mathrm{diag}(q_1, \dots, q_r) \times \mathrm{diag}(z_1, \dots, z_s)) \cdot
s_{\lambda}[X]
\end{equation}
where we take $q_1, \dots, q_r$ and $z_1, \dots, z_s$ to be nonzero complex numbers so that the product
$\mathrm{diag}(q_1, \dots, q_r) \times \mathrm{diag}(z_1, \dots, z_s)$ of diagonal matrices is an element
of $\GL_r \times \GL_s$ acting on $W^{\lambda}$.
The formal power series $\ch_{\GGG}(W)[Q_r + Z_s; X]$ may be written uniquely as
\begin{equation}
\ch_{\GGG}(W)[Q_r + Z_s; X] =
\sum_{\ell(\mu) \leq r} \sum_{\ell(\nu) \leq s} \sum_{\lambda \vdash n}
d_{\mu, \nu, \lambda} \cdot s_{\mu}[Q_r] \cdot s_{\nu}[Z_s] \cdot s_{\lambda}[X]
\end{equation}
for some  integers $d_{\mu, \nu, \lambda} \geq 0$.

\begin{rmk}
Although the indices $r$ and $s$ are suppressed in our product group notation
$\GGG = \GL_r \times \GL_s \times S_n$, they reappear in the $q$-alphabet and $z$-alphabet sizes 
in $\ch_{\GGG}(W)[Q_r + Z_s; X]$.  The notation $\ch_{\GGG}(W)[Q_r + Z_s; X]$ is therefore unambiguous.
\end{rmk}






Let $Q = (q_1, q_2, \dots )$ be an infinite list of $q$-variables. F. Bergeron showed \cite{BergeronAdv}
that the limit of the `purely bosonic' $\GGG$-characters of $R(n;r,0)$:
\begin{equation}
\EEE_n[Q;X] := \lim_{r \rightarrow \infty} \ch_{\GGG}(R(n;r,0))[Q_r + Z_0; X]
\end{equation}
is a well-defined formal power series in the variables $Q$ and $X$.
The following conjecture states that 
that the $\GGG$-characters of {\bf all} of the rings $R(n;r,s)$ may be determined from
$\EEE_n[Q;X]$ (which only has {\em a priori} knowledge of the $R(n;r,0)$).

\begin{conj}[F. Bergeron \cite{Bergeron}]
\label{supersymmetry-conjecture}
\emph{``Combinatorial Supersymmetry Conjecture'':}
For any integers $n, r, s \geq 0$ the $\GGG$-character of $R(n;r,s)$ is given by 
\begin{equation}
\ch_{\GGG}(R(n;r,s))[Q_r + Z_s; X] = 
\EEE_n[Q - \epsilon Z; X] \mid_{\substack{Q \rightarrow Q_r \\ Z \rightarrow Z_s}}.
\end{equation}
Here $\epsilon$ is  the plethystic epsilon and the subscript 
$Q \rightarrow Q_r, Z \rightarrow Z_r$ on the left-hand-side means to evaluate 
$q_i \rightarrow 0$ whenever $i > r$ and $z_j \rightarrow 0$ whenever $j > s$.
\end{conj}

Conjecture~\ref{supersymmetry-conjecture} implies that knowledge of all of the `purely bosonic' quotients
$R(n;r,0)$ for $r \geq 0$ would determine all of the the boson-fermion rings $R(n;r,s)$.  
This conjecture also implies that knowledge of all of the `purely fermionic' quotients 
$R(n;0,s)$ for $s \geq 0$ would determine all of the rings $R(n;r,s)$. 
Conjecture~\ref{supersymmetry-conjecture} therefore suggests that 
studying $R(n;0,s)$ will become very difficult as $s$ grows.

\begin{rmk}
\label{supersymmetry-remark}
The bigraded hook-shaped multiplicities $c_{\lambda}(q,t)$ in
Theorem~\ref{multiplicities} are in accordance with the prediction of Conjecture~\ref{supersymmetry-conjecture},
giving evidence for combinatorial supersymmetry.
\end{rmk}















\section{Proof idea: Exterior Lefschetz theory}

In this final section, we describe the main ideas of the proofs of 
Theorems~\ref{vector-space-structure} and \ref{w-module-structure}; for complete proofs 
see that full version \cite{KR} of this work.
The idea is to develop a (nontraditional) Lefschetz theory for the exterior algebra
$\wedge \{ \Theta_n, \Xi_n \}$.  We start by recalling classical Lefschetz theory.

Let $A = \bigoplus_{d = 0}^n A_d$ be a finite-dimensional graded commutative $\CC$-algebra.
The algebra $A$ satisfies {\em Poincar\'e Duality (PD)} if $A_n \cong \CC$ and if multiplication 
$A_d \otimes A_{n-d} \rightarrow A_n \cong \CC$ is a perfect pairing for each $0 \leq d \leq n$.
This implies that $\dim A_d = \dim A_{n-d}$. 
Assuming $A$ satisfies PD, an element $\ell \in A_1$ is a {\em (strong) Lefschetz element} if the map
$\ell^{n-2d} \times (-): A_d \rightarrow A_{n-d}$ is bijective for each $d < n/2$.
If $A$ has a Lefschetz element, it is said to satisfy the {\em Hard Lefschetz (HL)} property.

Algebras $A$ which satisfy PD and HL arise naturally in geometry. If $X$ is a 
compact complex manifold, the cohomology ring
$A = H^{\bullet}(X;\CC)$ satisfies PD and HL where we take $A_d := H^{2d}(X;\CC)$.

For some spaces $X$, PD and HL for $H^{\bullet}(X;\CC)$ may be understood combinatorially.
As an example, recall that the {\em Boolean poset} $B(n)$ consists of subsets of $\{1, \dots, n \}$ ordered 
by containment and that the $i^{th}$ rank $B(n)_i$ of $B(n)$ consists of $i$-element subsets of $\{1, \dots, n\}$.

\begin{thm}
\label{invertible-matrix}
For any $1 \leq i \leq j \leq n$, define an ${n \choose i} \times {n \choose j}$ matrix $M(n;i,j)$
with rows indexed by $B(n)_i$ and columns indexed by $B(n)_j$ by
\begin{equation*}
M(n;i,j)_{S,T} = \begin{cases}
1 & S \subseteq T \\
0 & else.
\end{cases}
\end{equation*}
For all $0 \leq i \leq n/2$, the matrix $M(n;i,n-i)$ is invertible.
\end{thm}

For example, when $n = 4$ and $i = 1$, Theorem~\ref{invertible-matrix} asserts that the matrix
\[  M(4;1,3) = 
\begin{blockarray}{ccccc}
& $\{1\}$ & $\{2\}$ & $\{3\}$ & $\{4\}$  \\
\begin{block}{c(cccc)}
$\{1,2,3\}$ & 1 & 1 & 1 & 0  \\
$\{1,2,4\}$ & 1 & 1 & 0 & 1  \\
$\{1,3,4\}$ &   1 & 0 & 1 & 1  \\
$\{2,3,4\}$ &  0 & 1 & 1 & 1 \\
\end{block}
\end{blockarray}
 \]
 is invertible. 
 
 The origins of Theorem~\ref{invertible-matrix} are difficult to trace. Stanley \cite{Stanley} gave a proof of
 Theorem~\ref{invertible-matrix} in the context of his theory of {\em differential posets}.
 Hara and Watanabe \cite{HW} gave another proof of Theorem~\ref{invertible-matrix} in which 
 they calculated the (nonzero) determinant of $M(n;i,n-i)$.  Hara and Watanabe interpreted 
 Theorem~\ref{invertible-matrix} in a geometric context by showing that the cohomology ring
 \begin{equation}
 H^{\bullet}(\PP^1 \times \cdots \times \PP^1; \CC) = \CC[x_1, \dots, x_n]/\langle x_1^2, \dots, x_n^2 \rangle
 \end{equation}
 of the $n$-fold product of the Riemann sphere $\PP^1$  with itself satisfies PD and HL with 
 $x_1 + \cdots + x_n$ serving as a Lefschetz element.
 
 The exterior algebra $\wedge \{ \Theta_n, \Xi_n \}$ has a natural bigrading
 \[\wedge \{ \Theta_n, \Xi_n \} = \bigoplus_{i,j = 0}^n \wedge \{ \Theta_n, \Xi_n \}_{i,j}.\]
 We have $\wedge \{ \Theta_n, \Xi_n \}_{n,n} \cong \CC$ and the multiplication map
 \begin{equation*}
 \wedge \{ \Theta_n, \Xi_n \}_{i,j} \otimes 
 \wedge \{ \Theta_n, \Xi_n \}_{n-i,n-j} \rightarrow 
 \wedge \{ \Theta_n, \Xi_n \}_{n,n} \cong \CC
 \end{equation*}
 is a perfect pairing for all $0 \leq i, j \leq n$. In this way, the algebra $\wedge \{ \Theta_n, \Xi_n \}$ 
 satisfies a bigraded kind of PD. The following result describes a kind of HL satisfied by 
 $\wedge \{ \Theta_n, \Xi_n \}$.
 
 \begin{thm}
 \label{lefschetz-theorem}
 Consider the element $\delta \in \wedge \{ \Theta_n, \Xi_n \}_{1,1}$ given by
 \begin{equation}
 \delta := \theta_1 \xi_1 + \theta_2 \xi_2 + \cdots + \theta_n \xi_n.
 \end{equation}
 Whenever $i + j \leq n$, the linear map
 \begin{equation}
 \delta^{n-i-j} \times (-): \wedge \{ \Theta_n, \Xi_n \}_{i,j} \longrightarrow
 \wedge \{ \Theta_n, \Xi_n \}_{n-j,n-i}
 \end{equation}
 is a bijection.
 \end{thm}
 
 Informally, Theorem~\ref{lefschetz-theorem} says that $\delta_n$ is a Lefschetz-like element
 for the exterior algebra $\wedge \{ \Theta_n, \Xi_n \}_{1,1}$.
 Theorem~\ref{lefschetz-theorem} is proven using Theorem~\ref{invertible-matrix}. 
 
 
 Using our realization of 
 $\Theta_n$ as a basis of the reflection representation $V$ on which our reflection group $W$ acts and 
 $\Xi_n$ as the dual basis of $V^*$, one sees that $\delta$ is fixed by the action of $W$
 (and, in fact, by the action of the full general linear group $\GL(V)$).
 We close by describing how Theorems~\ref{vector-space-structure} and 
 \ref{w-module-structure} are proven.
 
 \begin{proof} {\em (of Theorems~\ref{vector-space-structure} and \ref{w-module-structure}, sketch)}
A result of 
 Steinberg (see \cite[Thm. A, \S 24-3, p.250]{Kane}) 
 states that the exterior powers $\wedge^0 V, \wedge^1 V, \dots, \wedge^n V$ are inequivalent 
 irreducible $W$-modules.  From this one 
deduces that the defining ideal
 $\langle \wedge(V \oplus V^*)^{W}_+ \rangle$ of $FDR_W$ is  principal and generated by 
 the element $\delta$ of Theorem~\ref{lefschetz-theorem},  
so that $FDR_W = \wedge(V \oplus V^*)/\langle \delta \rangle$.
 
 Whenever a composition $f \circ g$ of two maps is a bijection, the map $f$ is surjective and the map 
 $g$ is injective. Theorem~\ref{lefschetz-theorem} therefore implies that the map
 \begin{equation}
 \delta \times (-): \wedge (V \oplus V^*)_{i,j} \longrightarrow 
 \wedge (V \oplus V^*)_{i+1,j+1}
 \end{equation}
 is surjective whenever $i + j \geq n$ and injective whenever $i + j < n$. In terms of the quotient 
$FDR_W = \wedge(V \oplus V^*)/\langle \delta \rangle$, the $W$-equivariance of 
$\delta$ means that $(FDR_W)_{i,j} = 0$ if $i + j > n$ and 
$[(FDR_W)]_{i,j} = [ \wedge(V \oplus V^*)_{i,j} ] - [ \wedge(V \oplus V^*)_{i-1,j-1} ]$ otherwise.
 \end{proof}

Theorem~\ref{lefschetz-theorem} is also used in the proof of Theorem~\ref{w-basis-theorem}
(giving a bigraded basis of $FDR_W$). 
Theorem~\ref{lefschetz-theorem} shows that the bigraded pieces of $FDR_W$
have the appropriate dimensions.



\acknowledgements{The authors are grateful to Fran\c{c}ois Bergeron,
Jim Haglund, Eugene Gorsky, Vic Reiner,
Richard Stanley, and Mike Zabrocki for many helpful conversations.}

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