%% if you are submitting an initial manuscript then you should have submission as an option here %% if you are submitting a revised manuscript then you should have revision as an option here %% otherwise options taken by the article class will be accepted \documentclass[submission]{FPSAC2026} %% 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} \newcommand{\symm}{\mathfrak{{S}}} \newcommand{\FF}{\mathbb{F}} \newcommand{\CC}{\mathbb{C}} \newcommand{\RR}{\mathbb{R}} \newcommand{\BBB}{\mathcal{B}} \newcommand{\AAA}{\mathcal{A}} \newcommand{\OP}{\mathcal{OP}} \newcommand{\xx}{\mathbf{x}} \newcommand{\yy}{\mathbf{y}} \newcommand{\Frob}{\mathrm{Frob}} \newcommand{\grFrob}{\mathrm{grFrob}} \newcommand{\Hilb}{\mathrm{Hilb}} \newcommand{\Stir}{\mathrm{Stir}} \newcommand{\Park}{\mathrm{Park}} \newcommand{\Der}{\mathrm{Der}} \newcommand{\Res}{\mathrm{Res}} \newcommand{\SYT}{\mathrm{SYT}} \newcommand{\maj}{\mathrm{maj}} \newcommand{\des}{\mathrm{des}} \newcommand{\sign}{\mathrm{sign}} \newcommand{\stair}{\mathrm{st}} \newcommand{\Fl}{\mathrm{Fl}} \newcommand{\ttheta}{\bm{\theta}} \newcommand{\aaaa}{\mathfrak{a}} \newcommand{\iiii}{\mathfrak{i}} \newcommand{\cccc}{\mathfrak{c}} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{definition}{Definition} \newtheorem{conjecture}{Conjecture} \usepackage{lipsum} \articlenumber{12} %% define your title in the usual way \title{The superspace coinvariant ring} %% define your authors in the usual way %% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details %% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author \author[R. Angarone, P. Commins, T. Karn, S. Murai, B. Rhoades, A. Wilson]{ Robert Angarone \thanks{\href{mailto:angar017@umn.edu}{angar017@umn.edu}.}\addressmark{1}, Patricia Commins \thanks{\href{mailto:angar017@umn.edu}{commi010@umn.edu}. P. Commins was partially supported by the NSF GRFP \#2237827}\addressmark{2}, Trevor Karn \thanks{\href{mailto:tkkarn@tamu.edu}{tkkarn@tamu.edu}.}\addressmark{3}, Satoshi Murai\thanks{\href{mailto:s-murai@waseda.jp}{s-murai@waseda.jp}. S. Murai was partially supported by KAKENHI 25K06943 and 23K17298.}\addressmark{4}, Brendon Rhoades\thanks{\href{mailto:bprhoades@ucsd.edu}{bprhoades@ucsd.edu}. B. Rhoades was partially supported by NSF Grant DMS-2246846}\addressmark{5}, \and Andy Wilson \thanks{\href{mailto:awils342@kennesaw.edu}{awils342@kennesaw.edu}. A. Wilson was partially supported by AMS-Simons PUI Grant 434651.}\addressmark{6}} %% then use \addressmark to match authors to institutions here \address{ \addressmark{1}School of Mathematics, University of Minnesota, Minneapolis, MN, USA \\ \addressmark{2}Institute for Computational and Experimental Research in Mathematics, Providence, RI, USA \\ \addressmark{3}Department of Mathematics, Texas A\&M University, College Station, TX, USA \\ \addressmark{4}Department of Mathematics, Waseda University, Tokyo, Japan \\ \addressmark{5}Department of Mathematics, UC San Diego, La Jolla, CA, USA \\ \addressmark{6}Department of Mathematics, Kennesaw State University, Marietta, GA, 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 $\FF$ be a field of characteristic 0 and let $\Omega_n$ be the ring of regular differential forms on $\FF^n$. The natural action of $\symm_n$ on $\FF^n$ induces an action on $\Omega_n$. The {\em superspace coinvariant ring} is the quotient $SR_n$ of $\Omega_n$ by the ideal generated by $\symm_n$-invariants with vanishing constant term. We calculate the bigraded character of $SR_n$, describe a vector space basis of $SR_n$, and give an `operator theorem' which characterizes the associated inverse system. Our results prove conjectures of Bergeron, Colmenarejo, Li, Machacek, Reiner, Sagan, Sulzgruber, Swanson, Wallach, and Zabrocki.} %% 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{superspace, coinvariant ring, hyperplane arrangement} %% 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; i.e., [xx, yy] in the text %% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex %% but you can use bibtex if you want \usepackage[backend=bibtex]{biblatex} \addbibresource{sample.bib} %% note the \printbibliography command at the end of the file which goes with these biblatex commands \begin{document} \maketitle %% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above \section{Introduction} \label{sec:Intro} This extended abstract outlines the results in three papers \cite{ACKMR, MRW, RW} on the superspace coinvariant ring $SR_n$. We review classical $\symm_n$-coinvariant theory before introducing the object of study. Let $\FF$ be a field of characteristic 0, let $\xx_n = (x_1,\dots,x_n)$ be a list of $n$ variables, and let $\FF[\xx_n] := \FF[x_1,\dots,x_n]$ be the rank $n$ polynomial ring. The symmetric group $\symm_n$ acts on $\FF[\xx_n]$ by subscript permutation. We write \begin{equation} I_n := ( \FF[\xx_n]^{\symm_n}_+) \end{equation} for the ideal generated by $\symm_n$-invariant polynomials with vanishing constant term and \begin{equation} R_n := \FF[\xx_n]/I_n \end{equation} for the corresponding quotient ring. The quotient $R_n$ is the {\em coinvariant ring} associated to the symmetric group; it is a graded $\symm_n$-module. The ring $R_n$ is among the most important representations in algebraic combinatorics. E. Artin proved \cite{Artin} that $R_n$ has vector space basis given by the `substaircase' monomials $\{x_1^{a_1} \cdots x_n^{a_n} \,:\, a_i < i \}$. In particular, the Hilbert series of $R_n$ is given by\footnote{We use the $q$-notation $[n]_q := \frac{1-q^n}{1-q}, [n]!_q := [n]_q [n-1]_q \cdots [1]_q,$ and ${n \brack k}_q := \frac{[n]!_q}{[k]!_q \cdot [n-k]!_q}.$} \begin{equation} \Hilb(R_n;q) = [n]!_q. \end{equation} Chevalley proved \cite{Chevalley} that there is an isomorphism of ungraded $\symm_n$-modules \begin{equation} R_n \cong_{\symm_n} \FF[\symm_n] \end{equation} so that $R_n$ is a graded refinement of the regular representation of $\symm_n$. Lusztig and Stanley \cite{Stanley} gave the following combinatorial formula for the graded character of $R_n$: \begin{equation} \grFrob(R_n;q) = \sum_{T \in \mathrm{SYT}(n)} q^{\mathrm{maj}(T)} \cdot s_{\lambda(T)}. \end{equation} Here $\mathrm{SYT}(n)$ is the family of standard Young tableaux with $n$ boxes, $\mathrm{maj}(T)$ is the major index of a tableau $T$, and $\lambda(T) \vdash n$ is the shape of $T$. Borel proved \cite{Borel} that $R_n$ presents the cohomology ring $H^*(G/B)$ of the type A$_{n-1}$ complete flag variety. Now let $\xx_n = (x_1,\dots,x_n), \yy_n = (y_1,\dots,y_n)$ be two lists of $n$ variables. Write $\FF[\xx_n,\yy_n]$ for the rank $2n$ polynomial ring with its `diagonal' action \begin{equation} w \cdot x_i := x_{w(i)}, \quad w \cdot y_i := y_{w(i)} \end{equation} of $\symm_n$. Let $DI_n \subseteq \FF[\xx_n,\yy_n]$ be the ideal generated by $\symm_n$-invariants with vanishing constant term. Garsia and Haiman defined \cite{HaimanConjectures} the {\em diagonal coinvariant ring} \begin{equation} DR_n := \FF[\xx_n,\yy_n]/DI_n. \end{equation} Haiman used the algebraic geometry of Hilbert schemes to prove \cite{HaimanVanishing} that the ungraded $\symm_n$-action on $DR_n$ is isomorphic to its permutation action on size $n$ parking functions, up to a sign twist. Haiman also proved \cite{HaimanVanishing} that the bigraded Frobenius image of $DR_n$ is given by \begin{equation} \grFrob(DR_n;q,t) = \nabla e_n \end{equation} where $\nabla$ is the Bergeron-Garsia {\em nabla operator} on symmetric functions. The rank $n$ {\em superspace ring} is the bigraded algebra \begin{equation} \Omega_n := \FF[x_1,\dots,x_n] \otimes_\FF \wedge \{ \theta_1,\dots,\theta_n\} \end{equation} of regular differential forms on $\FF^n$.\footnote{The `super' refers to supersymmetry in physics, where the $x_i$ correspond to states of bosons and the $\theta_i$ correspond to states of fermions.} The defining action of $\symm_n$ on $\FF^n$ induces a diagonal action \begin{equation} w \cdot x_i = x_{w(i)}, \quad w \cdot \theta_i = \theta_{w(i)} \end{equation} of $\symm_n$ on $\Omega_n$. Our object of study is as follows. \begin{definition} \label{def:srn} The {\em superspace coinvariant ideal} is the two-sided ideal $SI_n \subseteq \Omega_n$ generated by the space $(\Omega_n)_+^{\symm_n}$ of $\symm_n$-invariants with vanishing constant term. The {\em superspace coinvariant ring} is the quotient \begin{equation} SR_n := \Omega_n/SI_n. \end{equation} \end{definition} The quotient $SR_n$ is a bigraded $\symm_n$-module with respect to $x$-degree and $\theta$-degree. It was introduced by the Fields Institute Combinatorics Group in 2018.\footnote{N. Bergeron, L. Colmenarejo, S. X. Li, J. Machacek, R. Sulzgruber, and M. Zabrocki} Write $\OP_n$ for the family of ordered set partitions of $[n] := \{1,\dots.n\}$. For example, one has \[ \OP_3 = \left\{ \begin{array}{c c c c c c } (1 \mid 2 \mid 3), & (2 \mid 1 \mid 3), & (1 \mid 3 \mid 2), & (2 \mid 3 \mid 1), & (3 \mid 1 \mid 2), & (3 \mid 1 \mid 2), \cr (1 \, 2 \mid 3), & (1 \, 3 \mid 2), & (2\, 3 \mid 1), & (1 \mid 2 \, 3), & (2 \mid 1 \, 3), & (3 \mid 1 \, 2), \cr (1 \, 2 \, 3) & & & & & \end{array} \right\}. \] The Fields Group predicted that the structure $SR_n$ is governed by the combinatorics of $\OP_n$. \begin{conjecture} \label{conj:fields} {\em (Fields Conjectures \cite{Zabrocki})} The superspace coinvariant ring $SR_n$ enjoys the following properties. \begin{enumerate} \item The vector space dimension of $SR_n$ is $\# \OP_n$. \item The bigraded Hilbert series of $SR_n$ is $\Hilb(SR_n;q,z) = \sum_{k=1}^n z^{n-k} \cdot [k]!_q \cdot \Stir_q(n,k)$ where $q$ tracks $x$-degree, $z$ tracks $\theta$-degree, and $\Stir_q(n,k)$ is a $q$-Stirling number. %Andy: I might refer the reader to where we define the $q$-Stirling number (in Section 3). \item As ungraded $\symm_n$-modules, one has $SR_n \cong_{\symm_n} \FF[\OP_n] \otimes \sign$ where $\sign$ is the 1-dimensional sign representation of $\symm_n$. \item The bigraded Frobenius image of $SR_n$ is $\grFrob(SR_n;q,z) = \sum_{k=1}^n z^{n-k} \cdot C_{n,k}(\xx;q)$ where $C_{n,k}(\xx;q)$ is a symmetric function whose definition is given in Section~\ref{sec:Module}. %Andy: I might insert something in the previous sentence about how C_{n,k} ``is related to Macdonald eigenoperators'' or something like that. \end{enumerate} \end{conjecture} Shortly after Conjecture 1 appeared, other conjectures about $SR_n$ were proposed. \begin{itemize} \item Sagan and Swanson \cite{SS} gave a conjectural monomial basis of $SR_n$ extending Artin's substaircase monomial basis of $R_n$. \item Swanson and Wallach \cite{SW} made an `operator conjecture' on the inverse system $SI_n^\perp$ associated to $SR_n$. \item Reiner \cite{ReinerComm} made a conjecture on the restriction of $SR_n$ from $\symm_n$ to $\symm_{n-1}$ mirroring the usual combinatorial recursion relating ordered set partitions of $[n]$ to ordered set partitions of $[n-1]$. \end{itemize} The inscrutable Gr\"obner theory of the quotient ring $SR_n$ made these conjectures resistant to the `straightening' techniques often used to analyze quotient rings in algebraic combinatorics. Finding an elementary proof of any of these conjectures is an open problem. This extended abstract tells the story of how these conjectures were proven. % This extended abstract tells the story of how these conjectures were established. % Robbie: Is it better to use a word like `affirmed' or `proven,' instead of `established'? % Trevor: I agree and changed it. % Brendon: I changed it to 'proven'. It seemed the least ambiguous. % Trevor: I agree! \section{Background} \label{sec:Background} \subsection{Symmetric Functions and Representation Theory} We write $\Lambda = \bigoplus_{n \geq 0} \Lambda_n$ for the graded ring of symmetric functions in an infinite variable set $\xx = (x_1, x_2,\dots)$ over the ground field $\FF(q,t,z)$. Bases of the degree $n$ component $\Lambda_n$ are indexed by partitions $\lambda \vdash n$. We will use the elementary basis $\{ e_\lambda \}$ and the Schur basis $\{ s_\lambda \}$ of $\Lambda_n$. Let $\langle -,-\rangle$ be the Hall inner product on $\Lambda$ with respect to which the Schur basis is orthonormal. Irreducible representations of the symmetric group $\symm_n$ are in one-to-one correspondence with partitions of $n$. Write $V^\lambda$ for the irreducible $\symm_n$-module corresponding to $\lambda \vdash n$. If $V$ is any finite-dimensional $\symm_n$-module, there are unique multiplicities $c_\lambda \geq 0$ so that $V \cong \bigoplus_{\lambda \vdash n} c_\lambda V^\lambda$. The {\em Frobenius image} of $V$ is the symmetric function \begin{equation} \Frob(V) := \sum_{\lambda \vdash n} c_\lambda \cdot s_\lambda \end{equation} obtained by replacing each irreducible with its corresponding Schur function. More generally, let $V = \bigoplus_{i \geq 0} V_i$ be a graded representation of $\symm_n$ with each $V_i$ finite-dimensional. The {\em graded Frobenius image} of $V$ is the symmetric function \begin{equation} \grFrob(V;q) := \sum_{i\geq0} \Frob(V_i) \cdot q^i. \end{equation} Still more generally, if $V = \bigoplus_{i,j \geq 0} V_{i,j}$ is a bigraded $\symm_n$-module, %with commuting degree $i$ and anticommuting degree $j$, we write %Andy: Maybe just leave the bi-grading general, instead of ``commuting degree'' and ``anticommuting degree?'' %Trevor: +1 %Brendon: Agreed. \begin{equation} \grFrob(V;q,z) := \sum_{i,j \geq 0} \Frob(V_{i,j}) \cdot q^i z^j \end{equation} for the {\em bigraded Frobenius image} of $V$. The {\em Hilbert series} of a graded vector space $V = \bigoplus_{i \geq 0} V_i$ is given by \begin{equation} \Hilb(V;q) := \sum_{i\geq 0} \dim V_i \cdot q^i. \end{equation} Similarly, if $V = \bigoplus_{i,j \geq 0} V_{i,j}$ is a bigraded vector space %with commuting degree $i$ and anticommuting degree $j$, %Trevor - mirroring Andy's comment above. the {\em (bigraded) Hilbert series} of $V$ is \begin{equation} \Hilb(V;q,z) := \sum_{i,j \geq 0} \dim(V_{i,j}) \cdot q^i z^j. \end{equation} \subsection{Supercommutative Algebra} For $1 \leq i \leq n$, the partial derivative operator $\partial/\partial x_i$ acts on the first tensor factor of the superspace ring $\Omega_n = \FF[x_1,\dots,x_n] \otimes \wedge \{ \theta_1,\dots,\theta_n\}$. The {\em contraction operator} (or {\em fermionic partial derivative}) $\partial/\partial \theta_i$ acts on the second tensor factor of $\Omega_n$ as follows. Consider distinct indices $1 \leq j_1, \dots, j_r \leq n$ so that $\theta_{j_1} \cdots \theta_{j_r}$ is a nonzero exterior monomial. % Trevor - do we need to say both contraction operator and fermionic partial derivative above? could we just say fermionic partial derivative? % Brendon - these kinds of operators are often called `contraction', especially in homological contexts. I don't feel so strongly, but I was hoping that people working in e.g. Koszul theory might be interested. One has \begin{equation} \partial/\partial \theta_i : \theta_{j_1} \cdots \theta_{j_r} \mapsto \begin{cases} (-1)^{s-1} \theta_{j_1} \cdots \theta_{j_{s-1}} \theta_{j_{s+1}} \cdots \theta_{j_r} & \text{if $j_s=i$,} \\ 0 & \text{if $j_1,\dots,j_r \neq i.$} \end{cases} \end{equation} The operators $\partial/\partial x_i$ and $\partial/\partial \theta_i$ satisfy the same relations as $\Omega_n$, viz. \[ (\partial/\partial x_i) (\partial/\partial x_j) = (\partial/\partial x_j) (\partial/\partial x_i) \quad (\partial/\partial x_i) (\partial/\partial \theta_j) = (\partial/\partial \theta_j) (\partial/\partial x_i)\] \[(\partial/\partial \theta_i) (\partial/\partial \theta_j) = - (\partial/\partial \theta_j) (\partial/\partial \theta_i) \] for all $1 \leq i,j \leq n$. For any superspace element $f = f(x_1,\dots,x_n,\theta_1,\dots,\theta_n)$, we thus have a well-defined operator $\partial f: \Omega_n \to \Omega_n$ given by \[\partial f := f(\partial/\partial x_1,\dots,\partial/\partial x_n, \partial/\partial \theta_1,\dots,\partial/\partial\theta_n).\] Let $I \subseteq \Omega_n$ be a bihomogeneous ideal. The {\em inverse system} (or {\em harmonic space}) is the subspace $I^\perp \subseteq \Omega_n$ given by \begin{equation} I^\perp := \{ g \in \Omega_n \,:\, (\partial f)(g) = 0 \text{ for all } f \in I \}. \end{equation} One can show that $I^\perp$ is a bigraded subspace of $\Omega_n$ with the same Hilbert series as $\Omega_n/I$. If $G \subseteq GL_n(\FF)$ is a finite matrix group, then $G$ acts on $\Omega_n$ by linear substitutions. If $I \subseteq \Omega_n$ is stable under the action of $G$, one can show that $I^\perp$ is also stable under the action of $G$ and that one has an isomorphism $\Omega_n/I \cong_G I^\perp$ of bigraded $G$-modules. \section{Operator Theorem and Basis Transfer} \label{sec:Operator} There is a beautiful description of the inverse system $I_n^\perp$ attached to the classical coinvariant ideal $I_n \subseteq \FF[\xx_n]$. Write $\delta_n \in \FF[\xx_n]$ for the Vandermonde determinant \begin{equation} \delta_n := \prod_{1 \leq i