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\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{exam}[thm]{Example}
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\newcommand{\sgrp}{\mathbb{S}}
\newcommand{\alpxn}{\mathbf{x}_{n}}
\newcommand{\alpx}{\mathbf{x}}
\newcommand{\bN}{\mathbb{N}}
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\newcommand{\suchthat}{\;|\;}
\newcommand{\prim}{\mathrm{prim}}
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\title[Divisors, orbit harmonics and DT invariants]{Divisors, orbit harmonics, and DT invariants}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Reineke, Rhoades, Tewari]{M. Reineke\thanks{\href{mailto:markus.reineke@rub.de}{markus.reineke@rub.de}. M.R. was supported by DFG CRC-TRR 191 ``Symplectic structures in geometry, algebra
and dynamics''.}\addressmark{1}, B. Rhoades\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}. B.R. was supported by NSF Grant DMS-1953781.}\addressmark{2}, \and V. Tewari\thanks{\href{mailto:vvtewari@math.hawaii.edu}{vvtewari@math.hawaii.edu}. V.T. was supported by Simons Collaboration Grant \#855592.}\addressmark{3}}

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\address{\addressmark{1}{Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany} \\ \addressmark{2}{Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA}
\\
\addressmark{3}{Department of Mathematics, University of Hawaii at Manoa, Honolulu HI, 96822, USA}
}
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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{We give a combinatorial interpretation to Donaldson--Thomas invariants of symmetric quivers by using the method of orbit harmonics. 
Along the way we connect  Young subgroup actions on break divisors of graphs, quotients of the polynomial ring modulo power ideals of Postnikov--Ardila, and a construction of Efimov in the context of Cohomological Hall algebras.
}
 
%%% put your French abstract here, or comment this out if you don't have one
%\resume{Nous donnons une interpr\'etation combinatoire des invariants de Donaldson--Thomas des carquois sym\'etriques en faisant appel \`a la méthode des harmoniques d'orbite.
%Ce faisant, nous relions les actions des sous-groupes de Young sur les diviseurs de rupture des graphes, les id\'eaux de puissance de Postnikov--Ardila, et une construction d'Efimov dans le contexte des alg\`ebres cohomologiques de Hall.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{break divisors, Donaldson--Thomas invariants, orbit harmonics, power ideals}

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\newcommand{\hilb}{\mathrm{Hilb}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bQ}{\mathbb{Q}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mcp}{\mathcal{P}}
\newcommand{\mcd}{\mathcal{D}}
\newcommand{\brkd}{\mathrm{Break}}
\newcommand{\aut}{\mathrm{Aut}}
\newcommand{\bfx}[1]{\mathbf{x}_{#1}}
\newcommand{\edges}{\mathscr{E}} % edge
\newcommand{\vertices}{\mathscr{V}} % vertices
\newcommand{\cuts}{\mathscr{Cuts}}%cuts
\newcommand{\bonds}{\mathscr{B}} % set of bonds
\newcommand{\frob}{\mathrm{Frob}} % Frobenius
\newcommand{\grfrob}{\mathrm{grFrob}} % graded Frobenius

\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}
Donaldson--Thomas (DT) invariants of quivers with potential were introduced in the seminal work of Kontsevich--Soibelman \cite{KS} as a mathematical
definition of the string-theoretic concept of BPS state count.
We give a high-level introduction to DT invariants with the goal of emphasizing that the algebraic formalism hides an interesting combinatorial question, which is what we address.

In \cite{KS}, (quantum) DT invariants were introduced as follows: Attached to a quiver $Q$ on $k$ vertices is a \emph{cohomological Hall algebra} $\mc{H}$ which carries a $\bZ_{\geq 0}^k$-grading. If $Q$ is \emph{symmetric} then $\mc{H}$ is $\bZ_{\geq 0}^k\times \bZ$-graded. 
Kontsevich and Soibelman conjectured \cite[Conjecture 1]{KS} that in this case $\mc{H}$ is a free (super-)commutative algebra generated by a $\bZ_{\geq 0}^k\times \bZ$-graded vector space $V$ of the form $V^{\mathrm{prim}}\otimes \bQ[x]$.
Furthermore that, given a fixed \emph{dimension vector} $\gamma\in \bZ_{\geq 0}^k$, only finitely many $c_{\gamma,k}\coloneqq \dim(V^{\mathrm{prim}}_{\gamma,k})$ are nonzero.

Efimov \cite{Efi} proved this conjecture via an ingenious but mysterious construction for the spaces $V^{\mathrm{prim}}_{\gamma}$.
%His work was immediately used by Mozgovoy \cite{Moz} to establish a famous conjecture of Kac \cite{Kac} in the case of quivers \emph{with enough loops}. Kac's conjecture was settled in complete generality by Hausel--Letelier--Rodriguez-Villegas \cite{HLRV} in their groundbreaking paper which gave another algebraic interpretation to the $c_{\gamma,k}$.
The \emph{quantum DT invariant} $\Omega_{Q,\gamma}(q)$ is the Laurent \emph{polynomial}  tracking the $c_{\gamma,k}$ as $k$ varies over $\bZ$. Naturally its coefficients lie in $\bZ_{\geq 0}$. 
The \emph{numerical DT-invariant} $\mathrm{DT}_{Q,\gamma}$ is the evaluation of $\Omega_{Q,\gamma}(q)$ at $q=1$. We note that these numbers are called by various names in (mathematical physics) literature, e.g. LMOV invariants and BPS invariants.

At the level of Poincar\'e series, \cite[Conjecture 1]{KS} implies  a `sum=product' flavored equality common in combinatorics and number theory. For instance, the numerical DT-invariants $\mathrm{DT}_{Q,(n)}$ for a single vertex $(m+1)$-loop quiver $Q$ with dimension vector $(n)$ for $n\in \bZ_{\ge 1}$ may be defined by considering an \emph{Euler product} factorization:
\begin{align}
\label{eq:unclear}
\sum_{n\geq 0}\frac{1}{mn+1}\binom{(m+1)n}{n+1}t^n=\prod_{n\geq 1} (1-(-1)^{mn}t^n)^{-(-1)^{mn}\,   n\mathrm{DT}_{Q,(n)}}.
\end{align}
On the left-hand side is the generating function for Fuss--Catalan numbers well known to count $(m+1)$-ary trees. It is not immediate from \cref{eq:unclear} that the exponents $\mathrm{DT}_{Q,(n)}$ are nonnegative and integral. 

In what follows we give a combinatorial  interpretation for $\mathrm{DT}_{Q,\gamma}$ for a symmetric quiver $Q$ with at least one loop at each vertex and dimension vector $\gamma$.
Our route is winding and hinges on realizing Efimov's construction as an instance of a construction of Dahmen--Micchelli from the~1980s.

Let $X$ be a finite collection of nonzero vectors in $\mathbb{R}^n$.
Motivated by problems in approximation theory, Dahmen--Micchelli associated a pair of vector spaces $(\mcp(X),\mcd(X))$ to $X$.
Subsequent work \cite{AP10,HR11,Pos04} cast more light on these spaces. When the aforementioned collection of vectors is obtained from a finite connected graph $G$, work of 
Postnikov--Shapiro \cite{Pos04} lends a fresh perspective involving divisors on graphs.
This will be our point of entry, with $\mc{P}(X)$ becoming the Postnikov--Shapiro slim subgraph~space.

Throughout, graphs are finite connected graphs with multiple edges allowed but not self loops. Given such a graph $G$, we let $\vertices(G)$ (respectively $\edges(G)$) denote its vertex set (respectively multiset of edges).
%Our collection of vectors $X_G$ will correspond to elements of $E$.
An assignment $D:\vertices(G)\to \bZ$ is called a \emph{divisor}.
Baker--Norine \cite{BN} studied this notion in depth and shed light on rich combinatorics underlying it. Another place  where divisors play a role is tropical geometry, thanks to work of Mikhalkin--Zharkov  \cite{MZ08} who introduced break divisors in this context; see  \cite{ABKS14} for more on their combinatorics.  We care particularly about this class of divisors.


The \emph{genus} of $G$ is $g(G)= |\vertices(G)|-|\edges(G)|+1$. A divisor $D$ is \emph{effective} if $D(v)\geq 0$ for all $v\in \vertices(G)$. The \emph{degree} of $D$ is equal to $\sum_{v\in \vertices(G)}D(v)$.
We say that $D$ is  a \emph{break divisor} if its degree is $g(G)$ and furthermore, for every connected subgraph $H$ of $G$ the degree of $D$ restricted to $H$ weakly exceeds $g(H)$.
This description implies that the set  $\brkd(G)$ of break divisors on $G$ may be identified with integer lattice points of a polytope. It also implies that the automorphism group $\mathrm{Aut}(G)$ of $G$ acts by permutations on $\brkd(G)$.

With this elementary setup, one may ask an elementary question: \emph{Does the $\mathrm{Aut}(G)$ module $\brkd(G)$ carry interesting information?}
We hope to convince the reader that the answer is in the affirmative by giving an enumerative application to DT theory. %The space $\mcp$ of Dahmen--Micchelli will be crucial, and the role of $X$ will be essayed by a collection of vectors determined by elements of $E$.
Given a symmetric quiver $Q$ and dimension vector $\gamma$, we associate a connected graph $G_{Q,\gamma}$ that we call the \emph{covering graph}.
Let $S_{\gamma}$ denote the Young subgroup determined by $\gamma$.
Unraveling Efimov's construction using $G_{Q,\gamma}$ allows us to cast his results in the context of the space $\mc{P}(G_{Q,\gamma})$. Here is our chief result.
%In fact, $G_{Q,\gamma}$ is such that $S_{\gamma}$ is a subgroup of $\aut{G_{Q,\gamma}}$. 
%We summarize the discussion above tersely as our second main result. The reader is welcome to see \cref{sec:Efimov} for an in-depth discussion.
\begin{thm}
\label{thm:main_2}
The dimensions of the graded pieces of $\mathcal{P}(G_{Q,\gamma})^{S_{\gamma}}$ are the coefficients of the quantum DT invariant $\tilde{\Omega}_{\gamma}(q)$. In particular, the numerical DT invariant $\mathrm{DT}_{Q,\gamma}\coloneqq \tilde{\Omega}_{\gamma}(1)$ equals the number of $S_{\gamma}$-orbits on $\brkd(G)$.
\end{thm}
\cref{thm:main_2} gives, to the best of our knowledge, the first  manifestly nonnegative combinatorial interpretation for $\mathrm{DT}_{Q,\gamma}$ by describing it as an orbit count.
Our interpretation comes from invoking the theory of orbit harmonics (see \cref{thm:piecing_things}). In fact, this theory will give an interpretation of quantum DT invariant as the graded dimension of a space of $S_\gamma$-invariants in a certain polynomial ring quotient (see \cref{thm:graded_multiplicity_quantum_DT}).

%%We record this as our second main result, which may be of independent interest and makes no mention of DT invariants.
%%Briefly $\mc{I}(G)$ is a certain \emph{power ideal} determined from edge cuts in $G$, where $\mathsf{T}(G)$ is a graded ideal that results when applying orbit harmonics to the $\mathrm{Aut}(G)$ permutation module $\bQ[\brkd(G)]$.
%
%
%
%\begin{thm}
%\label{thm:main_1}
%We have the following isomorphisms and equalities of ungraded $\aut(G)$-modules
%\[
%\bQ[\brkd(G)]\cong \bQ[\bfx{n}]/\mathsf{T}(G) = \bQ[\bfx{n}]/\mc{I}(G) \cong \mc{P}(G),
%\]
%where the middle equality and right isomorphism are in the category of graded $\aut(G)$-modules.
%\end{thm}

%In a completely different direction, the case $G=K_n$ again provides some tantalizing hints by way of the following fact.
%The number of orbits under the $S_n$-action on $\brkd(K_n)$  equals the numerical Donaldson--Thomas (DT) invariant $\mathrm{DT}_{Q,(n)}$ where $Q$ is the $2$-loop quiver and the dimension vector is $(n)$; see \cite[Theorem 3.7]{KRT21}.
%\emph{What is the underlying reason for why this is the case, or is this coincidence purely numerological?}


%An algebraic interpretation for $\tilde{\Omega}_{\gamma}(q)$ distinct from that in \cref{thm:main_2} was given in the groundbreaking work of Hausel--Letellier--Rodriguez-Villegas \cite{HLRV}.
%Recently two other interesting interpretations for $\tilde{\Omega}_{\gamma}(q)$ have been given \cite{DFR,DM21}, both of which identify the coefficients as the dimensions of some graded piece in an algebra. 
%The objects involved are vertex algebras, and deriving our combinatorial interpretation from these interpretations does not appear to be straightforward.
%It would be an interesting endeavor relating these approaches to ours.

The full version of this extended abstract with proofs appears on the arXiv \cite{RRT22}.

\section{The Postnikov--Shapiro slim subgraph space}
For reasons of space, we discuss the objects in the title solely in the context they matter to us, referring the reader to \cite{DCP} and references therein for a detailed treatment. 
Some of the jargon is matroid-theoretic, and for good reason.


Let $G$ be a connected graph as before. 
We typically identify the vertex set with $[n]\coloneqq \{1,\dots,n\}$ where $n=|\vertices(G)|$.
Given a nonempty proper subset $S\subset \vertices(G)$, we denote by $G[S]$ the graph induced on vertices in $S$.
We define the \emph{edge cut} $\partial(S)$ of $G$ associated to $S$ to be the set of edges with one endpoint in $S$ and the other in $\bar{S}= \vertices(G) \setminus S$.
We let $d(S)\coloneqq |\partial(S)|$.
We denote the set of  edge cuts by $\cuts(G)$.
A minimal non-empty edge cut is called a \emph{bond}.
We denote the set of bonds by $\bonds(G)$.
%The following alternative characterization of bonds in connected graphs helpful given a nonempty proper $S\subset \vertices$, the cut $\partial(S)$ is a bond if and only if $G[S]$ and $G[\bar{S}]$ are both connected.

Suppose $H\subset \edges(G)$ is such that the subgraph of $G$ given by $G'\coloneqq (\vertices(G),\edges(G)\setminus H)$ is connected.
We say that $H$ (or the subgraph in $G$ determined by it) is \emph{slim}. Such subgraphs play a key role in two works that serve as our motivation, namely \cite{BR14, Pos04}.
We attach certain polynomials to subgraphs of $G$ and subsequently discuss the special role played by slim subgraphs.

%It is clear that if $Y$ is slim, then $X_G\setminus Y$ contains a basis for $\bR^n$.
%Indeed one only needs to take a spanning tree in the complement (which is connected) and throw in the vector $e_1+\cdots+e_n$ to get a basis. In particular, slim subgraphs do not give cocircuits.
%Recall that a cocircuit is a minimal subset $Y\subset X_G$ such that $X_G\setminus Y$ does not contain a basis.
%Clearly, $e_1+\cdots+e_n$ is a cocircuit.
%It is straightforward to see that all other cocircuits correspond to collections of vectors corresponding to edges of bonds in $G$.\footnote{Put differently, a bond is code for \emph{minimally non-slim}.}




For each edge $e\in \edges(G)$ joining vertices $i$ and $j$ where $i<j$, consider the linear form $p_{e}\coloneqq x_i-x_j$.
Given $H\subseteq \edges(G)$, define  the polynomial
\begin{align}
p_H=\prod_{e\in H}p_e.
\end{align}
Consider the polynomial ring $\bQ[p_e\suchthat e\in \edges(G)]$ which must equal $\bQ[x_i-x_j\suchthat i<j]$ as $G$ is connected. In this ring, the \emph{cocircuit ideal} $I(G)$ is defined as
\begin{align}
\label{eq:def_cocircuit_ideal}
I(G)\coloneqq \langle p_H\suchthat H\in\bonds(G)\rangle.
\end{align}
Note that we could have defined $I(G)$ as the ideal generated by \emph{all} $p_H$ where $H \in \cuts(G)$. Indeed $p_{H'}$ for $H'$ a cut is a multiple of $p_H$ for $H$ some bond.

%Note that it suffices to consider only the minimal cocircuits to generate $I(X)$.

Now define the \emph{(Postnikov--Shapiro) slim subgraph space} (also known as the \emph{central $P$-space}) as the $\bQ$-vector space associated to $G$ as follows:
\begin{align}
\mc{P}(G)\coloneqq \bQ\left\lbrace p_H\suchthat H \text{ slim subgraph of } G \right\rbrace.
\end{align}
%Put differently, $\mc{P}(G)$ is the span of all $p_H$ over subsets $H\subset \edges(G)$ such that $\edges(G)\setminus H$ contains a spanning tree of $G$.

\begin{exam}
\label{ex:1}
\emph{
Say $G$ is the graph on three vertices (identified with $[3]$) with two edges between vertices $1$ and $2$, and two between $2$ and $3$. Then it is easily checked that 
\begin{align*}
I(G)&=\langle (x_1-x_2)^2,(x_2-x_3)^2\rangle,\\
\mc{P}(G)&=\bQ\{1,x_1-x_2,x_2-x_3,(x_1-x_2)(x_2-x_3)\}.
\end{align*}
Observe that $\bQ[x_1-x_2,x_1-x_3,x_2-x_3]=I(G)\oplus \mc{P}(G)$, and that $\dim(\mc{P}(G))=4$. This is no coincidence, as we explain next.}
\end{exam}


Let $\mathsf{ST}(G)$ denote the set of \emph{spanning trees} of $G$, which are in bijection with the set of \emph{bases} in the collection of vectors $\{\mathbf{e}_i-\mathbf{e}_j\suchthat \{i<j\}\in \edges(G)\}$.
The vectors $\mathbf{e}_1$ through $\mathbf{e}_n$ denote the standard basis vectors for $\bR^n$.
The following fundamental results  then hold (see \cite[Theorem 3.8]{HR11} or \cite[Corollary 11.23]{DCP}):
\begin{enumerate}
\item $\bQ[x_i-x_j\suchthat i<j]=\mc{P}(G)\oplus I(G)$,
\item $\dim \mc{P}(G)=|\mathsf{ST}(G)|$.
\end{enumerate}
Let $\bQ[\alpxn]\coloneqq \bQ[x_1,\dots,x_n]$. 
By throwing in the linear form $x_1+\cdots+x_n$ into the mix, the first of these may be written as $\bQ[\alpx_{n}]=\mc{P}(G)\oplus \hat{I}(G)$, where $\hat{I}(G)$ is now the ideal in $\bQ[\alpxn]$ with the same generators as $I(G)$ as well as $x_1+\cdots+x_n$. 
Observe that the central $P$-space $\mc{P}(G)$ remains unchanged in both direct sum decompositions. We will henceforth abuse notation and treat $\hat{I}(G)$ and $I(G)$ as the same, with the ambient ring serving to distinguish them.

%Recall that with a graded vector space $A\coloneqq \bigoplus_{d\geq 0}A_d$ one can attach its \emph{Hilbert series} $\hilb(A)$ as follows:
%\begin{align}
%\hilb(A)=\sum_{d\geq 0}\dim(A_d)q^d.
%\end{align}
%Since $\mc{P}(X)$ is graded by degree, one may ask for its Hilbert series (or equivalently for that of the quotient $\bQ[\bfx{n}]/I(X)$). This is described in terms of the Tutte polynomial $T_{X}(x,y)$ of the linear matroid determined by $X$.
%One definition of $T_{X}(x,y)$ involves the notion of internal and external activity.
%We will only need the latter for the Hilbert series we are interested in.

\smallskip

%\subsection{The central zonotopal algebra}
An alternative perspective on $\mc{P}(G)$ than the one just offered will eventually allow us to describe it by the method of orbit harmonics.
$\mc{P}(G)$ is also the \emph{Macaulay-inverse} to a \emph{power ideal} determined by $G$. We proceed to make this precise. Like before the set of bonds $\bonds(G)$ will be important.
Given nonempty $S\subseteq [n]$, let $x_S\coloneqq \sum_{i\in S} x_i$.
Recall that for $S$ nonempty and proper, we denote by $d(S)$ the size of the edge cut $\partial(S)$.
Consider the power ideal $\mc{I}(G)$ in $\bQ[\alpxn]$ defined by
\[
\mc{I}(G)=\langle x_{1}+\cdots+x_n,\, x_S^{d(S)} \text{ where $\emptyset\neq S\subsetneq [n]$ such that $\partial(S)\in \bonds(G)$}\rangle
\]
Several nice properties of such ideals (and corresponding quotients) defined in this manner were brought forth by Ardila--Postnikov \cite{AP10}. The quotient $\bQ[\alpxn]/\mc{I}(G)$ is known as the \emph{central zonotopal algebra} attached to $G$.\footnote{We note that the ideals $\mc{I}(G)$ also appear in \cite{Pos04}, but their set of generators has many redundant elements, and Postnikov--Shapiro eliminate $x_n$ and work in a quotient of $\bQ[\alpx_{n-1}]$.}

\emph{How does $\mc{I}(G)$ relate to $\mc{P}(G)?$}
Let $\partial_i\coloneqq \frac{\partial}{\partial x_i}$.
We have (\cite[Theorem 3.8 (5)]{HR11},  or \cite[Theorem 11.25]{DCP}):
\begin{align}
\mc{P}(G) = \{f\in \bQ[\bfx{n}]\suchthat p(\partial_1,\dots,\partial_n)f=0 \text{ for all } p\in \mc{I}(G)\}.
\end{align}
Going back to \cref{ex:1}, we see for instance that the differential operator $(\frac{\partial}{\partial x_1}+\frac{\partial}{\partial x_2})^2$ corresponding to $(x_1+x_2)^2\in \mc{I}(G)$ annihilates the four polynomials that span $\mc{P}(G)$: three for degree reasons, and the fourth by a routine computation. Furthermore $\frac{\partial}{\partial x_1}+\frac{\partial}{\partial x_2}+\frac{\partial}{\partial x_3}$ annihilates any polynomial in the differences $x_i-x_j$ for $1\leq i<j\leq 3$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Orbit harmonics and break divisors}
\label{sec:orbit_harmonics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recall from the introduction that we denote the set of break divisors on $G$ by $\brkd(G)$. To motivate the connection of these divisors with the results in the preceding section, we note here the important equality $|\brkd(G)|=|\mathsf{ST}(G)|$ \cite[Theorem 4.25]{ABKS14}.

Under our usual identification of $\vertices(G)$ with $[n]$, elements of $\brkd(G)$ become certain lattice points in $\bR^{n}$. In fact, they are exactly the set of lattice points in a close cousin of a polytope attached to $G$.
Consider the \emph{graphical zonotope} $\mc{Z}_G\in\bR^n$ determined from $G$ by taking the Minkowski sum of line segments $[\mathbf{e}_i,\mathbf{e}_j]$ for every edge $\{i,j\}\in \edges(G)$.
Let $\Delta_{n-1,n}$ denotes the $(n-1)$-th hypersimplex obtained as the convex hull of the $S_n$-orbit of the point $(1^{n-1},0)\in \bR^n$.
This given we have \cite[Proposition 2.1]{KRT21}:
\[
 \brkd(G)=(\mc{Z}_G-\Delta_{n-1,n})\cap \bZ^n,
\]
where $\mc{Z}_G-\Delta_{n-1,n}$ denotes the \emph{trimmed graphical zonotope} obtained as a Minkowski~difference.
As mentioned in the introduction, $\aut(G)$ acts on $\brkd(G)$ by permutations. 
Thus we have a point set on which a linear group acts, and such a setup is ideal for the method of orbit harmonics. %While it is currently unclear what the result of doing so will be, we shall see that  the power ideal $\mc{I}(G)$ from before emerges naturally.

%\subsection{Orbit harmonics}
The method of {\em orbit harmonics} gives a technique for turning an ungraded permutation representation of some linear group $\mathscr{G}$
acting on a finite point
locus $Y$ into a graded $\mathscr{G}$-module.
Orbit harmonics was introduced by Kostant \cite{Kostant} and has seen subsequent application (for example)
by Garsia--Procesi \cite{GP92} in the context of Springer fibers and by
Haglund--Rhoades--Shimozono \cite{HRS18} in the context of Macdonald-theoretic delta
operators. 
%See \cite{Rh22} for a survey of recent developments.



%We begin with a concise description of the point-orbit method.
Consider a finite point set $Y \subset \bQ^n$ and let $\mathsf{I}(Y) \subseteq \bQ[\alpxn]$
be the ideal of polynomials which vanish on $Y$. The quotient of $\bQ[\alpxn]$ by the ideal
$\mathsf{I}(Y)$ has vector space dimension $\dim_{\bQ}\left(\bQ[\bfx{n}]/\mathsf{I}(Y)\right)=|Y|$.
If the set $Y$ is $\mathscr{G}$-stable for some group $\mathscr{G}\subset \mathrm{GL}_n(\bQ)$, then we have an isomorphism of ungraded
$\mathscr{G}$-modules
\[
\bQ[\bfx{n}]/\mathsf{I}(Y)\cong_{\mathscr{G}} \bQ[Y].
\]
 Orbit harmonics produces a quotient that will afford the structure of a {\em graded} $\mathscr{G}$-module.
 Given a nonzero polynomial $f \in \bQ[\alpxn]$, let $\tau(f)$ be the top degree homogeneous component of $f$.
 Define an ideal $\mathsf{T}(Y) \subseteq \bQ[\bfx{n}]$ by
 \[
 \mathsf{T}(Y) = \langle \tau(f) \,:\, f \in \mathsf{I}(Y), \, \, f \neq 0 \rangle.
 \]
The $\mathscr{G}$-module isomorphism given above extends to a chain
\[
\bQ[\bfx{n}]/\mathsf{T}(Y) \cong_{\mathscr{G}}
\bQ[\bfx{n}]/\mathsf{I}(Y)\cong_{\mathscr{G}} \bQ[Y],
\]
with the added feature that the left hand side is a graded $\mathscr{G}$-module.


%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Applying orbit harmonics to $\brkd(G)$}
\label{subsec:gp_to_brkd}
%%%%%%%%%%%%%%%%%%%%%%%%%%
We will apply orbit harmonics to the locus $Y = \brkd(G)$ with $\mathscr{G}=\aut(G)$. Remarkably, the ideal $\mathsf{T}(Y)$ which so arises will coincide with the power ideal $\mc{I}(G)$.
For brevity we refer to the ideals $\mathsf{I}(\brkd(G))$ and $\mathsf{T}(\brkd(G))$ by $\mathsf{I}(G)$ and $\mathsf{T}(G)$ respectively. We then have the following result.
%We briefly outline our strategy next.


%
%\begin{prop}
%\label{prop:GP_break_divisors}
%
%\end{prop}
%

\begin{thm}
\label{thm:piecing_things}
We have the equality of ideals $
\mathsf{T}(G)=\mc{I}(G).$
Consequently we have the following isomorphisms and equalities of ungraded $\aut(G)$-modules:
\[
\bQ[\brkd(G)]\cong \bQ[\bfx{n}]/\mathsf{T}(G) = \bQ[\bfx{n}]/\mc{I}(G) \cong \mc{P}(G).
\]
where the middle equality and right isomorphism are in the category of \emph{graded} $\aut(G)$-modules.
\end{thm}
\begin{proof}[Sketch of proof]
We begin by identifying a family of polynomials indexed by $\bonds(G)$ that vanish on $\brkd(G)$.
Subsequently we consider the ideal  $\tilde{\mathsf{T}}(G)$ generated by top degree homogeneous summands of the polynomials in the aforementioned family.
The inclusion $\tilde{\mathsf{T}}(G) \subset \mathsf{T}(G)$ on the one hand implies that
\[
\dim\left(\bQ[\bfx{n}]/\tilde{\mathsf{T}}(G)\right) \geq \dim\left(\bQ[\bfx{n}]/\mathsf{T}(G)\right).
\]
On the other hand, it transpires that $\tilde{\mathsf{T}}(G)$ equals the power ideal $\mc{I}(G)$ and thus the left hand side equals $|\mathsf{ST}(G)|$. Additionally, the right hand side equals $|\brkd(G)|$. Given that $|\mathsf{ST}(G)|= |\brkd(G)|$ we conclude that $\tilde{\mathsf{T}}(G)=\mathsf{T}(G)$.
\end{proof}

We demonstrate several key notions of this section in the following example. %The example also sheds light on $\mc{I}(G)$ appears in the application of orbit harmonics.

\begin{exam}
\label{ex:break_divisors_orbits}
\emph{
Let  $G$ be the complete bipartite graph $K_{2,3}$. Let us identify the vertex set with $[5]$ with $\{1,2\}$ and $\{3,4,5\}$ being the two partite sets.
Elements of $\brkd(G)$ are certain tuples $(d_1,d_2,d_3,d_4,d_5)\in \mathbb{Z}_{\geq 0}^5$ where $\sum_{1\leq i\leq 5}d_i=g(K_{2,3})=2$.
Any subgraph that is a tree gives a trivial constraint.  From  the $S_2\times S_3$ symmetry in $G$ it follows that given any break divisor we can permute $d_1$ and $d_2$ (respectively $d_3$, $d_4$, and $d_5$) to get another break divisor. 
Thus, up to  $S_2\times S_3$ symmetry, the only relevant inequality is $
d_1+d_2+d_3+d_4\geq 1
$. 
In summary we obtain the following four orbit representatives for $S_2\times S_3$-action on $\brkd(G)$: $(2,0,0,0,0)$, $(1,1,0,0,0)$, $(1,0,1,0,0)$, $(0,0,1,1,0)$.
We thus obtain $|\brkd(G)|=12$, which agrees with the number of spanning trees of $G$.}


\emph{Consider the bond $\partial(S)$ determined by choosing $S=\{1\}$. This determines the linear form $x_1$. The smallest (respectively largest) value that $x_1$ attains on $\brkd(G)$ is $0$ (respectively $2$). Thus the polynomial $x_1(x_1-1)(x_1-2)$ vanishes on $\brkd(G)$. This polynomial has $x_1^3$ as the top degree homogeneous summand, and this polynomial is indeed in $\mc{I}(K_{2,3})$ as well as $\mathsf{T}(K_{2,3})$. We leave it to the reader to verify that
\[
\mathsf{T}(K_{2,3})=\langle x_1+\cdots+x_5,x_i^3,x_j^2,(x_i+x_j)^3 \text{ where $i\in \{1,2\}, j\in \{3,4,5\}$} \rangle.
\]
The quotient $\bQ[\alpx_5]/\mathsf{T}(K_{2,3})$ has a monomial basis \[
\{1,x_1,x_2,x_3,x_4,x_1^2,x_2^2,x_1x_2,x_1x_3,x_2x_3,x_1x_4,x_2x_4\}.
\]
It can be shown that the elements $\{1,x_1+x_2,x_1^2+x_2^2,x_1x_2\}$ span the space of $S_2\times S_3$-invariants. We thus get
\[
\hilb((\bQ[\alpx_5]/\mathsf{T}(K_{2,3}))^{S_2\times S_3})=1+q+2q^2.
\]
As we shall soon see, this last polynomial is essentially a quantum DT invariant; see \cref{ex:revisiting_quantum_DT}. This shall follow once we make a connection to Efimov's work.
}
\end{exam}


We conclude this section with one final remark. 
In general, it follows from \cite{Pos04} that the quotient $\bQ[\bfx{n}]/\mc{I}(G)$ has a monomial basis indexed by $G$-parking functions. Thus, \cref{thm:piecing_things} provides an algebraic perspective on the fact that the set of $q$-reduced divisors on $G$ (essentially $G$-parking functions) has the same cardinality as $\brkd(G)$.   Indeed, this same quotient arises when applying orbit harmonics to $\brkd(G)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A construction of Efimov for quantum DT invariants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $A=(a_{ij})_{i,j\in [k]}$ be a symmetric matrix with nonnegative integer entries. We assume $a_{ii}\geq 1$ throughout.
The matrix $A$ determines a quiver $Q$ on $k$ vertices labeled $1$ through $k$, with $a_{ij}$ arrows from $i$ to $j$ for $i,j\in [k]$.
Note that $a_{ii}\geq 1$ ensures at least one loop at each vertex.
We further assume $A$ is such that $Q$ is connected.
Let $\gamma=(\gamma_1,\dots,\gamma_k)\in \bZ_{\geq 0}^k$ be a dimension vector.
We begin by describing the essential construction in \cite{Efi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Quantum Donaldson--Thomas (DT) invariants}
%\label{subsec:quantum_dt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Consider variables $x_{i,\alpha}$ for $1\leq i\leq k$ and $1\leq \alpha\leq \gamma_i$, and let $\sigma_{\gamma}$ be their sum.
Define the polynomial ring
\begin{align}
A_{\gamma}=\bQ[x_{i,\alpha}\suchthat 1\leq i\leq k,1\leq \alpha\leq \gamma_i].
\end{align}
Furthermore, let
\begin{align}
A_{\gamma}^{\prim}=\bQ[x_{j,\alpha_2}-x_{i,\alpha_1}\suchthat 1\leq i,j\leq k,1\leq \alpha_1\leq \gamma_i, 1\leq \alpha_2\leq \gamma_j].
\end{align}
Then we have $A_{\gamma}=A_{\gamma}^{\prim}\otimes \bQ[\sigma_{\gamma}]$.
Define $S_{\gamma}\coloneqq S_{\gamma_1}\times \cdots \times S_{\gamma_k}$.

Define $J_{\gamma}$ to be the smallest $S_{\gamma}$-stable $A_{\gamma}^{\prim}$-submodule of $A_{\gamma}^{\prim}$ such that the following holds: for any decomposition of $\gamma=\delta+\bar{\delta}$ where both $\delta$ and $\bar{\delta}$ are nonzero we have that
\begin{align}
  \label{eq:important_poly}
f_{\delta,\bar{\delta}}=\displaystyle\prod_{i\neq j\in[k]}\prod_{\alpha_1=1}^{\delta_i}\prod_{\alpha_2=\delta_j+1}^{\gamma_j}(x_{j,\alpha_2}-x_{i,\alpha_1})^{a_{ij}}\displaystyle\prod_{i\in[k]}\prod_{\alpha_1=1}^{\delta_i}\prod_{\alpha_2=\delta_i+1}^{\gamma_i}(x_{i,\alpha_2}-x_{i,\alpha_1})^{a_{ii}-1} \in J_{\gamma}.
\end{align}
 We shall later reinterpret $f_{\delta,\bar{\delta}}$ in terms of cuts (as the indexing hints) in a graph constructed from $(A,\gamma)$.

Letting $\mc{H}_{\gamma}^{\prim}\coloneqq (A_{\gamma}^{\prim})^{S_{\gamma}}$, consider the  decomposition \cite[p. 1139]{Efi}
\begin{align}
\label{eq:Efimov_decomposition}
\mc{H}_{\gamma}^\prim=V_{\gamma}^{\prim}\oplus J_{\gamma}^{S_{\gamma}}.
\end{align}
The quantum DT invariants of the quiver $Q$ with dimension vector $\gamma$ (assuming trivial potential and stability) arise as dimensions of the graded pieces of the graded vector space $V_{\gamma}^{\prim}$ as explained in \cite[Section 4]{Efi}. We describe the $\bZ$-grading employed.


Given dimension vectors $\gamma$ and $\delta$, the \emph{Euler form} $\chi_Q(\gamma,\delta)$  is defined as
\begin{align}
\chi_Q(\gamma,\delta)\coloneqq \sum_{1\leq i\leq k}\gamma_i\delta_i-\sum_{1\leq i,j\leq k}a_{ij}\gamma_i\delta_j.
\end{align}
Homogeneous polynomials $f\in \mc{H}_{\gamma}$ of degree $d$ get assigned the grading $2d+\chi_Q(\gamma,\gamma)$.
We let $V_{\gamma,m}^{\prim}$ be the space of elements in $V_{\gamma}^{\prim}$ with grading $m$.
Following \cite[Section 4]{Efi},  and as in the introduction, set $
c_{\gamma,m}\coloneqq \dim(V_{\gamma,m}^{\mathrm{prim}})$,
and consider the polynomial in $\bZ_{\geq 0}[q^{\pm \frac{1}{2}}]$:
\begin{align}
\Omega_{\gamma}(q)=\sum_{m\in \bZ}c_{\gamma,m}q^{m/2}.
\end{align}
These $\Omega_{\gamma}(q)$ are the \emph{quantum DT-invariants} of the quiver $Q$.

We get a necessary condition for when $V_{\gamma,m}^{\prim}$ is nonzero in \cite[Theorem 1.2]{Efi} (also presented as \cite[Theorem 3.10]{Efi}). 
It involves a statistic $N_{\gamma}(Q)$ \cite[Section 1]{Efi} whose precise definition is unnecessary for the purposes of this abstract. 
\begin{thm}[{\cite[Theorem 3.10]{Efi}}]
\label{thm:k_for_nonzero_spaces}
If $V_{\gamma,m}^{\mathrm{prim}}\neq 0$, then $\gamma\neq 0$, $
m\equiv \chi_Q(\gamma,\gamma) \mod 2$, and $\chi_Q(\gamma,\gamma)\leq m< \chi_Q(\gamma,\gamma)+2N_{\gamma}(Q).$
\end{thm}
In view of \cref{thm:k_for_nonzero_spaces} we may rewrite $\Omega_{\gamma}(q)$ as
\begin{align}
\Omega_{\gamma}(q)=q^{\frac{1}{2}\chi_Q(\gamma,\gamma)}\sum_{0\leq m\leq N_{\gamma}(Q)-1}c_{\gamma,2m+\chi_Q(\gamma,\gamma)}q^{m}.
\end{align}
Denote the sum on the right by $\tilde{\Omega}_{\gamma}(q)$. It lies in $\bZ_{\geq 0}[q]$ and its degree is bounded by $N_{\gamma}(Q)-1$. We abuse notation and refer to $\tilde{\Omega}_{\gamma}(q)$ as the quantum DT invariant as well.\footnote{As the careful reader may have realized, we already do so in the introduction.}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The covering graph construction}
\label{subsec:symmetric_matrix_to_graph}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The reader might find the preceding  construction both ingenious and mysterious.
It transpires that the space $V_{\gamma,m}^{\prim}$ is the space of $S_\gamma$-invariants for a slim subgraph space determined  naturally from the quiver and the dimension vector.
We proceed to describe this construction.


Our point of departure from Efimov is to consider an analogue of  \eqref{eq:Efimov_decomposition} where we do not  take $S_{\gamma}$-invariants.
We will construct an $S_\gamma$-stable  space $W_{\gamma}^{\prim}$ so that
\begin{align}
\label{eq:general_efimov}
A_{\gamma}^{\prim}=W_{\gamma}^{\prim}\oplus J_{\gamma}.
\end{align}
Our next construction is crucial to this end.

%Note that $W_{\gamma}^{\prim}$ naturally affords an $S_{\gamma}$-representation. We will realize this space as a central $P$-space.
%Of course, $V_{\gamma}^{\prim}=(W_{\gamma}^{\prim})^{S_{\gamma}}$.
% We proceed to show that $W_{\gamma}^{\prim}$ is the central $P$-space for the vector configuration of the sort considered in Subsection~\ref{subsec:graphic_matroid}.


Given $Q$ as before, we construct an undirected graph $G_{Q,\gamma}$:
Consider a set of vertices $v_{i,\alpha}$ for $1\leq i\leq k$ and $1\leq \alpha\leq \gamma_i$. For $i\in [k]$, the restriction of $G_{Q,\gamma}$ to the vertices $v_{i,1},\dots, v_{i,\gamma_i}$ is the clique on $\gamma_i$ vertices with $a_{ii}-1$ edges between any two distinct vertices. In particular, if $a_{ii}=1$, then we have a collection of $\gamma_i$ totally disconnected vertices.
For $i\neq j\in [k]$ we draw $a_{ij}$ edges between any vertex $v_{i,\alpha_1}$ and $v_{j,\alpha_2}$ for $1\leq \alpha_1\leq \gamma_i$ and $1\leq \alpha_2\leq \gamma_j$.
This determines $G_{Q,\gamma}$. 
Note that $S_{\gamma}$ is a subgroup of $\aut(G_{Q,\gamma})$.


\emph{We will assume throughout that $G_{Q,\gamma}$ is connected}. 
For the remainder of this section, we fix our symmetric quiver $Q$ and dimension vector $\gamma$, and we will drop them from notation. In particular, unless otherwise noted,  let $G\coloneqq G_{Q,\gamma}$.



We reinterpret the important element $f_{\delta,\bar{\delta}}\in J_{\gamma}$ from Equation~\eqref{eq:important_poly} as the polynomial $p_H$ for some cut in $G$. 
\begin{lem}
\label{lem:efimov_translated}
Consider a decomposition $\gamma=\delta+\bar{\delta}$ where $\delta,\bar{\delta}\neq \gamma$.
The following hold.
\begin{enumerate}
\item $f_{\delta,\bar{\delta}}=p_H$ for some $H\in \cuts(G)$.
\item For $\emptyset\neq S\subsetneq \vertices(G)$ such that $\partial(S)\in \bonds(G)$, we have that $p_{\partial(S)}\in J_{\gamma}$.
More specifically, there exists a decomposition $\gamma=\delta+\bar{\delta}$ such that $\sigma(f_{\delta,\bar{\delta}})=p_{\partial(S)}$ for some $\sigma\in S_{\gamma}$.
\end{enumerate}
\end{lem}
The preceding lemma simply states that Efimov's ideal $J_{\gamma}$ is precisely the ideal generated by $p_H$ for $H\in \bonds(G)$, i.e. $J_{\gamma}=I(G)$ where the latter is the cocircuit ideal  from Equation~\eqref{eq:def_cocircuit_ideal}. We thus obtain the following theorem that `combinatorializes' Efimov's construction and describes the DT invariants in terms of $\mc{P}(G)$.
\begin{thm}
\label{thm:graded_multiplicity_quantum_DT}
The following decomposition holds:
$A_{\gamma}^{\prim}=\mc{P}(G)\oplus J_{\gamma}.
$
Consequently we have the equality
\begin{align*}
\hilb(\mc{P}(G)^{S_{\gamma}})=\tilde{\Omega}_{\gamma}(q).
\end{align*}
From the chain of isomorphisms in \cref{thm:piecing_things} we have 
$\mathrm{DT}_{Q,\gamma}=\tilde{\Omega}_{\gamma}(1)=\dim(\bQ[\brkd(G)]^{S_{\gamma}})$, i.e. the number of $S_{\gamma}$-orbits on $\brkd(G)$.
\end{thm}
To demonstrate our main result, we revisit \cref{ex:break_divisors_orbits}, this time starting with a quiver and a dimension vector.

%Taking dimensions of the space of $S_{\gamma}$-invariants in the chain of isomorphisms in \cref{thm:piecing_things} yields our central result.
%\begin{thm}
%The numerical DT-invariant  $\mathrm{DT}_{Q,\gamma}$ equals the number of $S_{\gamma}$-orbits on $\brkd(G_{Q,\gamma})$.
%\end{thm}

\begin{exam}
\label{ex:revisiting_quantum_DT}
\emph{
Consider the quiver with two vertices, each with a self-loop, and two edges between the vertices, one in each direction. 
Pick $\gamma=(2,3)$.
The covering graph construction then yields $G_{Q,\gamma}$ as the bipartite graph $K_{2,3}$.
We reindex our variables $x_1= x_{11}$, $x_2=x_{12}$, $x_3=x_{21}$, $x_4=x_{22}$, and $x_5=x_{32}$.
We let $S_2\times S_3$ act on $\bQ[x_1,\dots,x_5]$ by letting $S_2$ (respectively $S_3$) act on $x_1,x_2$ (respectively $x_3,x_4,x_5$).
}

\emph{
 $\mc{P}(G)$ is spanned by $p_H$ for slim subgraphs $H$ which may be obtained as the $S_{\gamma}$-orbit of elements in $\{1,x_1-x_3, (x_1-x_3)(x_1-x_4), (x_1-x_3)(x_2-x_4) \}$.
The space $\mc{P}(G)^{S_\gamma}$ has basis elements
\[
\{1,\sum_{\sigma\in S_{\gamma}}\sigma\cdot(x_1-x_3), \sum_{\sigma\in S_{\gamma}} \sigma\cdot (x_1-x_3)(x_1-x_4), \sum_{\sigma\in S_{\gamma}}\sigma\cdot (x_1-x_3)(x_2-x_4))\}.
\]
Explicitly, other than the constant polynomial $1$, up to normalization these equal
\begin{align*}
&3(x_1+x_2)-2(x_3+x_4+x_5)\\
&3(x_1^2+x_2^2)-2(x_1+x_2)(x_3+x_4+x_5)+2(x_3x_4+x_3x_5+x_4x_5)\\
& 3x_1x_2-(x_1+x_2)(x_3+x_4+x_5)+(x_3x_4+x_3x_5+x_4x_5).
\end{align*}
We thus infer that
\[
\tilde{\Omega}_{\gamma}(q)=1+q+2q^2,
\]
and therefore that $\tilde{\Omega}_{\gamma}(1)=4$. Going back to \cref{ex:break_divisors_orbits}, this agrees with the number of $S_2\times S_3$-orbits on $\brkd(K_{2,3})$. We also encountered the polynomial $1+q+2q^2$ in that same example. Back then we obtained it as $\hilb((\bQ[\alpx_5]/\mc{I}(K_{2,3}))^{S_2\times S_3})$.
}
\end{exam}


\section{Two applications}
We close with two representation-theoretic results of independent interest, with motivation coming from \cite{BR14}.
Like before, we set $G\coloneqq G_{Q,\gamma}$.

\smallskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{The top degree of $\mc{P}(G)$ as an $S_\gamma$-module.}
%\label{subsec:top_degree}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Our second result generalizes a result of \cite[Theorem 5]{BR14} which identifies the top degree graded piece of $\mc{P}(K_n^m)$ with $\mathrm{Lie}_n\otimes \varepsilon_n$.
%Here $\mathrm{Lie}_n$ is the well-known \emph{Lie representation} of $S_n$ with the property that upon restriction to $S_{n-1}$, one recovers the regular representation. In particular, the dimension of $\mathrm{Lie}_n$ is $(n-1)!$.
%One description of $\mathrm{Lie}_n$ is as the action on $S_n$ on the multilinear part of the free Lie algebra on $n$ symbols.
%
The graph $G$ (with $\vertices(G)=[n]$) determines a hyperplane arrangement $\mc{A}_G$ in $\bC^n$ by considering the union of hyperplanes $x_i-x_j=0$ for edges $\{i,j\}\in E(G)$.
The \emph{de Rham cohomology ring} $H^*(\bC^n\setminus \mc{A}_G)$ over $\bC$ of the complement $\bC^n\setminus \mc{A}_G$  is the exterior algebra over the generators $\mathrm{d}\log(\beta)=\frac{d\beta}{\beta}$ where $\beta$ ranges over the linear forms that cut out $\mc{A}_G$.
Consider the collection of vectors $X_G=\{\mathbf{e}_i-\mathbf{e_j}\suchthat \{i<j\}\in \edges(G)\}$, and endow it with a total order.
The top degree graded piece of $H^*(\bC^n\setminus \mc{A}_G)$, denoted by $H^*(\bC^n\setminus \mc{A}_G)_{\mathrm{top}}$, has a basis indexed by so-called \emph{unbroken bases}. Such bases also form a basis for the top degree of $\mc{P}(G)$, denoted by $\mc{P}(G)_{\mathrm{top}}$. This equality in dimension of the two spaces is a shadow of an $S_{\gamma}$-isomorphism between two vector spaces. The following result generalizes \cite[Theorem 5]{BR14}.
%This is easily seen from the expression~\eqref{eq:hilb_p(x)} describing $\hilb(\mc{P}(G))$ as a specialization of the Tutte polynomial of $\hat{X}_G$.
%Thus we see that
%\begin{align}
%\label{eq:equality_of_dimensions}
%\dim(\mc{P}(G)_{\mathrm{top}})=\dim(H^*(\bC^n\setminus \mc{A}_G)_{\mathrm{top}}).
%\end{align}

%The quantity in~\eqref{eq:equality_of_dimensions} is known to equal the number of bounded regions in the complement $\bR^n\setminus \mc{A}_G$ of the \emph{real} arrangement.


%In fact, following Berget--Rhoades \cite{BR14}, we can say more.
\begin{prop}
\label{prop:top_degree}
We have the following $S_{\gamma}$-isomorphism:
\[
\mc{P}(G)_{\mathrm{top}}\cong H^*(\bC^n\setminus \mc{A}_G)_{\mathrm{top}}\otimes \varepsilon_{\gamma_1}^{a_{11}-1}\otimes \cdots \otimes \varepsilon_{\gamma_k}^{a_{kk}-1}.
\]
Here $a_{ii}-1$ is one less than the number of loops at vertex $i$ in $Q$, and $\varepsilon_{\gamma_i}$ denotes the \emph{sign representation} of $S_{\gamma_i}$.
\end{prop}


%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\textbf{The $(m+1)$-loop quiver.}
%\label{subsec:loop quiver}
%%%%%%%%%%%%%%%%%%%%%%%%
Let $m$ be a positive integer.
We revisit the case of the $(m+1)$-loop quiver $Q$ with dimension vector $\gamma=(n)$ where $n\geq 1$.
The covering graph $G\coloneqq G_{Q,\gamma}$ is the \emph{complete multipartite graph} $K_n^{m}$, i.e. the graph on $n$ vertices with $m$ edges between any two distinct vertices.
 $\aut(G)$ is clearly the symmetric group $S_n$.
\cref{thm:piecing_things} then gives us (isomorphic) spaces with dimension $m^{n-1}n^{n-2}$.

Berget--Rhoades \cite{BR14} consider the $S_n$-modules $\mc{P}(K_n^m)$ and establish several interesting results.
While the graded Frobenius characteristics of these modules  remains elusive, we are able to shed light on related matters. In fact these modules encode a large class of DT invariants; see \cite[Section 5.4]{RRT22} for more.
%We consider one such example



 \cref{thm:piecing_things} allows us to connect the space $\mc{P}(K_n^{m})$ to $\bQ[\brkd(K_n^m)]$ and, among other things, resolve \cite[Conjecture 3.3]{KT21} and recover one of the main results in \cite{BR14}.
\begin{corollary}
\label{cor:p-space-corollary}
The following hold.
\begin{enumerate}
\item We have the isomorphism $\mc{P}(K_n^m)\cong_{S_n} \bQ[\brkd(K_n^m)]$ of ungraded representations. In particular, the \emph{Frobenius characteristic} $\frob(\mc{P}(K_n^m))$ is $h$-positive.
\item The graded multiplicity of the trivial representation of $S_n$ in $\mc{P}(K_n^m)$ is given by the quantum DT invariant $\tilde{\Omega}_{(n)}(q)$.
At $q=1$ we obtain
\begin{align*}
\label{eq:explicit_m_loop_quiver}
\dim(\mc{P}(K_n^m)^{S_n})=\mathrm{DT}_{Q,(n)}=
\frac{1}{mn^2}\sum_{d|n}(-1)^{mn+\frac{mn}{d}}\mu(d)\binom{\frac{(m+1)n}{d}-1}{\frac{n}{d}}.
\end{align*}
\item On the level of graded Frobenius characteristics, we have \[\grfrob(\mc{P}(K_n)\downarrow_{S_{n-1}}^{S_n})=\grfrob(\mathrm{PF}_{n-1}),\] where $\mathrm{PF}_{n-1}$ is the parking function representation of $S_{n-1}$, with basis indexed by parking functions and grading given by the sum of entries in a parking function.
\end{enumerate}
\end{corollary}


%
%\acknowledgements{V.T. is very grateful to Matja\v{z} Konvalinka for previous collaboration. Further thanks go to Spencer Backman, Chris Eur, Philippe Nadeau, Marino Romero, and Chi Ho Yuen.}

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