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\documentclass[finalversion]{FPSAC2025}
\articlenumber{110}
%\usepackage{prefix}

%\usepackage[utf8]{inputenc}
\usepackage{amsfonts,amsbsy,amstext}
\usepackage{graphicx}
%\usepackage[colorlinks,allcolors=blue]{hyperref}
%\usepackage[nameinlink,noabbrev,capitalise]{cleveref}
\usepackage{enumerate}
\usepackage{aliascnt}%make lemma numbering agree with theorem numbering
\usepackage{tikz,tikz-cd,tkz-euclide,tkz-graph}
%\usepackage{biblatex}
\usepackage{xcolor,forest}
\usepackage{accents,color,enumerate,float,verbatim}
\usepackage{bbm}
%\linespread{1.10}
\usepackage{comment}
%\usepackage[margin=1.5in]{geometry}
\usepackage{enumitem}
%\usepackage[font=small]{caption}
\usepackage{fancyhdr}
\restylefloat{table}
\DeclareMathOperator\cis{cis}
\usepackage{multicol}
\usepackage{nicefrac}

%COMMANDS
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\BB}{\mathbb{B}}
\newcommand{\bigO}{\mathcal{O}} %big o notation

\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} %floor
\renewcommand{\brace}[1]{\left\{ #1 \right\}} %{value}
\renewcommand{\brack}[1]{\left[ #1 \right]}   %[value]
\newcommand{\paren}[1]{\left( #1 \right)}     %(value)
\newcommand{\range}[2]{#1 \;\dots\; #2} % a ... b
\newcommand{\rangeS}[2]{\brace{#1 \;\dots\; #2}} %{a ... b} (range+set)

\DeclareMathOperator{\Img}{Im}
\newcommand{\newword}[1]{\emph{#1}}

%CHIP FIRING COMMANDS
\newcommand{\sstab}{\mathrm{sstab}} %superstable entries
\newcommand{\crit}{\mathrm{crit}}   %critical entries
\newcommand{\critgroup}{\mathcal{K}}  %critical group
\newcommand{\grouping}{\square}  %preimage groupings square notation
\newcommand{\flcm}{\mathrm{flcm}} %fractional gcds
\newcommand{\cmax}{\vec c_{\mathrm{max}}}

\newcommand{\switchEquiv}{\sim}
\newcommand{\firingMEquiv}{\sim_M}
\newcommand{\firingLEquiv}{\sim_L}

\newcommand{\LM}{LM^{-1}}
\newcommand{\ML}{ML^{-1}}
\newcommand{\Mi}{M^{-1}}
\newcommand{\Minv}{\Mi}
\newcommand{\Li}{L^{-1}}
\newcommand{\Linv}{\Li}

\newcommand{\switchMat}{\hat E} %the matrix \hat E that we vertex switch with

\newcommand{\fracket}{F}
\newcommand{\involution}{\mu}
\newcommand{\flip}{\mathrm{flip}}
\newcommand{\duality}{\chi} %Signed duality


%SHORTHANDS (use these ones)
\newcommand{\frp}[1]{\brace{#1}} %fractional part
\newcommand{\set}[1]{\brace{#1}} %make a set
\newcommand{\xor}{\oplus}
\newcommand{\bigxor}{\bigoplus}

\newcommand{\sst}{\sstab}
\newcommand{\cm}{\cmax}

\newcommand{\Meq}{\firingMEquiv}
\newcommand{\Leq}{\firingLEquiv}

\newcommand{\SK}{\critgroup} 
\newcommand{\KG}{\critgroup\paren{G}}
\newcommand{\KGP}{\critgroup\paren{G_\phi}}

\newcommand{\fkt}{\fracket}
\newcommand{\ivo}{\involution}
\newcommand{\dual}{\chi}
\newcommand{\fl}[1]{\flip}
\newcommand{\sfl}{\sflip}

\newcommand{\Ehat}{\switchMat}

%VECTORS
\newcommand{\cc}{\vec{c}}
\newcommand{\dd}{\vec{d}}
\newcommand{\ee}{\vec{e}}
\newcommand{\rr}{\vec{r}}
\renewcommand{\ss}{\vec{s}}
\newcommand{\vv}{\vec{v}}
\newcommand{\xx}{\vec{x}}

% MACROS



%\setlength{\parindent}{0pt}
%\setlength{\parskip}{2mm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{question}[theorem]{Question}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}

% make theorems referenced properly with \Cref
%\newcommand{\newcreftheorem}[2]{
%	\newaliascnt{#1}{theorem}
%	\newtheorem{#1}[#1]{#2}
%	\aliascntresetthe{#1}
%}
%\newcreftheorem{lemma}{Lemma}
%\newcreftheorem{corollary}{Corollary}
%\newcreftheorem{definition}{Definition}
%\newcreftheorem{algorithm}{Algorithm}
%\newcreftheorem{remark}{Remark}
%\newcreftheorem{question}{Question}
%\newcreftheorem{proposition}{Proposition}
%\newcreftheorem{example}{Example}



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%% note that the class file already loads {amsmath, amsthm, amssymb}




%\numberwithin{equation}{section}

\usepackage{lipsum}

%% define your title in the usual way
\title[On z-superstable and critical configurations of chip-firing pairs]{On z-superstable and critical configurations of chip-firing pairs}

%% 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[Zach Benton, Jane Kwak, Suho Oh, Mateo Torres, Mckinley Xie]{
\texorpdfstring{Zach Benton \addressmark{1}, Jane Kwak \addressmark{2}, Suho Oh \addressmark{3}, Mateo Torres \addressmark{4},\\ \and Mckinley Xie \addressmark{5}}{Zach Benton, Jane Kwak, Suho Oh, Mateo Torres, and Mckinley Xie} }


%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Stanford University, Stanford, CA \\ \addressmark{2} Department of Mathematics, UCLA, Los Angeles, CA \\
\addressmark{3} Department of Mathematics, Texas State University, San Marcos, TX \\
\addressmark{4} College of Arts and Sciences, University of Delaware, Newark, DE \\
\addressmark{5} Department of Mathematics, Texas A\&M, College Station, TX
}

%% 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{It is well known that there is a duality map between the superstable configurations and the critical configurations of a graph. This was extended to all M-matrices in (Guzmán and Klivans, 2015). We show a natural way to extend this to all $(L,M)$-chip firing pairs introduced in (Guzmán and Klivans, 2016). In addition, we study various properties of $z$-superstable configurations and critical configurations of $(L,M)$-chip firing pairs.}

%% put your French abstract here, or comment this out if you don't have one
\resume{Il est bien connu qu’il existe une dualité entre les configurations superstables et les configurations critiques d’un graphe. Dans (Guzmán et Klivans, 2015), ce résultat a été étendu à toutes les M-matrices. Nous démontrons une façon naturelle d’étendre cette dualité à toutes les paires $(L,M)$ qui sont « chip-firing », introduites dans (Guzmán et Klivans, 2016). De plus, nous étudions diverses propriétés des configurations $z$-superstables et des configurations critiques des paires « chip-firing ».}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Chip-firing, Critical group, Avalanche finite, M-matrices, Chip-firing pairs}

%% 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
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\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{bibliography.bib}
\appto\bibfont{\setlength{\emergencystretch}{.5em}}
%% 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
\input{introandprelim}

\section{The Central Duality}

In this section we establish the duality between the superstable configurations and critical configurations of $(L,M)$-pairs, that extends the canonical duality between superstable and critical configurations of $M$-matrices. We are mainly going to be dealing with the pre-images of the configurations in $R^{+}$.

\subsection{The involution \texorpdfstring{$\ivo$}{mu} and the duality map.}
We define an involution on the set of superstable configurations of $M$. Given any vector $\vec{f}$, recall that we used $\floor{\vec{f}}$ to denote the vector obtained from $f$ by taking the floor at every coordinate. We are going to let $\{\vec{f}\}$ denote $f - \floor{\vec{f}}$.

\begin{definition}
\label{def:ivomap}
    For any chip-firing pair $(L,M)$, we define the map $\ivo : \sst(M) \rightarrow \sst(M)$:
    \[
    \ivo(\vec s) = \begin{cases}
     \vec s &\text{if $\{\LM2\vec s\:\} = \{\LM\cm\}$, } \\
    \sstab(\cmax-\vec s) &\text{otherwise.}
    \end{cases}
    \]
\end{definition}

It is clear from definition that the above map is indeed an involution:

\begin{proposition}
    The map $\ivo$ is an involution.
\end{proposition}

In the special case where $L=M$ (this recovers the usual chip-firing on $M$-matrices, and in particular when $M$ is the Laplacian of a graph, the classical chip-firing) the above involution is simply the identity map: since all superstable and critical preimages are integer vectors.

\begin{lemma}
\label{lem:equivsameblock}
Let $\vec{a},\vec{b}$ be integer vectors such that they are equivalent under $M$. Let $\vec{f}$ be a vector such that every entry $f_i$ satisfies $0 \leq f_i < 1$. Then $\vec{a}+\vec{f} \in R^{+}$ if and only if $\vec{b}+\vec{f} \in R^{+}$.
\end{lemma}

%**Example here demonstrating that if a,b are equivalent they their block structure is the same

\begin{example}
    We use the same $(L,M)$ pair coming from a signed graph studied in previous examples. From the configuration $\vec a = (1, 2, 0)$, we can fire vertex 2 to obtain the configuration $\vec b = (2, 0, 1)$, so these configurations are firing-equivalent under $M$. Take some non-negative rational vectors where the entries are bounded above by $1$, say  $\vec f_1 = (\frac 1 3, \frac 2 3, 0)$ and $\vec f_2 = (\frac 2 3, \frac 1 3, 0)$.

    \begin{center}
    \begin{tabular}{c | c}
        Preimage & Image under $LM^{-1}$ \\
        \hline
        $\vec a + \vec f_1 = (\frac 4 3, \frac 8 3, 0)$ & $(8, 7, 0)$ \\
        $\vec b + \vec f_1 = (\frac 7 3, \frac 2 3, 1)$ & $(7, 5, 1)$ \\
        \hline
        $\vec a + \vec f_2 = (\frac 5 3, \frac 7 3, 0)$ & $(8, \frac{27}{4}, 0)$\\
        $\vec b + \vec f_2 = (\frac 8 3, \frac 1 3, 1)$ & $(7, \frac{19}{4}, 1)$
    \end{tabular}
    \end{center}

    Notice from the table that we have $\vec{a} + \vec f_1$ and $\vec{b} + \vec f_1$ are both in $R^{+}$ whereas $\vec{a}+ \vec f_2$ and $\vec{b} + \vec f_2$ are both not in $R^{+}$, which is consistent with \Cref{lem:equivsameblock}.
\end{example}

What the above lemma suggests is that we should group up the superstable/critical preimages having the same floor (which we call a \newword{bucket}), and try to map the superstable configurations of $M$ to critical configurations of $M$ where the bucket structure is the same.


\begin{theorem}
    Let $(L,M)$ be any chip-firing pair. The map $\dual : \sst(L,M) \rightarrow \crit(L,M)$ given by $\ss \mapsto \cm-\ivo(\floor{\ss}) + \frp{\ss}$ is a bijection.
\end{theorem}

\begin{example}
        Consider the signed graph from \Cref{ex:usualbad}.
        Start with the superstable configuration $(5,4,0) \in S^{+}$ of the chip-firing pair. Then the corresponding superstable preimage is $\vec{s} = (\frac{4}{3}, \frac{7}{6},0)$.

        We can check that $\{LM^{-1}2\floor{\vec s}\} = (0, \frac{1}{2}, 0) \neq (0,0,0) = \{LM^{-1}\cmax\}$, so $\ivo (\floor{\vec s}) = \sstab(\cmax - \floor{\vec s})$. For this graph, we have $\cmax = (2,1,2)$, so $\ivo (\floor{\vec s}) = (0,0,1)$ as 
        {\parfillskip=0pt \par} \noindent
        $M^{-1}(\cmax - \floor{\vec s} - (0,0,1)) \in \ZZ$.

        Then \[\dual(\vec s) = \cmax - \ivo(\floor{\vec s}) + \{\vec s\} = (2,1,2) - (0,0,1) + \paren{\frac{1}{3}, \frac{1}{6}, 0} = \paren{\frac{7}{3}, \frac{7}{6}, 1}\]
        gives a critical preimage in $R^+$. Its image in $S^{+}$ is $(8,6,1)$, so we have mapped the superstable configuration $(5,4,0)$ to the critical configuration $(8,6,1)$.

        It can be verified that $\dual$ recovers all of the critical preimages (and hence the critical configurations) in the table from \Cref{ex:usualbad} starting from all superstable preimages.
    \end{example}

\begin{remark}
    The duality $\chi$ recovers the classical duality for chip-firing on graphs when $L$ is the (reduced) Laplacian of a signed graph and $M$ is the (reduced) Laplacian of the underlying unsigned graph. It also recovers the duality for $M$-matrices when $L = M$.
\end{remark}
    % **Example (usual graphs, toy graph)
    % Example of a signed graph case, using our map


\begin{example}
    Take the $M$-matrix coming from the unsigned graph in \Cref{ex:unsignedkyle}. Now let us study the $(M,M)$ chip-firing pair. We have that $(0,0,1)$ and $(1,1,0)$ are superstable configurations of $M$. Since the preimages of all superstable and critical configurations are obviously integral ($MM^{-1}$ is the identity matrix), they are fixed points in our involution $\ivo$.

    Then
    \[\dual(0,0,1) = \cm-\ivo(\floor{(0,0,1)}) + \frp{(0,0,1)} = (2,1,2) - (0,0,1) + (0,0,0) = (2,1,1)\]
    \[\dual(1,1,0) = \cm-\ivo(\floor{(1,1,0)}) + \frp{(1,1,0)} = (2,1,2) - (1,1,0) + (0,0,0) = (1,0,2)\]
    This aligns with the usual duality between superstable and critical configurations on graphs.
\end{example}
    % just start w superstables in R+ (pick 2), show where the two superstables go using this map, then show the whole list of superstable and criticals and say that it can be verified that the map works for all of them (lol)
    
  %  In any all-positive graph, every superstable in $R^+$ is integral, so every superstable is a fixed point under $\ivo$, so this recovers the usual duality.

\subsection{Counting the number of fixed points of \texorpdfstring{$\ivo$}{mu}.}

Let $F^M_0$ stand for the set of superstable preimages of an $(L,M)$ chip-firing pair such that the all entires are integers. Then this turns out to be a subgroup of the critical group of $M$. Then from this we can compute the number of fixed points of $\ivo$:

\begin{theorem}\label{numsols}
    The number of fixed points of the involution map $\ivo$ 
    is equal to either $0$ or $|F^M_0|d$, where $d$ is the number of elements of $\mathcal{K}(M) / F^M_0$ with order at most 2.
\end{theorem}

\begin{example}
Let's revisit the running example of a $(L,M)$-pair coming from a signed graph.
    \begin{center}
        \begin{tabular}{c | c | c}
        $\sstab(M)$ & Its image (under $LM^{-1}$) & Is it a fixed point of $\ivo$? \\
        \hline
        $(0, 0, 0)$ & $(0, 0, 0)$ & Yes \\
        $(0, 0, 1)$ & $(1, \frac 3 4, 1)$ & No \\
        $(0, 0, 2)$ & $(2, \frac 3 2, 2)$ & Yes \\
        $(0, 1, 0)$ & $(2, 2, 0)$ & Yes \\
        $(0, 1, 1)$ & $(3, \frac{11}{4}, 1)$ & No \\
        $(1, 0, 0)$ & $(2, \frac 5 4, 1)$ & No \\
        $(1, 1, 0)$ & $(4, \frac{13}{4}, 1)$ & No \\
        $(2, 0, 0)$ & $(4, \frac 5 2, 2)$ & Yes \\
        \hline
        $\cm = (2, 1, 2)$ & $(8, 6, 4)$
        \end{tabular}
    \end{center}

    %From the above table, we see that $\cm \in F^M_0$. Therefore, any of the $4$ superstable preimages $\ss$ that lie in an order-1 or order-2 S-fracket will be a fixed point, since these configurations satisfy $\cm - 2\ss \in F^M_0$.

    We have $\mathcal{K}(M) = \ZZ_8$ and $|F^M_0| = 2$. Therefore, $\mathcal{K}(M) / F^M_0 \cong \ZZ_4$, which has 2 elements of order at most 2. Using $\Cref{numsols}$ with $d = 2$ and $|F^M_0| = 2$, we get $4$ fixed points in $\ivo$.
\end{example}

%The remainder of this subsection will be on other more specific observations and corollaries of \Cref{numsols} and \Cref{cor:cyclic_fracket_size}.

%\begin{proposition}\label{oddorder}
 %   If $\cm$ has odd order in $\mathcal{K}(M) / F^M_0$, then the number of fixed points of $\ivo$ is nonzero.
%\end{proposition}

%If $\mathcal{K}(M) / F^M_0$ is cyclic and $\cm$ has even order, we have an exact criterion to check whether the number of fixed points is zero or not.

%\begin{corollary}
 %   Suppose $\mathcal{K}(M) / F^M_0$ is cyclic and $\cm$ has even order in $\mathcal{K}(M) / F^M_0$. Then, the number of fixed points of $\ivo$ is nonzero if and only if $\frac{|\mathcal{K}(M) / F^M_0|}{ord(\cm)}$ is even.
%\end{corollary}


\acknowledgements{This research was conducted as part of an NSF funded REU at Texas State University during Summer 2024. We are grateful to these organizations for their financial support and for providing a fruitful work environment.}

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\printbibliography



\end{document}
