%% 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]{FPSAC2019}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{102}
\addbibresource{102.bib}

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

\usepackage{enumerate, amsfonts, mathtools, thmtools, wasysym, graphics,
xcolor, frcursive,xparse,comment,bbm}
\usepackage{graphicx, xspace, ifthen, rotating, pict2e}
\usepackage{multirow,multicol,mathtools}
\usepackage{tikz}
\usetikzlibrary{matrix, positioning}
\usepackage{ytableau}
\usepackage[enableskew]{youngtab}
\usepackage[cmtip,all]{xy}
\usepackage{young}
\usepackage{subcaption}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}

\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{question}[theorem]{Question}

\numberwithin{equation}{section}

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\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}}
\newcommand{\defn}[1]{{\color{darkred}\emph{#1}}} % emphasis of a definition

\newcommand{\wt}{\operatorname{wt}}
\newcommand{\et}{\overline{e}}
\newcommand{\ft}{\overline{f}}
\newcommand{\st}{\overline{s}}
\newcommand{\epst}{\overline{\varepsilon}}
\newcommand{\phit}{\overline{\varphi}}
\newcommand{\sage}{{\sc SageMath }}

\usepackage{lipsum}

%% define your title in the usual way
%\title[Characterization of queer supercrystals]{Characterization of queer supercrystals}
\title{Queer supercrystals in \sage}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Wencin Poh \and
Anne Schilling
\thanks{Partially supported by NSF grants DMS--1760329 and DMS--1764153.
\href{mailto:anne@math.ucdavis.edu}{anne@math.ucdavis.edu}}}


%@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{Department of Mathematics, University of California, Davis, CA 95616, U.S.A}

%% put the date of submission here
\received{\today}

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%\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{We describe the implementation of queer supercrystals. Our code is documented with test suites and has been
integrated into {\sc SageMath}.
Through computer explorations using this implementation, we provide a 
counterexample to Assaf's and Oguz' conjecture that their local queer axioms uniquely characterize the queer supercrystal. 
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{Rien}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Crystal graphs, queer Lie superalgebras, Stembridge axioms, \sage}

%% 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, ie. [xx, yy] in the text

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%\usepackage[backend=bibtex]{biblatex}
%\addbibresource{fpsac-demo.bib}
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\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: \sage and queer supercrystals}
\label{section.introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\sage~\cite{combinat,sage} is a free open-source mathematical software licensed under GPL. It is based on Python, 
supports object-oriented programming, and has interfaces to many open source packages such as GAP, matplotlib, Numpy, 
R, and SciPy. Since its first release in 2005, \sage has an active pool of developers worldwide and supports computations 
in diverse mathematical areas such as algebra, combinatorics, number theory, numerical analysis, graph theory, and statistics. 

Lie superalgebras~\cite{Kac.1977} arose in physics in theories that unify bosons and fermions. They are essential in 
modern string theories~\cite{GGRS.1983} and appear in other areas of mathematics, such as the projective representations 
of the symmetric group. The crystal basis theory has been developed for various quantum 
superalgebras~\cite{BKK.2000,Grantcharov.2010,Grantcharov.2014, Grantcharov.2015,Grantcharov.2017,
Kwon.2015, Kwon.2016}. In this paper, we are interested in the queer superalgebra $\mathfrak{q}(n)$
(see for example~\cite{ChengWang.2012}). A theory of highest weight crystals for the queer superalgebra $\mathfrak{q}(n)$
was recently developed by Grantcharov et al.~\cite{Grantcharov.2010,Grantcharov.2014, Grantcharov.2015}. They provide 
an explicit combinatorial realization of the highest weight crystal bases in terms of semistandard decomposition tableaux
and show how these crystals can be derived from a tensor product rule and the vector representation. They also 
use the tensor product rule to derive a Littlewood--Richardson rule. 
Choi and Kwon~\cite{ChoiKwon.2017} provide a new characterization of Littlewood--Richardson--Stembridge tableaux 
for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. 
Independently, Hiroshima~\cite{Hiroshima.2018} and Assaf and Oguz~\cite{AssafOguz.2018,AssafOguz.2018a}
defined a queer crystal structure on semistandard shifted tableaux, extending the type $A$ crystal structure
of~\cite{HawkesParamonovSchilling.2017} on these tableaux. 

The authors, in joint work with Gillespie and Hawkes~\cite{GHPS.2018}, provided a characterization of the queer 
supercrystals in analogy to Stembridge's characterization of crystals associated to classical 
simply-laced root systems~\cite{Stembridge.2003}.
In the initial stages of this work, there was no implementation of queer supercrystals. As there is an extensive 
infrastructure to perform computations with other crystals within \sage\!, it was a natural choice to
use the existing infrastructure in \sage to code the queer supercrystals. The implementation of queer
supercrystals was achieved in the \sage ticket~\cite{PohSchilling-ticket.2018}.

Assaf and Oguz~\cite{AssafOguz.2018,AssafOguz.2018a} conjecture a local characterization of queer crystals in the 
spirit of Stembridge~\cite{Stembridge.2003}, which involves local relations between the odd crystal operator
$f_{-1}$ with the type $A_{n-1}$ crystal operators $f_i$ for $1\leqslant i <n$. However, through computer explorations
using the code in~\cite{PohSchilling-ticket.2018},  we provide a counterexample to~\cite[Conjecture~4.16]{AssafOguz.2018a}, 
which conjectures that these local axioms uniquely characterize the queer supercrystals.

The remainder of this paper is organized as follows. In \cref{section.definition queer supercrystal},
we give the definition of queer supercrystals. In \cref{section.functionalities+application}, we describe the main features 
and methods that the user can access from our implementation of queer supercrystals. As an application, we also outline 
how a counterexample to Assaf's and Oguz's conjecture is discovered using this implementation in the same section. 
Finally, in \cref{section: sustainability}, we elaborate how our implementation of queer supercrystals was integrated 
and tested by the \sage community as well as provide references for documentation of our implementation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Definition of queer supercrystals}
\label{section.definition queer supercrystal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

An \defn{(abstract) crystal} of type $A_n$ is a nonempty set $B$ together with the maps
\begin{equation}
	e_i, f_i \colon B \to B \sqcup \{0\} \qquad \text{for $i \in I$ and} \qquad
        \wt \colon B \to \Lambda,
\end{equation}
where $\Lambda = \mathbb{Z}^{n+1}_{\geqslant 0}$ is the weight lattice of the root of type $A_n$ and $I=\{1,2,\ldots,n\}$
is the index set, subject to several conditions. Denote by $\alpha_i =\epsilon_i -\epsilon_{i+1}$ for $i \in I$ the simple 
roots of type $A_n$, where $\epsilon_i$ is the $i$-th standard basis vector of $\mathbb{Z}^{n+1}$. Then we require:
\begin{itemize}
\item[\textbf{A1.}]  For $b,b'\in B$, we have $f_ib=b'$ if and only if $b=e_ib'$. In this case $\wt(b') =\wt(b)-\alpha_i$.
\end{itemize}
For $b\in B$, we also define $\varphi_i(b) = \max\{k \in \mathbb{Z}_{\geqslant 0} \mid f_i^k(b) \neq 0\}$ and
$\varepsilon_i(b) = \max\{k \in \mathbb{Z}_{\geqslant 0} \mid e_i^k(b) \neq 0\}$.
For further details, see for example~\cite[Definition 2.13]{BumpSchilling.2017}.

There is an action of the symmetric group $S_n$ on a type $A_n$ crystal $B$ given by the operators 
\begin{equation}
\label{equation.si}
	s_i(b) = \begin{cases}
	f_i^k(b) & \text{if $k\geqslant 0$,}\\
	e_i^{-k}(b) & \text{if $k<0$,}
	\end{cases}
\end{equation}
for $b\in B$, where $k=\varphi_i(b)-\varepsilon_i(b)$.
An element $b\in B$ is called \defn{highest weight} if $e_i(b)=0$ for all $i \in I$. Similarly, $b$ is called
\defn{lowest weight} if $f_i(b)=0$ for all $i \in I$. 
For a subset $J \subseteq I$, we say that $b$ is $J$-highest weight
if $e_i(b)=0$ for all $i \in J$ and similarly $b$ is $J$-lowest weight if $f_i(b)=0$ for all $i \in J$.
We are now ready to define an abstract queer crystal.

\begin{definition} \cite[Definition 1.9]{Grantcharov.2014}
\label{definition.abstract queer}
An \defn{abstract $\mathfrak{q}(n+1)$-crystal} is a type $A_n$ crystal $B$ together with the maps
$e_{-1},f_{-1} \colon B \to B \sqcup \{0\}$ satisfying the following conditions:
\begin{enumerate}
\item[\textbf{Q1.}] $\wt(B) \subset \Lambda$;
\item[\textbf{Q2.}] $\wt(e_{-1} b) = \wt(b) + \alpha_1$ and $\wt(f_{-1} b) = \wt(b) - \alpha_1$;
\item[\textbf{Q3.}] for all $b,b'\in B$, $f_{-1}b = b'$ if and only if $b = e_{-1} b'$;
\item[\textbf{Q4.}] if $3\leqslant i \leqslant n$, we have
\begin{enumerate}
\item the crystal operators $e_{-1}$ and $f_{-1}$ commute with $e_i$ and $f_i$;
\item if $e_{-1}b \in B$, then $\varepsilon_i(e_{-1}b) = \varepsilon_i(b)$ and $\varphi_i(e_{-1}b) = 
\varphi_i(b)$.
\end{enumerate}
\end{enumerate}
\end{definition}

Given two $\mathfrak{q}(n+1)$-crystals $B_1$ and $B_2$, Grantcharov et al.~\cite[Theorem 1.8]{Grantcharov.2014}
provide a crystal on the tensor product $B_1 \otimes B_2$, which we state here in reverse convention.
It consists of the type $A_n$ tensor product rule (see for example~\cite[Section 2.3]{BumpSchilling.2017})
and the \defn{tensor product rule} for $b_1 \otimes b_2 \in B_1 \otimes B_2$
\begin{equation}
\label{equation.tensor product}
	e_{-1} (b_1 \otimes b_2) = \begin{cases}
	b_1 \otimes e_{-1} b_2 & \text{if $\wt(b_1)_1 = \wt(b_1)_2 =0$,}\\
	e_{-1} b_1 \otimes b_2 & \text{otherwise,}
	\end{cases}
\end{equation}
and similarly for $f_{-1}$.
The crystals of interest are the \defn{crystals of words} $\mathcal{B}^{\otimes \ell}$, where $\mathcal{B}$ 
is the $\mathfrak{q}(n+1)$-queer crystal of letters depicted in \cref{figure.queer letter}.

\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance = 1.5cm, auto]
  \node[draw] (1)  {$1$};
  \node[draw] (2) [right of=1] {$2$};
  \node[draw] (3) [right of=2]{$3$};
  \node     (A) [right of=3]{$\ldots$};
  \node[draw] (n+1) [right of=A]{$n+1$};
  \draw[->,transform canvas={yshift=1mm},blue,thick] (1) edgenode[above,blue]{$1$} (2);
  \draw[->,dashed,transform canvas={yshift=-1mm},blue,thick] (1) edgenode[below,blue]{$-1$} (2);
  \draw[->,darkred,thick] (2) edgenode[above,darkred]{$2$} (3);
  \draw[->,darkgreen,thick] (3) edgenode[above,darkgreen]{$3$} (A);
  \draw[->] (A) edgenode[above]{$n$} (n+1);
\end{tikzpicture}
\end{center}
\caption{$\mathfrak{q}(n+1)$-queer crystal of letters $\mathcal{B}$ \label{figure.queer letter}}
\end{figure}

In addition to the queer crystal operators $f_{-1},f_1,\ldots,f_n$ and $e_{-1},e_1,\ldots,e_n$, we define the crystal operators
for $1<i\leqslant n$
\begin{equation}
\label{equation.-i}
	f_{-i} := s_{w_i^{-1}} f_{-1} s_{w_i} \qquad \text{and} \qquad e_{-i} := s_{w_i^{-1}} e_{-1} s_{w_i},
\end{equation}
where $s_{w_i}=  s_2 \cdots s_i s_1 \cdots s_{i-1}$ and $s_i$ is the reflection along the $i$-string in the crystal defined
in~\eqref{equation.si}. 

By~\cite[Theorem 1.14]{Grantcharov.2014}, with all operators $e_i,f_i$ for $i\in \{\pm 1,\pm 2,\ldots,\pm n\}$ each 
connected component of $\mathcal{B}^{\otimes \ell}$ has a unique highest weight vector.

The operators $f_i$ for $i \in I_0=\{1,2,\ldots,n\}$ have an easy combinatorial description on $b\in \mathcal{B}^{\otimes \ell}$
given by the \defn{signature rule}, which can be directly derived from the tensor product rule (see for
example~\cite[Section 2.4]{BumpSchilling.2017}). One can consider $b$ as a word in the alphabet $\{1,2,\ldots,n+1\}$. 
Consider the subword of $b$ consisting only of the letters $i$ and $i+1$. Pair (or bracket) any consecutive letters $i+1 , i$ 
in this order, remove this pair, and repeat. Then $f_i$ changes the rightmost unpaired $i$ to $i+1$; if there is no such letter 
$f_i(b)=0$. Similarly, $e_i$ changes the leftmost unpaired $i+1$ to $i$; if there is no such letter $e_i(b)=0$.

From~\eqref{equation.tensor product}, one may also derive a simple combinatorial rule for $f_{-1}$ and $e_{-1}$.
Consider the subword $v$ of $b\in \mathcal{B}^{\otimes \ell}$ consisting of the letters $1$ and $2$.
The crystal operator $f_{-1}$ on $b$ is defined if the leftmost letter of $v$ is a $1$, in which case it turns it into a $2$.
Otherwise $f_{-1}(b)=0$. Similarly, $e_{-1}$ on $b$ is defined if the leftmost letter of $v$ is a $2$, in which case it turns it 
into a $1$. Otherwise $e_{-1}(b)=0$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Main Functionalities and Applications}
\label{section.functionalities+application}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Description of Features}
The user can construct the crystal of words $\mathcal{B}^{\otimes \ell}$ via tensors of the standard queer supercrystal 
$\mathcal{B}$ of type $\mathfrak{q}(n+1)$ in \sage~\cite{PohSchilling-ticket.2018,}. As with the case of classical crystals, for 
every $b\in\mathcal{B}^{\otimes \ell}$ the user can query $\wt(b)$, $e_i(b)$, $f_i(b)$, $\varepsilon_i(b)$ and $\varphi_i(b)$ 
for each $i\in \{\pm 1,\pm 2,\ldots, \pm n\}$.

Additionally, the user may specify a subcrystal either by restricting to a certain subset of indices in the index set or by 
specifying a set of generators. This is useful when one wants to focus on say, $J$-highest weight elements, for 
$J\subset \{\pm 1,\pm 2,\ldots,\pm n\}$ for computer explorations. Moreover, if dot2tex is installed, the user may also 
view the crystal graph and import the LaTeX commands for rendering the crystal graph.

\begin{example} \label{example.highest_weights}
One can retrieve a list of $\{1,2,-1,-2\}$-highest weight elements in $\mathcal{B}^{\otimes 6}$, where $\mathcal{B}$ is a standard crystal of type $\mathfrak{q}(3)$, using the following commands:

\begin{verbatim}
sage: Q = crystals.Letters(['Q',3]); Q
The queer crystal of letters for q(3)
sage: T = tensor([Q]*6)
sage: L = [t for t in T if all(t.epsilon(i)==0 for i in Q.index_set())]
sage: L
[[1, 1, 1, 1, 1, 1],
 [1, 1, 1, 1, 2, 1],
 [1, 1, 1, 2, 1, 1],
 [1, 1, 2, 1, 1, 1],
 [1, 1, 2, 1, 2, 1],
 [1, 1, 2, 2, 1, 1],
 [1, 2, 1, 1, 1, 1],
 [1, 2, 1, 1, 2, 1],
 [1, 2, 1, 2, 1, 1],
 [1, 2, 1, 3, 2, 1],
 [1, 2, 2, 1, 1, 1],
 [1, 2, 3, 1, 2, 1]]
\end{verbatim}
\end{example}

\begin{example}
\label{example.full crystal}
To view a crystal or to obtain the latex code of the crystal graph, one may type:
\begin{verbatim}
sage: Q = crystals.Letters(['Q',3])
sage: T = tensor([Q]*2)
sage: view(T)
sage: latex(T)
\end{verbatim}
The output is given in \cref{figure.full crystal}. Note that the crystal graph contains the additional arrows
$f_{-i'} := s_{w_0} e_{-(n+1-i)} s_{w_0}$ for $i \in I_0$, where $w_0$ is the long word in the symmetric group $S_{n+1}$. 
The crystal operators $f_{-i'}$ are labeled by $\overline{i+n}$, whereas $f_{-i}$ are labeled by $\overline{i}$ in \sage\!.
\end{example}

\begin{figure}
\begin{center}
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%%
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  \draw [dashed,green,->] (node_0) ..controls (139.56bp,278.31bp) and (132.31bp,261.41bp)  .. (124.2bp,248.0bp) .. controls (122.22bp,244.73bp) and (119.9bp,241.37bp)  .. (node_1);
  \draw (144.2bp,259.0bp) node {$\overline{3}$};
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  \draw (13.198bp,115.0bp) node {$1$};
  \draw [dashed,blue,->] (node_2) ..controls (39.726bp,134.04bp) and (37.012bp,116.0bp)  .. (45.198bp,104.0bp) .. controls (53.246bp,92.201bp) and (68.002bp,85.958bp)  .. (node_5);
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  \draw (92.198bp,115.0bp) node {$\overline{3}$};
%
\end{tikzpicture}
\end{center}
\caption{The crystal $\mathcal{B}^{\otimes 2}$ of type $\mathfrak{q}(3)$ of \cref{example.full crystal}.
\label{figure.full crystal}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Application: Discovery of a Counterexample}
\label{subsection.application}
In~\cite[Definition 4.11]{AssafOguz.2018a}, Assaf and Oguz give a definition of regular queer crystals.
In essence, their axioms are rephrased in the following definition, where $\tilde{I}:=I_0 \cup \{-1\}$.

\begin{definition}[\defn{Local queer axioms}]
\label{definition.local queer axioms}
Let $\mathcal{C}$ be a graph with labeled directed edges given by $f_i$ for $i \in I_0$ and $f_{-1}$.
If $b'=f_jb$ for $j \in \tilde{I}$, define $e_j$ by $b=e_j b'$.
\begin{enumerate}
\item[{\bf LQ1.}] The subgraph with all vertices but only edges labeled by $i\in I_0$ is a type $A_n$ Stembridge
crystal~\cite{Stembridge.2003}.
\item[{\bf LQ2.}] $\varphi_{-1}(b),\varepsilon_{-1}(b) \in \{0,1\}$ for all $b\in \mathcal{C}$.
\item[{\bf LQ3.}] $\varphi_{-1}(b)+\varepsilon_{-1}(b)>0$ if $\wt(b)_1 + \wt(b)_2>0$.
\item[{\bf LQ4.}] Assume $\varphi_{-1}(b)=1$ for $b\in \mathcal{C}$.
\begin{enumerate}
\item If $\varphi_1(b)>2$, we have
$f_1 f_{-1}(b) = f_{-1} f_1(b)$, $\varphi_1(b) = \varphi_1(f_{-1}(b))+2$, and
$\varepsilon_1(b) = \varepsilon_1(f_{-1}(b))$.
\item If $\varphi_1(b)=1$, we have $f_1(b) = f_{-1}(b)$.
\end{enumerate}
\item[{\bf LQ5.}] Assume $\varphi_{-1}(b)=1$ for $b\in \mathcal{C}$.
\begin{enumerate}
\item If $\varphi_2(b)>0$, we have $f_2 f_{-1}(b) = f_{-1} f_2(b)$, $\varphi_2(b) = \varphi_2(f_{-1}(b))-1$, and
$\varepsilon_2(b) = \varepsilon_2(f_{-1}(b))$.
\item If $\varphi_2(b)=0$, we have
\begin{equation*}
\begin{aligned}
	\varphi_2(b) &= \varphi_2(f_{-1}(b))-1=0, & \text{or} \quad \varphi_2(b) &= \varphi_2(f_{-1}(b))=0,\\
	\varepsilon_2(b) &= \varepsilon_2(f_{-1}(b)), & \varepsilon_2(b) &= \varepsilon_2(f_{-1}(b))+1.
\end{aligned}
\end{equation*}
\end{enumerate}
\item[{\bf LQ6.}] Assume that $\varphi_{-1}(b)=1$ and $\varphi_i(b)>0$ with $i\geqslant 3$ for $b\in \mathcal{C}$. Then
$f_i f_{-1}(b) = f_{-1} f_i(b)$, $\varphi_i(b) = \varphi_i(f_{-1}(b))$, and $\varepsilon_i(b) = \varepsilon_i(f_{-1}(b))$.
\end{enumerate}
\end{definition}

Axioms {\bf LQ4} and {\bf LQ5} are illustrated in \cref{figure.local queer}.

\begin{figure}
\centering
\scalebox{0.7}{
\begin{tikzpicture}[node distance = 1.5cm, auto]
  \tikzstyle{place}=[circle,fill=black,scale=1]
  \tikzstyle{minusone}=[->,dashed,blue,thick]
  \tikzstyle{one}=[->,blue,thick]
  \tikzstyle{two}=[->,darkred,thick]
  \node [place] (A1) {};  
  \node [place] (B1) [right= 1cm of A1] {};
  \draw [minusone] (A1) edgenode{$-1$} (B1);
  \node [place] (A2) [below= 1cm of A1] {};  
  \node [place] (B2) [right= 1cm of A2] {};
  \draw [minusone] (A2) edgenode{$-1$} (B2);
  \draw [one] (A1) edgenode[left]{$1$} (A2);
  \draw [one] (B1) edgenode[right]{$1$} (B2);
  \node [place] (A3) [below= 1cm of A2] {};  
  \node [place] (B3) [right= 1cm of A3] {};
  \draw [minusone] (A3) edgenode{$-1$} (B3);
  \draw [one] (A2) edgenode[left]{$1$} (A3);
  \draw [one] (B2) edgenode[right]{$1$} (B3);
  \node [place] (A4) [below= 1cm of A3] {};  
  \node [place] (B4) [right= 1cm of A4] {};
  \draw [minusone] (A4) edgenode{$-1$} (B4);
  
  \draw [loosely dotted,blue,thick] (A3) -- (A4);
  \draw [loosely dotted,blue,thick] (B3) -- (B4);
  
  \node [place] (A5) [below= 1cm of A4] {};  
  \node [place] (B5) [right= 1cm of A5] {};
  \draw [minusone] (A5) edgenode{$-1$} (B5);
  \draw [one] (A4) edgenode[left]{$1$} (A5);
  \draw [one] (B4) edgenode[right]{$1$} (B5);
  
  \node [place] (A6) [below= 1cm of A5] {};
  \draw [one] (A5) edgenode[left]{$1$} (A6);
  \node [place] (A7) [below= 1cm of A6] {};
  \draw [minusone,transform canvas={xshift=-1mm}] (A6) edgenode[left]{$-1$} (A7);
  \draw [one,transform canvas={xshift=1mm}] (A6) edgenode[right]{$1$} (A7);


  \node [place] (C1) [right= 2cm of B1] {};  
  \node [place] (D1) [right= 1cm of C1] {};
  \draw [minusone] (C1) edgenode{$-1$} (D1);
  \node [place] (C2) [below= 1cm of C1] {};  
  \node [place] (D2) [right= 1cm of C2] {};
  \draw [minusone] (C2) edgenode{$-1$} (D2);
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  \draw [two] (D1) edgenode[right]{$2$} (D2);
  \node [place] (C3) [below= 1cm of C2] {};  
  \node [place] (D3) [right= 1cm of C3] {};
  \draw [minusone] (C3) edgenode{$-1$} (D3);
  \draw [two] (C2) edgenode[left]{$2$} (C3);
  \draw [two] (D2) edgenode[right]{$2$} (D3);
  \node [place] (C4) [below= 1cm of C3] {};  
  \node [place] (D4) [right= 1cm of C4] {};
  \draw [minusone] (C4) edgenode{$-1$} (D4);
  
  \draw [loosely dotted,red,thick] (C3) -- (C4);
  \draw [loosely dotted,red,thick] (D3) -- (D4);
  
  \node [place] (C5) [below= 1cm of C4] {};  
  \node [place] (D5) [right= 1cm of C5] {};
  \draw [minusone] (C5) edgenode{$-1$} (D5);
  \draw [two] (C4) edgenode[left]{$2$} (C5);
  \draw [two] (D4) edgenode[right]{$2$} (D5);
  
  \node [place] (D6) [below= 1cm of D5] {};
  \draw [two] (D5) edgenode[left]{$2$} (D6);
  
  \node [place] (E2) [right= 1cm of D2] {};  
  \node [place] (F2) [right= 1cm of E2] {};
  \draw [minusone] (E2) edgenode{$-1$} (F2);
  \node [place] (E3) [below= 1cm of E2] {};  
  \node [place] (F3) [right= 1cm of E3] {};
  \draw [minusone] (E3) edgenode{$-1$} (F3);
  \draw [two] (E2) edgenode[left]{$2$} (E3);
  \draw [two] (F2) edgenode[right]{$2$} (F3);
  \node [place] (E4) [below= 1cm of E3] {};  
  \node [place] (F4) [right= 1cm of E4] {};
  \draw [minusone] (E4) edgenode{$-1$} (F4);
  
  \draw [loosely dotted,red,thick] (E3) -- (E4);
  \draw [loosely dotted,red,thick] (F3) -- (F4);
  
  \node [place] (E5) [below= 1cm of E4] {};  
  \node [place] (F5) [right= 1cm of E5] {};
  \draw [minusone] (E5) edgenode{$-1$} (F5);
  \draw [two] (E4) edgenode[left]{$2$} (E5);
  \draw [two] (F4) edgenode[right]{$2$} (F5);
  
  \node [place] (F6) [below= 1cm of F5] {};
  \draw [two] (F5) edgenode[left]{$2$} (F6);
  
  \draw [minusone] (D6) edgenode{$-1$} (F6);
\end{tikzpicture}
}
\caption{Illustration of axioms \textbf{LQ4} (left) and \textbf{LQ5} (right). The $(-1)$-arrow at the bottom of the right
figure might or might not be there.}
\label{figure.local queer}
\end{figure}

\begin{proposition}[\cite{AssafOguz.2018a}]
\label{proposition.local queer axioms}
The queer crystal of words $\mathcal{B}^{\otimes \ell}$ satisfies the axioms in \cref{definition.local queer axioms}.
\end{proposition}

In~\cite[Conjecture 4.16]{AssafOguz.2018a}, Assaf and Oguz conjecture that every regular queer crystal is a 
normal queer crystal. In other words, every connected graph satisfying the local queer axioms
of \cref{definition.local queer axioms} is isomorphic to a connected component in some
$\mathcal{B}^{\otimes \ell}$. Using the implementation of crystals of words in \sage\cite{PohSchilling-ticket.2018}, 
we provide a counterexample to this claim~\cite{GHPS.2018}.
The $\{1,2,-1,-2\}$-highest weight elements within $\mathcal{B}^{\otimes 6}$ are listed in 
\cref{example.highest_weights}. Now one may ask whether there is any choice in the $-1$ arrows after
retaining all edges labeled $-1$ that are asserted by {\bf LQ2}, but cannot be deduced from the top by axioms 
{\bf LQ4} to {\bf LQ6}. The answer is that there is choice, which led to the counterexample depicted in 
\cref{figure.counterexample}.

In \cref{figure.counterexample}, the $I_0$-components of the $\mathfrak{q}(3)$-crystal of highest weight $(4,2,0)$ are 
shown. Some of the $f_{-1}$-arrows are drawn in green. The remaining arrows can be filled in using the axioms
of \cref{figure.local queer} in a consistent manner. If the dashed green arrow from $331131$ to $332131$ and the 
dashed green arrow from $331132$ to $332132$ are replaced by the dashed purple arrow from $331131$ to
$331231$ and the dashed purple arrow from $331132$ to $332231$, respectively, all axioms of 
\cref{definition.local queer axioms} are still satisfied with the remaining $f_{-1}$-arrows filled in. 
However, the $I_0$-component with highest weight element $132121$ has become disconnected and hence the two 
crystals are not isomorphic.

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%
\end{tikzpicture}}}
\caption{Counterexample to the unique characterization of the local queer axioms of 
\cref{definition.local queer axioms}.
\label{figure.counterexample}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ease of use and sustainability}
\label{section: sustainability}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The standard queer supercrystals and their tensor product rule have been implemented and integrated into \sage 
within the classes Crystal of Letters and Tensor Product of Crystal Elements, respectively. 
\sage is a free open-source mathematical software licensed under GPL and is available for download at 
\url{http://www.sagemath.org/} or within the cloud service CoCalc \url{https://cocalc.com/}.

The trac ticket for this implementation was created during SageDays@ICERM in July 2018 
(see~\cite{PohSchilling-ticket.2018}), received a positive review by a reviewer and was integrated into \sage\!.
This implementation is available in the current stable release of \sage (v8.6)~\cite{sage}.

Documentation of this implementation is available in the \sage online documentation for combinatorics as well as locally 
by appending a ? to a particular method or attribute as in the example shown below:
\begin{verbatim}
sage: Q = crystals.Letters(['Q',3])
sage: t = Q(1)
sage: t.e?
\end{verbatim}
Furthermore, within the docstring of the implemented methods, the user can access examples of usage within \sage\!.
These examples also serve as a test suite and are run on a regular basis by \sage to ensure that the code continues
to work in future version of \sage\!.

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\printbibliography



\end{document}
