%% 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[final]{FPSAC2022}
%% 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}

 

%% define your title in the usual way
\title{Highest Weight Crystals for Schur $Q$-Functions}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[E.~Marberg and K.~H.~Tong]{Eric Marberg\thanks{\href{mailto:emarberg@ust.hk}{emarberg@ust.hk}. Supported by RGC grants ECS 26305218 and GRF 16306120.}\addressmark{1} \and Kam Hung Tong\thanks{\href{mailto:khtongad@connect.ust.hk}{khtongad@connect.ust.hk}}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology}

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

%% leave this blank until submitting a revised version
%\revised{}
\abstract{Work of Grantcharov \textit{et al.}\ develops a theory of abstract crystals for the queer Lie superalgebra $\mathfrak{q}_n$. Such $\mathfrak{q}_n$-crystals form a monoidal category in which the connected normal objects have unique highest weight elements and characters that are Schur $P$-polynomials. We introduce a modified form of this category, with an extra crystal operator and a different tensor product, whose connected normal objects again have unique highest weight elements but now possess characters that are Schur $Q$-polynomials. The crystals in this category have some interesting features not present for ordinary $\mathfrak{q}_n$-crystals. For example, there is an action of the hyperoctahedral group exchanging highest and lowest weight elements. There are natural examples of normal $\mathfrak{q}_n$-crystal structures on shifted tableaux and factorized reduced words. We describe extended forms of these structures that give examples in our new category.
}



%% 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{\lipsum[1]}

%% put your French abstract here, or comment this out if you don't have one


%% put your keywords here, or comment this out if you don't have them yet
\keywords{crystals, Schur $Q$-functions, queer Lie superalgebras, shifted tableaux, involution words}


%% 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

%% 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{schur-q-crystals-fpsac.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands




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}

\def\OEG{\textsf{OEG}}
\def\fks{{\fk s}}
\def\shword{\textsf{shword}}

\def\minfpf{1_\fpf}

\def\pair{\textsf{pair}}


\def\fO{f^\O}
\def\eO{e^\O}

\def\fSp{f^\Sp}
\def\eSp{e^\Sp}

%\def\break{\ \ {\arraycolsep=0.8pt\def\arraystretch{0.8} \barr{c} \\ \\ \\ \\ \\[-6pt] \isim \earr}\ \ }
%\newcommand{\fbreak}{\  {\tiny\arraycolsep=0.8pt\def\arraystretch{0.8} \barr{c}\\ \\ \\ \\ \\ \\ \\ \\[-3pt] \fsim \earr} \ }

\def\pipe{ \hs / \hs }


\def\ttimes{\textcircled{$\times$}}

\def\cS{\textsf{Perm}}
\def\cSe{\textsf{Even}}

\def\dbl{\textsf{dbl}}
\def\invert{\textsf{inv}}






\def\fkSO{\fkS^{\O}}
\def\fkSS{\fkS^{\Sp}}


\def\T{\mathsf{T}}

\def\colform{\mathsf{colform}}
\def\rowform{\mathsf{rowform}}
\def\rowcontent{\mathsf{rowcontent}}


\def\BB{\mathbb{B}}

\def\concat{\mathsf{concat}}
\def\cyc{\mathsf{cyc}}
\def\marked{\mathsf{marked}}
\def\unprime{\mathsf{unprime}}

\def\rf{\mathrm{RF}}
\def\brf{\mathrm{BRF}}

\def\RF{\rf_n}
\def\KRF{\rf^\K_n}
\def\ORF{\rf^\O_n}
\def\SpRF{\rf^\Sp_n}
\def\BRF{\brf_n}
\def\BORF{\brf^\O_n}
\def\BSpRF{\brf^\Sp_n}
\def\BKRF{\brf^\K_n}
\def\CS{\mathsf{CS}}






\def\athreediagram{
\begin{tikzpicture}
\node[label=below:$a$,inner sep=0mm] (a) at (-2,0) {
$\circ$
};
  \node[label=below:$x$,inner sep=0mm] (b) at (-1,0) {
$\circ$
};
  \node[label=below:$b$,inner sep=0mm] (c) at (0,0) {
$\circ$
};
\draw
(a)   --  (b)
(b) -- (c)
;
\end{tikzpicture}
}


\def\bthreediagram{
\begin{tikzpicture}
\node[label=below:$a$,inner sep=0mm] (a) at (-2,0) {
$\circ$
};
  \node[label=below:$x$,inner sep=0mm] (b) at (-1,0) {
$\circ$
};
  \node[label=below:$b$,inner sep=0mm] (c) at (0,0) {
$\circ$
};
\draw[double] (b) -- node [anchor=south] {4} (c);
\draw
(a)   --  (b) 
;
\end{tikzpicture}
}

\def\hthreediagram{
\begin{tikzpicture}
\node[label=below:$a$,inner sep=0mm] (a) at (-2,0) {
$\circ$
};
  \node[label=below:$x$,inner sep=0mm] (b) at (-1,0) {
$\circ$
};
  \node[label=below:$b$,inner sep=0mm] (c) at (0,0) {
$\circ$
};
\draw[double] (b) --  node [anchor=south] {5} (c);
\draw
(a)   --  (b) 
;
\end{tikzpicture}
}


\def\dfourdiagram{
\begin{tikzpicture}
\node[label=below:$a$,inner sep=0mm] (a) at (-2,0) {
$\circ$
};
  \node[label=below:$x$,inner sep=0mm] (b) at (-1,0) {
$\circ$
};
  \node[label=above:$c$,inner sep=0mm] (c) at (-1,0.8) {
$\circ$
};
  \node[label=below:$b$,inner sep=0mm] (d) at (0,0) {
$\circ$
};
\draw
(a)   --  (b) 
(b) -- (c)
(b) -- (d)
;
\end{tikzpicture}
}


\def\itwodiagram{
\begin{tikzpicture}
  \node[label=below:$a$,inner sep=0mm] (b) at (-1,0) {
$\circ$
};
  \node[label=below:$b$,inner sep=0mm] (c) at (0,0) {
$\circ$
};
\draw[double] (b) --  node [anchor=south] {$m$} (c);
\end{tikzpicture}
}


\def\aonediagram{
\begin{tikzpicture}
\tikzstyle{vertex}=[circle,draw=black]
  \node[label=below:$a$,inner sep=1mm] (b) at (-1,0) {$\circ$};
\end{tikzpicture}
}














\def\simA{\xleftrightarrow{\ \cR\ }}
\def\fsim{\xleftrightarrow{\ \cRfpf\ }}
%\def\isim{\xleftrightarrow{\ \iR\ }}
\def\wfpf{1_{\mathsf{fpf}}}
\def\Ifpf{I^{\mathsf{fpf}}}

\def\cRfpf{\cR_{\mathsf{fpf}}}
\def\cAfpf{\mathcal{A}_{\mathsf{fpf}}}
\def\ellfpf{\ell_{\mathsf{fpf}}}

\def\iR{\cR_{\mathsf{inv}}}
\def\cA{\mathcal{A}_{\mathsf{inv}}}
\def\ellhat{\ell_{\mathsf{inv}}}
\def\s{\mathbf{s}}
\def\hb{\hs |\hs}

\def\zweight{\weight_0}

\def\dbl{\mathsf{double}}
\def\invert{\mathsf{invert}}

\def\fkd{\mathfrak{d}}

\def\ShTab{\mathsf{ShTab}}
\def\unprimediag{\unprime_{{\text{\emph{diag}}}}}
\def\cycdiag{\cyc_{{\text{\emph{diag}}}}}
\def\primes{\mathsf{primes}}
\def\entries{\mathsf{entries}}
\def\dbump{\Delta^{\mathsf{bump}}}
\def\tpi{\tau}

\def\ck{\textsf{ock}}
\def\shk{\mathsf{shk}}
\def\op{\underline{\shk}}


\def\double{\mathsf{double}}
\def\halve{\mathsf{halve}}
\def\SW{\mathsf{SW}}

\def\simeqSW{\overset{\textsf{shK}}\simeq}
\def\simSW{\overset{\textsf{shK}}\sim}

\def\simA{\equiv}
%\def\isim{\mathop{\widehat \equiv}}
\def\cA{\mathcal{A}}
\def\Incr{\mathsf{Incr}}

\def\inverse{\mathsf{inverse}}
\def\cseq{\mathsf{cseq}}

\def\regionSW{\raisebox{-0.5pt}{\tikz{\filldraw (0.25,0.25) circle (1pt);  \draw (0.25,0.25) -- (.25,0) -- (0,.25) -- (0.25,0.25);}}}
\def\regionNE{\raisebox{-0.5pt}{\tikz{\filldraw (0,0) circle (1pt);  \draw (0,0) -- (.25,0) -- (0,.25) -- (0,0);}}}


\def\wpath{\mathsf{path}^\leq}
\def\spath{\mathsf{path}^<}
\def\rwpath{\mathsf{rpath}^\leq}
\def\rspath{\mathsf{rpath}^<}
\def\cwpath{\mathsf{cpath}^\leq}
\def\cspath{\mathsf{cpath}^<}

\def\diag{\text{\emph{diag}}}
\def\init{\text{\emph{init}}}
\def\entriesdiag{\mathsf{diag}}

\def\free{\text{\emph{free}}}

\def\W{W^{\mathsf{BC}}}
\def\isim{\mathbin{\widehat \equiv}}



 
%% For typesetting code listings                                                
\usepackage{listings}
\lstdefinelanguage{Sage}[]{Python}
{morekeywords={False,sage,True},sensitive=true}
\lstset{
  frame=single,
  showtabs=False,
  showspaces=False,
  showstringspaces=False,
  commentstyle={\ttfamily\color{dgreencolor}},
  keywordstyle={\ttfamily\color{dbluecolor}\bfseries},
  stringstyle={\ttfamily\color{dgraycolor}\bfseries},
  language=Sage,
  basicstyle={\footnotesize\ttfamily},
  aboveskip=0.75em,
  belowskip=0.75em,
  xleftmargin=.15in,
}
\definecolor{dblackcolor}{rgb}{0.0,0.0,0.0}
\definecolor{dbluecolor}{rgb}{0.01,0.02,0.7}
\definecolor{dgreencolor}{rgb}{0.2,0.4,0.0}
\definecolor{dgraycolor}{rgb}{0.30,0.3,0.30}
\newcommand{\dblue}{\color{dbluecolor}\bf}
\newcommand{\dred}{\color{dredcolor}\bf}
\newcommand{\dblack}{\color{dblackcolor}\bf}
%% End code listings

\def\content{\mathsf{content}}
\newcommand{\eric}[1]{\todo[size=\tiny,inline,color=green!30]{#1 \\ \hfill --- E.}}
\newcommand{\terry}[1]{\todo[size=\tiny,inline,color=blue!30]{#1 \\ \hfill --- T.}}


\def\decr{\mathsf{decrement}}
\def\incr{\mathsf{incremenent}}

\def\std{\mathsf{standardize}}
\def\destd{\mathsf{destandardize}}
\def\qq{\q^+}
\def\SShTab{\mathsf{ShTab}^+}
\def\Words{\mathsf{Words}}

\def\ORF{\mathsf{Incr}^+}
\def\unpaired{\mathsf{unpaired}}

\def\Thighest{T^{\mathsf{highest}}}
\def\Tlowest{T^{\mathsf{lowest}}}
\def\TTlowest{  \widehat T^{\mathsf{lowest}}}

\def\sF{\mathscr{F}}

\usepackage{bbm}
\def\One{\mathbbm{1}}


\def\fkg{\mathfrak{g}}
\def\CCCq{\CC(\hspace{-0.5mm}(q)\hspace{-0.5mm})}



\begin{document}

\maketitle

\section{Introduction}


 
\defn{Crystals} are an abstraction for the \defn{crystal bases} of 
%integrable highest weight 
quantum group representations.
% of  quantum groups $U_q(\fkg)$ associated to symmetrizable Kac--Moody Lie algebras $\g$. 
Invented by Kashiwara \cite{Kashiwara1990,Kashiwara1991} and Lusztig \cite{Lusztig1990a,Lusztig1990b} in the 1990s,
crystals may be viewed concretely as
directed acyclic graphs with labeled edges, along with a map assigning weight vectors  to each vertex, satisfying certain axioms.
Isomorphisms of crystals correspond to weight-preserving graph isomorphisms,
while subcrystals correspond to unions of weakly connected graph components.

For each finite-dimensional Lie superalgebra $\fkg$ there is a category of (abstract) $\fkg$-crystals.
The structure of $\fkg$
imposes different requirements for the weight map and edge labels.
These categories have some common features. There is always a direct sum $\oplus$ operation for crystals
corresponding to the disjoint union of directed graphs.
There is also a more subtle notion of a crystal tensor product $\otimes$.
% that lets us produce a $\g$-crystal $\cB\otimes \cC$ from two given crystals.
%formally analogous to the tensor product of $U_q(\fkg)$-representations.
There is also a character map $\ch$ assigning to each finite crystal its weight-generating function.
Finally, there is a \defn{standard crystal} $\BB$ corresponding to the vector representation of the quantum group $U_q(\fkg)$.


These ingredients are enough to define a full subcategory of \defn{normal $\g$-crystals}:
this consists of the $\g$-crystals whose connected components are each isomorphic to a subcrystal
of $\BB^{\otimes m} $ for some $m\geq 0$.
Such crystals form the smallest monoidal subcategory containing the standard crystal that is closed under isomorphisms, direct sums, and passage to subcrystals.

Defined in this way, the normal $\g$-crystals are typically
 the abstract $\g$-crystals that correspond directly to crystal bases of finite-dimensional integrable $U_q(\g)$-modules.
 This connection implies some desirable properties:
 for example, that each connected normal crystal has a unique \defn{highest weight element} whose weight determines the crystal's isomorphism class.

 
Sections~\ref{intro1-sect} and \ref{intro2-sect} review how this works in two cases that have been well-studied, when $\g=\gl_n$ is the complex general linear Lie algebra
 and when $\g=\q_n$ is the \defn{queer Lie superalgebra}.
 Section~\ref{intro3-sect} contains our main results, which establish similar formal properties of a new category of what we call \defn{$\qq_n$-crystals}.    
 In the categories of normal $\gl_n$- and $\q_n$-crystals,
the connected objects have  characters that are \defn{Schur polynomials} and \defn{Schur $P$-polynomials}, respectively.
In our new category of normal $\qq_n$-crystals, the connected objects have characters
\defn{Schur $Q$-polynomials}.  

Schur $P$- and $Q$-polynomials were first defined in work of Schur on the projective representations of the symmetric group
but have applications in many other areas;
see Definition~\ref{pq-def}. 
 One application of $\gl_n$- and $\q_n$-crystals in combinatorics is to show that certain power series are \defn{Schur positive} and \defn{Schur $P$-positive}. A similar application of $\qq_n$ crystals is to demonstrate the stronger property of \defn{Schur $Q$-positivity}.


 
 
 
 
 

 \section{Results}
 
 Throughout, $n \geq 2$ is a fixed positive integer and $[n] := \{1,2,\dots, n\}$.
 \subsection{Crystals for Schur functions}\label{intro1-sect}
 
 
 Let $\cB$ be a nonempty set with a \defn{weight function} $
\weight \colon \cB\to \ZZ^n
$ and an auxiliary element $0 \notin \cB$. %
For each $i \in [n-1]$, assume that maps
$
e_i,f_i \colon \cB \to \cB \sqcup \{0\}$ are given.  
We define $\varepsilon_i, \varphi_i \colon \cB \to \NN\sqcup\{\infty\}$ by 
$\varepsilon_i(b) := \max\left\{ k\in \NN : e_i^k(b)\neq0 \right \}$ and
$\varphi_i(b) := \max\left\{ k\in \NN : f_i^k(b) \neq 0\right \}.$
Write $\e_1,\e_2,\dots,\e_n$ for the standard basis of $\ZZ^n$.


\begin{definition}%[{See \cite[\S2.2]{BumpSchilling}}]
\label{crystal-def}
The set $\cB$ is an \defn{(abstract) $\gl_n$-crystal} 
%relative to the weight map $\weight$ and the operators $e_i$ and $f_i$
if 
for all $b,c \in \cB$ and $i \in [n-1]$:
% the following holds:
\ben
\item[(S1)]  One has $e_i(b) = c$ if and only if $f_i(c) = b$, in which case
$\weight(c)- \weight(b) = \e_i - \e_{i+1}.$

\item[(S2)]  Both $\varepsilon_i(b)$ and 
$\varphi_i(b) $ are finite and  $\varphi_i(b) - \varepsilon_i(b) = \weight(b)_i - \weight(b)_{i+1}$.
\een
%A map $\phi : \cB \to \cC$ between $\gl_n$-crystals is an \defn{isomorphism}
%if $\phi$ is a weight-preserving bijection that commutes with the raising and lowering operators,
%where it is understood that $\phi(0) =0$.
\end{definition}

The maps $e_i$ and $f_i$ are called \defn{raising} and \defn{lowering (crystal) operators}.
An \defn{isomorphism} of $\gl_n$-crystals (or any of the other types of crystal appearing later) is a weight-preserving bijection that commutes with all crystal operators.



The \defn{crystal graph} of $\cB$ is the directed graph with vertex set $\cB$
and edges $b \xrightarrow{\ i\ } c$ whenever $c = f_i(b)$.
A weakly connected component of this graph %of a  $\gl_n$-crystal
%inherits its own $\gl_n$-crystal structure and 
is called a \defn{full subcrystal}.
%
\begin{example}\label{bb-ex}
%\begin{example}
The \defn{standard $\gl_n$-crystal} $\BB_n$ has
crystal graph
  \begin{center}
    \begin{tikzpicture}[xscale=1.8, yscale=1,>=latex]]
      \node at (0,0) (T0) {$\boxed{1}$};
      \node at (1,0) (T1) {$\boxed{2}$};
      \node at (2,0) (T2) {$\boxed{3}$};
      \node at (3,0) (T3) {${\cdots}$};
      \node at (4,0) (T4) {$\boxed{n}$};
      %
      \draw[->,thick]  (T0) -- (T1) node[midway,above,scale=0.75] {$1$};
      \draw[->,thick]  (T1) -- (T2) node[midway,above,scale=0.75] {$2$};
      \draw[->,thick]  (T2) -- (T3) node[midway,above,scale=0.75] {$3$};
      \draw[->,thick]  (T3) -- (T4) node[midway,above,scale=0.75] {$n-1$};
     \end{tikzpicture}
  \end{center}
  and weight map  $\weight(\boxed{i}):=\e_i$ for $i \in [n]$.
%
\end{example}


If  $\cB$ and $\cC$ are nonempty sets then define $\cB \otimes \cC := \{ b\otimes c : b\in \cB,\ c\in \cC\}$.
An essential feature of all categories of crystals is the existence of a nontrivial tensor product.
%be the set of formal tensors of elements of $\cB$ with elements of $\cC$.
%The next definition follows the ``anti-Kashiwara'' convention.

\begin{theorem}[{See \cite[\S2.3]{BumpSchilling}}]\label{thmdef1}
Let $\cB$ and $\cC$ be $\gl_n$-crystals.
Then $\cB \otimes \cC$ has a unique $\gl_n$-crystal structure with
weight map 
$\weight(b\otimes c) := \weight(b) + \weight(c)$ and lowering operators
\[
%e_i(b\otimes c) := \begin{cases}
%b \otimes e_i(c) &\text{if }\varepsilon_i(b) \leq \varphi_i(c) \\
%e_i(b) \otimes c &\text{if }\varepsilon_i(b) > \varphi_i(c)
%\end{cases}
%\quand
f_i(b\otimes c) := \begin{cases}
b \otimes f_i(c) &\text{if }\varepsilon_i(b) < \varphi_i(c) \\
f_i(b) \otimes c &\text{if }\varepsilon_i(b) \geq \varphi_i(c)
\end{cases}
\]
%for all $i \in [n-1]$,  
where  it is understood that $b\otimes 0 = 0\otimes c = 0$.
If  $\cD$ is another $\gl_n$-crystal
then the map $(b\otimes c) \otimes d \mapsto b\otimes(c\otimes d)$ 
is a  $\gl_n$-crystal isomorphism $(\cB\otimes\cC) \otimes \cD \cong \cB\otimes(\cC\otimes \cD)$.
\end{theorem}

 
 Let $\One$ be a $\gl_n$-crystal with  a single element, whose weight is $0 \in \ZZ^n$.
A
 $\gl_n$-crystal is \defn{normal} 
if each of its full subcrystals is isomorphic to a full subcrystal of $\BB_n^{\otimes m}$ for some $m\in \NN := \{0,1,2,\dots\}$,
where we identify $\BB_n^0 $ with $ \One$.

The \defn{character} of a finite $\gl_n$-crystal $\cB$ is the Laurent polynomial
$\ch (\cB) := \sum_{b \in \cB} x^{\weight(b)}$ where  $x^{\weight(b)} :=   \prod_{i \in [n]} x_i^{\weight(b)_i}$.
If $\cB$ is a $\gl_n$-crystal then
a \defn{$\gl_n$-highest} (respectively, \defn{$\gl_n$-lowest}) \defn{weight element} $b \in \cB$
is an element with  $e_i (b) = 0$ (respectively, $f_i(b) = 0$) for all $i$.
The following theorem is well-known and the prototype for subsequent results.



\begin{theorem}[{See \cite[Theorems~3.2 and 8.6]{BumpSchilling}}]
\label{gl-thm}
If $\cB$ is a connected normal $\gl_n$-crystal,
then $\cB$ has a unique $\gl_n$-highest weight element, whose 
 weight  $\lambda \in \ZZ^n$  is a partition such that $\ch(\cB)=s_\lambda(x_1,x_2,\dots,x_n)$ is a {Schur polynomial}.
For each partition $\lambda \in \ZZ^n$,
there is a connected normal $\gl_n$-crystal with highest weight $\lambda$,
and finite normal $\gl_n$-crystals with the same character are isomorphic.
\end{theorem}

 \subsection{Crystals for Schur $P$-functions}\label{intro2-sect}
 
 
Suppose  
$\cB$ is a $\gl_n$-crystal with maps $e_{\overline 1},f_{\overline 1} \colon \cB \to \cB\sqcup\{0\}$,
to be called the \defn{queer raising and lowering operators}.
The definitions of    $\varepsilon_{i},\varphi_{i} \colon \cB \to \NN\sqcup\{\infty\}$
extend to when $i=\overline 1$.
The following definition is a slight variation of \cite[Def. 1.9]{GJKKK}:

\begin{definition}%[See \cite{GJKKK,GJKKK15}]%\cite[Def. 1.9]{GJKKK}}]
\label{q-def}
The $\gl_n$-crystal $\cB$ is an \defn{(abstract) $\q_n$-crystal} 
if 
%relative to the operators $e_{\overline 1}$ and $f_{\overline 1}$
%if the weight map satisfies $\weight(\cB) \subset \NN^n$ and 
for all $b,c \in \cB$:
\ben
\item[(P1)] %Let $b,c \in \cB$. Then
One has $e_{\overline 1}(b)=c$ if and only if $f_{\overline 1}(c) = b$,
in which case 
$\weight(c)- \weight(b) = \e_1 - \e_2$
as well as 
$\varepsilon_i(b)=\varepsilon_i(c)$
and
$\varphi_i(b) = \varphi_i(c)$ for all $i \in \{3,4,\dots,n-1\}$.

\item[(P2)] If $i \in  \{3,4,\dots,n-1\}$ then $e_i$ and $f_i$ commute
 with $e_{\overline 1}$ and $f_{\overline 1}$.
% and it holds that $\varepsilon_i(b)=\varepsilon_i(c)$
%and
%$\varphi_i(b) = \varphi_i(c)$ whenever $e_{\overline 1}(b)=c$.
%under the convention that
%$e_{\overline 1}(0) = f_{\overline 1}(0) = e_i(0) = f_i(0)=0$

\item[(P3)] %If $b \in \cB$ then
If $\weight(b)_1=\weight(b)_2= 0$ then $(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b) =0$, and otherwise $(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b) =1$.
%, with strict inequality if and only if 
%$\weight(b)_1=\weight(b)_2= 0$.
\een
%A map $\phi : \cB \to \cC$ between $\q_n$-crystals is an \defn{isomorphism}
%if $\phi$ %is an isomorphism of $\gl_n$-crystals that also commutes with the corresponding queer raising and lowering operators.
% is a weight-preserving bijection that commutes with $e_i$ and $f_i$ for all $i \in \{\overline 1,1,2,\dots,n-1\}$.
\end{definition}



The \defn{crystal graph} of a $\q_n$-crystal 
has additional 
edges $b \xrightarrow{\ \overline 1\ } c$ whenever $c = f_{\overline 1}(b)$.
A weakly connected component of this graph
%inherits its own $\q_n$-crystal structure and
is called a \defn{full $\q_n$-subcrystal}.
%
%
\begin{example}
The \defn{standard $\q_n$-crystal} $\BB_n$ has
 crystal graph
  \begin{center}
    \begin{tikzpicture}[xscale=1.8, yscale=1,>=latex]]
      \node at (0,0) (T0) {$\boxed{1}$};
      \node at (1,0) (T1) {$\boxed{2}$};
      \node at (2,0) (T2) {$\boxed{3}$};
      \node at (3,0) (T3) {${\cdots}$};
      \node at (4,0) (T4) {$\boxed{n}$};
      %
      \draw[->,thick,dashed,color=darkred]  (T0.15) -- (T1.165) node[midway,above,scale=0.75] {$\overline 1$};
      \draw[->,thick]  (T0.345) -- (T1.195) node[midway,below,scale=0.75] {$1$};
      \draw[->,thick]  (T1) -- (T2) node[midway,above,scale=0.75] {$2$};
      \draw[->,thick]  (T2) -- (T3) node[midway,above,scale=0.75] {$3$};
      \draw[->,thick]  (T3) -- (T4) node[midway,above,scale=0.75] {$n-1$};
     \end{tikzpicture}
  \end{center}
     and weight map  $\weight(\boxed{i}):=\e_i$ for $i \in [n]$.
\end{example}

Grantcharov \emph{et al.}\ prove the following in \cite{GJKKK,GJKKK15}.
Our description of the tensor product here follows the ``anti-Kashiwara'' convention, which is opposite that of \cite{GJKKK,GJKKK15}.
%\cite[Theorem 1.8]{GJKKK} and \cite[Theorem 2.7]{GJKKK15}.

\begin{theorem}[{See \cite{GJKKK,GJKKK15}}]
\label{q-tensor-thm}
Suppose
 $\cB$ and $\cC$ are $\q_n$-crystals.
Then the $\gl_n$-crystal
$\cB \otimes \cC$
has a unique $\q_n$-crystal structure with queer lowering operator
%whose queer crystal operators are given by 
\[%\label{q-otimes}
\ba
 f_{\overline 1}(b\otimes c) := \begin{cases} 
 b \otimes f_{\overline 1}(c)&\text{if }\weight(b)_1 = \weight(b)_2 = 0,%\text{if }(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b)=0
 \\
  f_{\overline 1}(b) \otimes c
&\text{otherwise}. %\text{if }(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b)=1
 \end{cases}
%  \\[-12pt]
%\\
%  e_{\overline 1}(b\otimes c) := \begin{cases} 
% b \otimes e_{\overline 1}(c)
%&\text{if }\weight(b)_1 = \weight(b)_2 = 0%\text{if }(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b)=0
% \\
%  e_{\overline 1}(b) \otimes c
%&\text{otherwise} %\text{if }(\varepsilon_{\overline 1} + \varphi_{\overline 1})(b)=1
% \end{cases}
 \ea
\]
%for $b \in \cB$ and $c \in \cC$, where it is again understood that $b\otimes 0 = 0\otimes c = 0$.
%when $n\geq 2$,
%where it is understood that $b\otimes 0 = 0\otimes c = 0$.
If  $\cD$ is another $\q_n$-crystal
then the bijection $(b\otimes c) \otimes d \mapsto b\otimes(c\otimes d)$ 
is a  $\q_n$-crystal isomorphism $(\cB\otimes\cC) \otimes \cD \cong \cB\otimes(\cC\otimes \cD)$.
\end{theorem}

A $\q_n$-crystal (exactly like $\gl_n$-crystals) is \defn{normal} 
if each of its full $\q_n$-subcrystals is isomorphic to a full $\q_n$-subcrystal of $\BB_n^{\otimes m}$ for some $m\in \NN$. The notion of highest and lowest weights for $\q_n$-crystals is more involved than for $\gl_n$-crystals, however.

Let $\cB$ be a $\q_n$-crystal.
Fix $i \in [n-1]$ and $b \in \cB$ and let $k :=  \varphi_i(b) - \varepsilon_i(b)$.
Define a map $\sigma_i \colon \cB \to \cB$ by setting 
$ \sigma_i (b) := e_i^{-k} (b)$ if $k\leq0$ and $\sigma_i (b) := f_i^{k} (b)$ if $k\geq 0$.
Inductively define $e_{\overline i} \colon \cB \to \cB \sqcup\{0\}$ for $i\in \{2,3,\dots,n-1\}$ by
$ e_{\overline i} :=  \sigma_{i-1}   \sigma_i   e_{\overline{i-1}}   \sigma_i   \sigma_{i-1}$
and set 
\[ \sigma_{w_0} := (\sigma_1) (\sigma_2\sigma_1)(\sigma_3\sigma_2\sigma_1) \cdots  (\sigma_{n-1} \cdots \sigma_2\sigma_1).\]
%Let  $I := [n-1]$ and 
%$\overline{I} := \{\overline i : i \in I\}$.
A \defn{$\q_n$-highest weight element} $b \in \cB$
is an element with  $e_i (b) = 0$ for $i=\overline 1,1,\dots,\overline{n-1}, n-1$.
A \defn{$\q_n$-lowest weight element} $b \in \cB$
is an element such that $\sigma_{w_0}(b)$ is $\q_n$-highest weight.



 






The following
contains several results in \cite{GJKKK}; see, \textit{e.g.}, \cite[Theorem~2.5 and Corollary~4.6]{GJKKK}.
%The fact that the characters of connected normal $\q_n$-crystals are Schur $P$-polynomials
%is not explicitly stated in \cite{GJKKK} but can be deduced using \cite[Theorem 2.17]{Serrano}.

\begin{theorem}[\cite{GJKKK}]
\label{main-q-thm}
If $\cB$ is a connected normal $\q_n$-crystal,
then $\cB$ has a unique $\q_n$-highest weight element, whose 
 weight  $\lambda \in \NN^n$  is a strict partition such that $\ch(\cB)=P_\lambda(x_1,x_2,\dots,x_n)$
 is a {Schur $P$-polynomial}.
For each strict partition $\lambda \in \NN^n$,
there is a connected normal $\q_n$-crystal with highest weight $\lambda$,
and 
finite normal $\q_n$-crystals with the same character are isomorphic.
\end{theorem}




  \subsection{Crystals for Schur $Q$-functions}\label{intro3-sect}
  
  This section contains our main new results.
  Suppose $\cB$ is a $\q_n$-crystal with additional maps $e_{0},f_{0}\colon \cB \to \cB\sqcup\{0\}$.
The definitions of    $\varepsilon_{i},\varphi_{i} \colon \cB \to \NN\sqcup\{\infty\}$
extend to when $i=0$.


\begin{definition}
\label{qq-def}
The $\q_n$-crystal $\cB$ is an \defn{(abstract) $\qq_n$-crystal}
%relative to the operators $e_0$ and $f_0$
 if for all $b,c \in \cB$:
\ben
\item[(Q1)]  One has  $e_{0}(b)=c$ if and only if $f_{0}(c) = b$, 
in which case $\weight(b) = \weight(c)$ as well as
$\varepsilon_i(b)=\varepsilon_i(c)$
and
$\varphi_i(b) = \varphi_i(c)$ for all $i \in [n-1]$ and also for $i=\overline 1$ if $n\geq 2$. % \left\{ \overline 1, 1,2,\dots,n-1\right\}$.

 

\item[(Q2)] If $i \in \{2,3,\dots,n-1\}$ then $e_i$ and $f_i$ commute with  $e_{0}$ and $f_{0}$.
%under the convention that
%$e_{0}(0) = f_{0}(0) = e_i(0) = f_i(0)=0.$

\item[(Q3)]
%If $b \in \cB$ then $(\varepsilon_{0} + \varphi_{0})(b) =0$ when $\weight(b)_1= 0$ and otherwise $(\varepsilon_{0} + \varphi_{0})(b) =1$.
If $\weight(b)_1= 0$ then $(\varepsilon_{0} + \varphi_{0})(b) =0$, and otherwise $(\varepsilon_{0} + \varphi_{0})(b) =1$.
 
 

\een
%A map $\phi : \cB \to \cC$ between $\qq_n$-crystals is an \defn{isomorphism}
%%if $\phi$ is an isomorphism of $\q_n$-crystals that commutes with the additional operators $e_0$ and $f_0$.
%if $\phi$ is a weight-preserving bijection that commutes with $e_i$ and $f_i$ for all $i \in \{\overline 1,0,1,2,\dots,n-1\}$.
\end{definition}

We draw the \defn{crystal graph} of a $\qq_n$-crystal by adding arrows $b \xrightarrow{\ 0\ } f_0(b)\neq 0$ 
 to the $\q_n$-crystal graph.
 This graph's weakly connected components 
are called \defn{full $\qq_n$-subcrystals}.

\begin{example}\label{bb-ex3}
The \defn{standard $\qq_n$-crystal} $\BB^+_n$ has
 crystal graph
  \begin{center}
    \begin{tikzpicture}[xscale=1.8, yscale=1,>=latex]]
      \node at (0,0) (T0) {$\boxed{1'}$};
      \node at (1,0) (T1) {$\boxed{2'}$};
      \node at (2,0) (T2) {$\boxed{3'}$};
      \node at (3,0) (T3) {${\cdots}$};
      \node at (4,0) (T4) {$\boxed{n'}$};
      %
      \node at (0,1.4) (U0) {$\boxed{1}$};
      \node at (1,1.4) (U1) {$\boxed{2}$};
      \node at (2,1.4) (U2) {$\boxed{3}$};
      \node at (3,1.4) (U3) {${\cdots}$};
      \node at (4,1.4) (U4) {$\boxed{n}$};
      %
      \draw[->,thick,dashed,color=darkred]  (T0.15) -- (T1.165) node[midway,above,scale=0.75] {$\overline 1$};
      \draw[->,thick]  (T0.345) -- (T1.195) node[midway,below,scale=0.75] {$1$};
      \draw[->,thick]  (T1) -- (T2) node[midway,above,scale=0.75] {$2$};
      \draw[->,thick]  (T2) -- (T3) node[midway,above,scale=0.75] {$3$};
      \draw[->,thick]  (T3) -- (T4) node[midway,above,scale=0.75] {$n-1$};
%
      \draw[->,thick,dashed,color=darkred]  (U0.15) -- (U1.165) node[midway,above,scale=0.75] {$\overline 1$};
      \draw[->,thick]  (U0.345) -- (U1.195) node[midway,below,scale=0.75] {$1$};
      \draw[->,thick]  (U1) -- (U2) node[midway,above,scale=0.75] {$2$};
      \draw[->,thick]  (U2) -- (U3) node[midway,above,scale=0.75] {$3$};
      \draw[->,thick]  (U3) -- (U4) node[midway,above,scale=0.75] {$n-1$};
      %
      \draw[->,thick,dotted,color=blue]  (U0) -- (T0) node[midway,left,scale=0.75] {$0$};
     \end{tikzpicture}
  \end{center}
    and weight map  $\weight(\boxed{i})=\weight(\boxed{i'})=\e_i$ for $i \in [n]$.
\end{example}


The tensor product for $\qq_n$-crystals is more complicated than for $\q_n$-crystals.

\begin{theorem}\label{qq-thmdef}
Let $\cB$ and $\cC$ be $\qq_n$-crystals.
Then the $\gl_n$-crystal $\cB\otimes \cC$
has a unique $\qq_n$-crystal structure
with lowering operators
\[
%  e_{0}(b\otimes c) := \begin{cases} 
% e_0(b) \otimes c&\text{if }\weight(b)_1 \neq 0
% \\
%b \otimes e_0(c) & \text{if }\weight(b)_1  = 0
% \end{cases}
%\quand
 f_{0}(b\otimes c) := \begin{cases} 
 f_0(b) \otimes c&\text{if }\weight(b)_1 \neq 0,
 \\
b \otimes f_0(c) & \text{if }\weight(b)_1 = 0,
 \end{cases}
\]
and
\[
%\ba
%  e_{\overline 1}(b\otimes c) &:= \begin{cases} 
% b \otimes e_{\overline 1}(c)
% &\text{if }\weight(b)_1 = \weight(b)_2 = 0
% \\
% f_0  e_{\overline 1} (b) \otimes e_0(c) 
% &\text{if $\weight(b)_1 = 0$, $f_0e_{\overline 1}(b) \neq 0$, and $e_0(c)\neq 0$}
%\\
% e_0  e_{\overline 1} (b) \otimes f_0(c) 
% &\text{if $\weight(b)_1 = 0$, $e_0e_{\overline 1}(b)\neq 0$, and $f_0(c) \neq 0$}
% \\
%  e_{\overline 1}(b) \otimes c
%&\text{otherwise}
% \end{cases}
%  \\[-12pt]
%\\
 f_{\overline 1}(b\otimes c) := \begin{cases} 
 b \otimes f_{\overline 1}(c)&\text{if }\weight(b)_1 = \weight(b)_2 = 0,
 \\
  f_{\overline 1}f_0(b) \otimes e_0(c)
 &\text{if }\weight(b)_1 = 1, \text{ $f_{\overline 1}f_0(b)\neq 0 $, and $e_0(c)\neq 0$},
\\
  f_{\overline 1}e_0(b) \otimes f_0(c)
 &\text{if }\weight(b)_1 = 1, \text{ $f_{\overline 1}e_0(b) \neq 0$, and $f_0(c) \neq 0$},
  \\
    f_{\overline 1}(b) \otimes c &\text{otherwise}.
 \end{cases}
\]
%where it is understood that $b\otimes 0 = 0\otimes c = 0$.
If  $\cD$ is another $\qq_n$-crystal
then the bijection $(b\otimes c) \otimes d \mapsto b\otimes(c\otimes d)$ 
is a  $\qq_n$-crystal isomorphism $(\cB\otimes\cC) \otimes \cD \cong \cB\otimes(\cC\otimes \cD)$.
\end{theorem}





\begin{example}\label{qq-bb-bb-ex}
The crystal graph of $\BB^+_2\otimes \BB^+_2$ is 
 \begin{center}
    \begin{tikzpicture}[xscale=2.0, yscale=1.2,>=latex]]
      \node at (0,0) (22') {$\boxed{2}\otimes \boxed{2'}$};
     \node at (0,1.4) (12') {$\boxed{1}\otimes \boxed{2'}$};
      \node at (0,2.8) (1'2') {$\boxed{1'}\otimes \boxed{2'}$};
      \node at (0,4.2) (2'2') {$\boxed{2'}\otimes \boxed{2'}$};
 %
      \node at (1.4,0) (21') {$\boxed{2}\otimes \boxed{1'}$};
     \node at (1.4,1.4) (11') {$\boxed{1}\otimes \boxed{1'}$};
      \node at (1.4,2.8) (1'1') {$\boxed{1'}\otimes \boxed{1'}$};
      \node at (1.4,4.2) (2'1') {$\boxed{2'}\otimes \boxed{1'}$};
 %
      \node at (2.8,0) (21) {$\boxed{2}\otimes \boxed{1}$};
     \node at (2.8,1.4) (11) {$\boxed{1}\otimes \boxed{1}$};
      \node at (2.8,2.8) (1'1) {$\boxed{1'}\otimes \boxed{1}$};
      \node at (2.8,4.2) (2'1) {$\boxed{2'}\otimes \boxed{1}$};
 %
      \node at (4.2,0) (22) {$\boxed{2}\otimes \boxed{2}$};
     \node at (4.2,1.4) (12) {$\boxed{1}\otimes \boxed{2}$};
      \node at (4.2,2.8) (1'2) {$\boxed{1'}\otimes \boxed{2}$};
      \node at (4.2,4.2) (2'2) {$\boxed{2'}\otimes \boxed{2}$};
      %
      \draw[->,thick]  (12.280) -- (22.80) node[midway,right,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (12.260) -- (22.100) node[midway,left,scale=0.75] {$\overline 1$};
      \draw[->,thick,dotted,color=blue]  (12) -- (1'2) node[midway,right,scale=0.75] {$ 0$};
      %
      \draw[->,thick]  (11) -- (12) node[midway,above,scale=0.75] {$ 1$};
      \draw[->,thick,dotted,color=blue]  (11) -- (1'1) node[midway,right,scale=0.75] {$ 0$};
      \draw[->,thick,dashed,color=darkred]  (11) -- (21) node[midway,right,scale=0.75] {$\overline 1$};
      %
      \draw[->,thick,dotted,color=blue]  (21) -- (21') node[midway,above,scale=0.75] {$ 0$};
      %
      \draw[->,thick,dotted,color=blue]  (2'1) -- (2'1') node[midway,above,scale=0.75] {$ 0$};      
      %
      \draw[->,thick,dotted,color=blue]  (11') -- (1'1') node[midway,right,scale=0.75] {$ 0$}; 
      \draw[->,thick]  (11') -- (12') node[midway,above,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (11') -- (2'1) node[midway,left,scale=0.75] {$\overline 1$\ \ };
      %
      \draw[->,thick,dotted,color=blue]  (12') -- (1'2') node[midway,right,scale=0.75] {$ 0$};
      \draw[->,thick]  (12'.280) -- (22'.80) node[midway,right,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (12'.260) -- (22'.100) node[midway,left,scale=0.75] {$\overline 1$};
      %
      \draw[->,thick]  (1'2.80) -- (2'2.280) node[midway,right,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (1'2.100) -- (2'2.260) node[midway,left,scale=0.75] {$\overline 1$};
      %
      \draw[->,thick]  (1'2'.80) -- (2'2'.280) node[midway,right,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (1'2'.100) -- (2'2'.260) node[midway,left,scale=0.75] {$\overline 1$};
      %
     \draw[->,thick]  (1'1') -- (1'2') node[midway,above,scale=0.75] {$ 1$};
     \draw[->,thick,dashed,color=darkred]  (1'1') -- (2'1') node[midway,right,scale=0.75] {$ \overline 1$};
     %
     \draw[->,thick]  (1'1) -- (1'2) node[midway,above,scale=0.75] {$ 1$};
      \draw[->,thick,dashed,color=darkred]  (1'1) -- (21') node[midway,right,scale=0.75] {\ $\overline 1$};
     \end{tikzpicture}
  \end{center}
There are two full $\qq_2$-subcrystals,  which are isomorphic.
\end{example}

%The $1$-element $\gl_n$-crystal $\One$ may be regarded as a $\qq_n$-crystal.
A $\qq_n$-crystal is \defn{normal} 
if each of its full $\qq_n$-subcrystals is isomorphic to a full $\qq_n$-subcrystal of $(\BB^+_n)^{\otimes m}$ for some $m\in \NN$,
where $(\BB^+_n)^0  := \One$.
 

 
 
 
 


\begin{example}
For $i \in \ZZ$ set $i' := i - \frac{1}{2}$.
Define $\cW^+_n(m)$ to be the set of words  of length $m$ with all letters in $  \{1'<1<2'<2<\dots<n'<n\}$. 
We identify $\cW^+_n(m) \cong (\BB_n^+)^{\otimes m}$ by viewing 
$w =w_1w_2\cdots w_m $ as $ \boxed{w_1} \otimes \boxed{w_2} \otimes \cdots \otimes \boxed{w_m}$.
This gives $\cW^+_n(m)$ a normal $\qq_n$-crystal structure, which we describe below. It suffices to explain the $f_i$ operators.

Let $w =w_1w_2\cdots w_m \in \cW^+_n(m)$ and $i \in [n-1]$.
Consider the word formed by replacing each letter $w_k \in \{i',i\}$ by a right parenthesis ``)''
and each letter $w_j \in \{ i+1,i+1'\}$ by a left parenthesis ``(''.
%
The word $f_i(w)$ is constructed from $w$ by adding $1$ to the letter in $w$ in the position of the last unpaired ``)''; if no such position exists then $f_i(w) =0$.
For example,
 %if $w=131'22'131'2 \in \cW^+_3(9)$ and $i=1$  then 
%$\unpaired_i(w) = (1,3,9)$
%and 
 $f_1(13{\color{red}1'}22'131'2 ) =13{\color{red}2'}22'131'2 $.

To describe $f_{\overline 1}(w)$, 
let $j \in [m]$ be minimal with $w_j\in\{1',1\}$.
If no such $j$ exists or  $w_i\in\{2,2'\}$ for some $i \in [j-1]$, then $f_{\overline 1}(w)=0$.
Otherwise let $k \in \{ j+1,j+2,\dots,m\}$ be minimal with $w_k \in \{1',1\}$.
%[(F1)] 
If no such $k$ exists or if  $w_j = w_k$ then
 $f_{\overline 1}(w)$ is formed from $w$ by adding $1$ to $w_j$.
%  \[ w = 3{\color{red}1'}21'1 \mapsto 3{\color{red}2'}21'1 = f_{\overline 1}(w).\]
 %
%[(F2)]
  If $w_j=1$ and $ w_k=1'$ then  $f_{\overline 1}(w)$ is formed from $w$ by changing $w_j$ to $2'$ and $w_k$ to $1$.
%  \[ w = 3{\color{red}1}2{\color{red}1'}1 \mapsto 3{\color{red}2'}2{\color{red}1}1 = f_{\overline 1}(w).\]
%  
 %[(F3)] 
 If $w_j=1'$ and $ w_k=1$ then  $f_{\overline 1}(w)$ is formed from $w$ by changing $w_j$ to $2$ and $w_k$ to $1'$.
 For example, $f_{\overline 1}(3{\color{red}1'}21'1 ) =  3{\color{red}2'}21'1 $
 and $f_{\overline 1}(3{\color{red}1}2{\color{red}1'}1 ) =  3{\color{red}2'}2{\color{red}1}1 $.
%Finally: % we have the definitions of $f_{0}(w)$ and $e_{0}(w)$:

 Finally, to describe $f_0(w)$, let $j\in[m]$ be minimal with $w_j \in \{1',1\}$.
If no such $j$ exists or $w_j = 1'$ then $f_0(w) = 0$. Otherwise $f_0(w)$ is formed from $w$ by changing $w_j$ to $1'$.
For example, $f_{0}(3{\color{red}1}21'1) = 3{\color{red}1'}21'1$ and $f_{0}(3{1'}21'1) =0$.
\end{example}


%To define highest and lowest weight elements for $\qq_n$-crystals, we need a few more operators.
Assume $\cB$ is a $\qq_n$-crystal. 
For each $i \in [n]$ let 
$ e_{0}^{[i]}    := \sigma_{i-1} \cdots \sigma_2 \sigma_1 e_0 \sigma_1 \sigma_2 \cdots \sigma_{i-1}$
and
$
f_{0}^{[i]}  := \sigma_{i-1} \cdots \sigma_2 \sigma_1 f_0 \sigma_1 \sigma_2 \cdots \sigma_{i-1}
$.
%where  $\sigma_i$ is defined as in \eqref{weyl-action-eq}.
%This means that $e_{0}^{[1]} = e_0$ and $f_{0}^{[1]} = f_0$.
A \defn{$\qq_n$-highest weight element}
$b \in \cB$ is a 
$\q_n$-highest weight element
with $e_{0}^{[i]}(b)  = 0$ for all $i \in [n] $.
A \defn{$\qq_n$-lowest weight element}
$b \in \cB$ is a 
$\q_n$-lowest weight element
with $f_{0}^{[i]}(b)  = 0$ for all $i\in [n] $.

Since  $\BB_n^+\cong \BB_n \sqcup \BB_n$ as $\gl_n$-crystals,
a  normal $\qq_n$-crystal is normal as a $\gl_n$-crystal.
%
However, normal $\qq_n$-crystals
are not always normal as $\q_n$-crystals. %; see Example~\ref{abnormal-remark}.
This means that results like Theorem~\ref{main-q-thm} do not directly imply 
 similar properties of normal $\qq_n$-crystals. 
In the particular, the proof of the following extension of Theorem~\ref{main-q-thm} is nontrivial:


\begin{theorem}\label{main-thm}
If $\cB$ is a connected normal $\qq_n$-crystal,
then $\cB$ has a unique $\qq_n$-highest weight element, whose 
 weight  $\lambda \in \NN^n$  is a strict partition with $\ch(\cB)=Q_\lambda(x_1,x_2,\dots,x_n)  $
 is a {Schur $Q$-polynomial}.
For each strict partition $\lambda \in \NN^n$,
there is a connected normal $\qq_n$-crystal with highest weight $\lambda$,
and 
finite normal $\qq_n$-crystals with the same character are isomorphic.
\end{theorem}

If $\cB$ is a normal $\gl_n$-crystal then there is a unique action of the symmetric group
$S_n$ on $\cB$ in which the simple transposition $s_i = (i,i+1)$ acts as $\sigma_i$.
The reverse permutation $w_0 =n\cdots 321 \in S_n$ acts as $\sigma_{w_0}$
and interchanges highest and lowest weight elements.


On normal $\qq_n$-crystals, there is a natural action of hyperoctahedral group extending this $S_n$-action.
This is an interesting feature of $\qq_n$-crystals that is not present for $\q_n$-crystals.
Suppose $\cB$ is a $\qq_n$-crystal. The formula for $\sigma_i \colon \cB\to\cB$
makes sense when $i=0$. 
Let $\W_n$ denote the group whose elements are the permutations $w$ of $\ZZ$
satisfying $w(-i) = -w(i)$ for all $i$ and $w(i) = i$ for all $i>n$. 
This is the finite Coxeter group of type $\mathsf{BC}_n$, with simple generators 
 $t_0 := (-1,1)$ and $t_i := (i,i+1)(-i,-i-1)$ for $i \in [n-1]$.


\begin{theorem}\label{bc-action-prop}
Let $\cB$ be a normal $\qq_n$-crystal.
There exists a unique action of $\W_n$ on $\cB$
in which $t_0$ and $t_i$ for $i \in [n-1]$
act as $\sigma_0$ and $\sigma_i$, respectively.
Moreover, the operator $ \sigma_{w^+_0} :=  (\sigma_0) (\sigma_1\sigma_0)(\sigma_2\sigma_1\sigma_0) \cdots  (\sigma_{n-1} \cdots \sigma_1\sigma_0)$
 swaps 
$\qq_n$-highest and $\qq_n$-lowest weight elements.
\end{theorem}


\section{Constructions}\label{constr-sect}

Here, we describe explicit constructions realizing all connected normal $\qq_n$-crystals.
 
\subsection{Crystal operators on increasing factorizations}


 

Let $S_\ZZ$ be the group of permutations of $\ZZ$ that fix all but finitely many points.
This is a Coxeter group generated by the simple transpositions $s_i = (i,i+1)$ for $i \in \ZZ$.


Fix  $\pi \in S_\ZZ$.  A \defn{reduced word} for $\pi $ is an integer sequence $a_1a_2\cdots a_l$ of shortest possible length such that $\pi=s_{a_1}s_{a_2}\cdots s_{a_l}$. Let $\cR(\pi)$ denote the set of such words.
For any set of words $\cW$ let $\Incr_n(\cW)$ be the set of $n$-tuples $a=(a^1,a^2,\cdots ,a^n)$ 
where each $a^i$ is a strictly increasing (possibly empty) primed word such that
%the concatenation 
$\concat(a) := a^1a^2\cdots a^n$ is in $\cW$.


Morse and Schilling  identify a $\gl_n$-crystal structure on $\Incr_n(\cR(\pi))$ in \cite{MorseSchilling}, which we recall below.
The weight of $a=(a^1,a^2,\cdots ,a^n)$  is $\weight(a) := (\ell(a^1), \ell(a^2),\dots,\ell(a^n))$.


\begin{definition}
Let $v$ and $w$ be strictly increasing words with letters in $\frac{1}{2}\ZZ$.
Form a set of paired letters $\pair(v,w)$ by iterating over the letters in $w$ from largest to smallest; at each iteration,
the current letter $w_j$ is paired with the smallest unpaired letter $v_i$ with $\lceil v_i\rceil >\lceil w_j\rceil $ (if such a letter exists)
and $(v_i,w_j)$ is added to $\pair(v,w)$.
If $v=1,3,4,5,8,10',11$ and $w=2',6,9,12,13$ (where $i' := i-\frac{1}{2}$) then 
$\pair(v,w) = \{ (10',9), (8,6),(3,2')\}.$
\end{definition}

For each $i \in [n-1]$ let $f_i \colon \Incr_n(\cR(\pi)) \to \Incr_n(\cR(\pi)) \sqcup \{0\}$
be the operator given as follows. 
Let
 $a=(a^1,a^2,\dots,a^n) \in \Incr_n(\cR(\pi))$.
%
If every letter in $a^i$ is the first term of an element of $ \pair(a^i,a^{i+1})$
then $f_i(a) := 0$.
Otherwise, let $x$ be the largest unpaired letter in $a^i$,  
let $y$ be the smallest integer not in $a^{i+1}$ 
with $y\geq  x $, and form  $f_i(a)$ from $a$ by removing $x$ from $a^i$ and adding $y$ to $a^{i+1}$
in the unique position that gives an increasing word.

A permutation is \defn{vexillary} if it is $2143$-avoiding.
The \defn{ shape} $\lambda(\pi)$ of $\pi$ is the transpose
of the partition sorting the numbers $c_i := |\{ j \in \ZZ : \text{$ i < j$ and $\pi(j) < \pi(i)$}\}|$ for $i \in \ZZ$.
%We denote this partition by $\lambda(\pi)$.
The set $\Incr_n(\cR(\pi))$ is nonempty if and only if $\lambda(\pi)$  has at most $n$ parts, so is in $\NN^n$.

\begin{theorem}[See \cite{MorseSchilling}]
For each $\pi \in S_\ZZ$, there exists a unique $\gl_n$-crystal structure on 
$\Incr_n(\cR(\pi))$ with the weight function and lowering operators $f_i$ defined above.
This $\gl_n$-crystal is normal.
Whenever $\pi$ is vexillary and $\lambda(\pi) \in \NN^n$, the $\gl_n$-crystal $\Incr_n(\cR(\pi))$ is connected with  unique highest weight  $\lambda(\pi)$.
\end{theorem}




There is an extension of this result for $\qq_n$-crystals.
The \defn{Demazure product} is the unique associative product $\circ \colon S_\ZZ \times S_\ZZ\to S_\ZZ$ %, called the \defn{Demazure product},
such that $\pi \circ s_i = \pi$ if $\pi(i) > \pi(i+1)$ and $\pi\circ s_i = \pi s_i$ if $\pi(i)<\pi(i+1)$ for each $i \in \ZZ$.
 Let 
 $I_\ZZ := \{ \pi \in S_\ZZ : \pi=\pi^{-1}\}$. % be the set of \defn{involutions} in $S_\ZZ$. 
 If $z \in I_\ZZ$ and $i \in \ZZ$, then 
$
 s_i \circ z \circ s_i$ is either $   z $ when $z(i) > z(i+1)$,
 $zs_i = s_iz$ when $z(i)=i$ and $z(i+1)=i+1$,
 or  $s_izs_i$ otherwise.
This implies that $I_\ZZ = \{ \pi^{-1} \circ \pi : \pi \in S_\ZZ\}$.


An \defn{involution word} for  $z \in I_\ZZ$ is word $a_1a_2\cdots a_n$
of shortest possible length such that 
$z=s_{a_n}\circ \cdots   \circ s_{a_2}\circ s_{a_1} \circ 1 \circ s_{a_1}  \circ s_{a_2} \circ \cdots\circ s_{a_n}.$
%Equivalently, this is a reduced word for any permutation $\pi \in S_\ZZ$ of minimal length such that $z = \pi^{-1}\circ \pi$.
Write $\iR(z)$ for the set of such words.
A \defn{commutation} for  $a_1a_2\cdots a_n  \in \iR(z)$ is an index $i \in [n]$ such that
both $a_i$ and $1+a_{i}$ are fixed points of 
 the involution $s_{a_{i-1}}\circ \cdots \circ s_{a_2} \circ s_{a_1} \circ 1 \circ s_{a_1} \circ s_{a_2} \circ\cdots \circ s_{a_{i-1}}.$
 The number of commutations is the same for every involution word 
for $z \in I_\ZZ$. 
%and given by
%the \defn{absolute length}  $\ell_{\mathsf{abs}}(z) := |\{ i \in \ZZ : i < z(i)\}|$.

A \defn{primed word} is a word $w $ with letters in $\frac{1}{2}\ZZ = \{ \cdots < 1'<1<2'<2<\cdots\}$,
where we set $i' := i -\frac{1}{2}$. 
Form $\unprime(w)$ from a primed word $w $ by adding $\frac{1}{2}$ to all primed letters.
A \defn{primed involution word} for
$z\in I_\ZZ $ is a primed word whose unprimed form is in $\iR(z)$ and whose primed letters occur at commutations.
%
Write $\iR^+(z)$ for the set of such words.
For example, if $z = (1,3)(2,4)$ then $\iR^+(z) = \left\{ 
132, 13'2, 1'32, 1'3'2, 312, 31'2, 3'12, 3'1'2
\right\}$.








Fix  $z \in I_\ZZ$. 
%There are similar $\q_n$- and $\qq_n$-crystal structures on the sets $\Incr_n(\iR(z))$ and $\Incr_n(\iR^+(z))$.
For each $i \in [n-1]$ we define the lowering operator $f_i \colon \Incr_n(\iR^+(z)) \to \Incr_n(\iR^+(z)) \sqcup \{0\}$  as follows. 
Let
 $a=(a^1,a^2,\dots,a^n) \in \Incr_n(\iR^+(z))$.
 If every letter in $a^i$ is the first term of an element of $ \pair(a^i,a^{i+1})$
then $f_i(a) := 0$.
Otherwise, let $x  \in \frac{1}{2}\ZZ$ be the largest unpaired letter in $a^i$,  
let $y  \in \ZZ$ be the smallest integer not in $\unprime(a^{i+1})$ 
with $y\geq \lceil x\rceil $, and construct an $n$-tuple of strictly increasing words $f_i(a)$
by applying the following procedure to $a$:
\begin{itemize}
\item If  $x$ is primed
then remove $x$ from $a^i$ and add $y'$ to $a^{i+1}$:
\[ 
 a=(\dots,1{\color{red}3'}459,347',\dots) \mapsto (\dots,1459,34{\color{red}5'}7',\dots)=f_i(a).\]
\item If $x $ is unprimed
then remove $x$ from $a^i$, add $y$ to $a^{i+1}$,
and for each integer $x \leq v < y$
with $v+1 \in a^i$ and $v' \in a^{i+1}$, replace $v+1 \in a^i$ by $v+1'$ and $v' \in a^{i+1}$ by $v$:
\[ 
a = (\dots,1{\color{red}3}4{\color{red}5}69,3{\color{red}4'}58,\dots) \mapsto (\dots,14{\color{red}5'}69,3{\color{red}4}5{\color{red}6}8,\dots) = f_i(a).\]
\end{itemize}

We also define a map $f_{\overline 1} \colon \Incr_n(\iR^+(z)) \to \Incr_n(\iR^+(z)) \sqcup \{0\}$.
%Again suppose  $a=(a^1,a^2,\dots,a^n) \in \ORF_n(z)$.
If $a^1=\emptyset$ or if the first letter of $a^1$ is not strictly smaller than every letter in $a^2$,
then  $f_{\overline 1}(a) := 0$.
If $a^1$ has at least two letters and the first two of these are not both primed or unprimed,
then reverse the primes on these letters and move the modified first letter of $a^1$ to the start of $a^2$:
\[ a = ({\color{red}1'3}4,25,\dots) \mapsto (3'4,{\color{red}1}25,\dots) = f_{\overline 1}(a).\]
Otherwise, move the first letter of $a^1$ to the start of $a^2$:
 \[ a = ({\color{red}1'}3'4,25,\dots) \mapsto (3'4,{\color{red}1'}25,\dots) =   f_{\overline 1}(a).\]
 
 Finally, let $f_0 \colon \Incr_n(\iR^+(z)) \to \Incr_n(\iR^+(z)) \sqcup \{0\}$ be the operator such that
 if $a^1$ is empty or begins with a primed letter then $f_{0}(a):=0$, and
otherwise $f_0(a)$ is formed from $a$ by adding a prime to the first letter of $a^1$.

 
The \defn{involution shape} of $z \in I_\ZZ$ is the transpose
of the partition sorting the numbers $m_i := \{ j \in \ZZ : \text{$z(j) \leq  i < j$ and $z(j) < z(i)$}\}$ for $i \in \ZZ$.
We denote this partition by $\mu(z)$.
The set $\Incr_n(\iR^+(z))$ is nonempty if and only if $\mu(z)$ has at most $n$ parts.

\begin{theorem}
For each $z \in I_\ZZ$, there exists a unique $\qq_n$-crystal structure on 
$\Incr_n(\iR^+(z))$ with the weight function and lowering operators $f_i$ defined above.
This $\qq_n$-crystal is normal.
The subset $\Incr_n(\iR(z)) \subset \Incr_n(\iR^+(z))$
is a union of full $\q_n$-subcrystals, so inherits a $\q_n$-crystal structure, which is also normal.
When $z \in I_\ZZ$ is vexillary and $\mu(z) \in \NN^n$, 
both $\Incr_n(\iR(z))$  
and  $\Incr_n(\iR^+(z))$ are connected (as $\q_n$- and $\qq_n$-crystals, respectively)
with unique highest weight $\mu(z)$, which is always a strict partition.
\end{theorem}

The $\q_n$-crystal $\Incr_n(\iR(z))$ has been studied previously in \cite{Hiroshima,Marberg2019b}.
The complementary subset $ \Incr_n(\iR^+(z)) \setminus  \Incr_n(\iR(z))$
is another union of full $\q_n$-crystals, but this is not typically normal or connected as a $\q_n$-crystal.





\subsection{Crystal operators on shifted tableaux}\label{pq-sect}


Assume $\mu = (\mu_1> \mu_2> \dots \geq 0)$ is a strict partition.
Let $\ell(\mu) = |\{ i > 0: \mu_i >0\}|$.
The \defn{shifted diagram} of  $\mu$
is the set $\SD_\mu:= \{ (i,i+j-1) :  i\in  [\ell(\mu)] \text{ and } j \in [\mu_i]\}.$
A \defn{shifted tableau} of shape $\mu$ is a  map $ \SD_\mu \to \frac{1}{2}\ZZ$.
We draw tableaux in French notation, so that %row indices increase from bottom to top. If
\be\label{tableau-ex}
T = \ytab{ \none & 2' & 2 & 4' \\ 1' & 1 & 1 & 4'}
\ee
is a shifted tableau of shape $\mu=(4,3)$. The \defn{(main) diagonal} of a shifted tableau
is the set of positions $(i,j)$ in its domain with $i=j$.

 
A shifted tableau is \defn{semistandard} if its entries are all positive and its rows and columns are weakly increasing,
such that no primed entry is repeated in any row and no unprimed entry is repeated in any column.
The example 
in \eqref{tableau-ex}
is semistandard.
%
We write $\SShTab_n(\mu)$
 for the set of semistandard shifted tableaux of shape $\mu$ with all entries in $\{1'<1<\dots<n'<n\}$,
and $ \ShTab_n(\mu)$ for the subset of elements in $ \SShTab_n(\mu)$
with no primed entries on the diagonal.


Let $\weight(T) := (a_1,a_2,\dots,a_n)$
where $a_i$ is the number of entries in $T$ equal to $i'$ or $i$.

\begin{definition}\label{pq-def}
The \defn{Schur $P$- and Schur $Q$-polynomials} of a strict partition $\mu \in \NN^n$ are 
\[ 
P_\mu(x_1,x_2,\dots,x_n) = \sum_{T \in \ShTab_n(\mu)} x^{\weight(T)}
\quand
Q_\mu(x_1,x_2,\dots,x_n) = \sum_{T \in \SShTab_n(\mu)} x^{\weight(T)}.
\]
These polynomials are both symmetric. As $\mu$ varies, they are linearly independent.
\end{definition}

Fix $z \in I_\ZZ$.
The correspondence defined below was introduced in \cite{M2021a}. Here, when we refer to ``interchanging the primes''  on two elements of $  \frac{1}{2}\ZZ$,
we mean the operation   that adds a prime to one number while removing the prime from the other if the two are not both primed or unprimed,
and otherwise leaves both numbers unchanged. 


\begin{definition}%[{See \cite[Defs. 3.1 and 3.8]{M2021a}}]
\label{oeg-def}
Suppose  $a \in  \Incr_n(\iR^+(z))$ and $w=w_1w_2\cdots w_m =\concat(a)$.
Let $\emptyset = T_0,T_1,\dots,T_m$ be the sequence of shifted tableaux
in which
 $T_i$ for $i \in [m]$ is formed by inserting $w_i$ into $T_{i-1}$ according to the following procedure:
\ben
\item Start by inserting $w_i$ into the first row.
At each stage, an entry $x$ is inserted into a row or column.
Let $y$ and $\widetilde y$ be 
the first entries in the same row or column with $\lceil x\rceil \leq \lceil y\rceil $
and $\lceil x\rceil < \lceil \widetilde y \rceil $.

\item If no such entries exist then $ x $ is added to the end of the row or column,
with the exception that if $x$ is added to a diagonal position then its value is changed to $\lceil x \rceil$.
The process to form $T_i$ \defn{ends in column insertion} if we are inserting into a column or 
if $ \lceil x \rceil \neq x$  is added to the diagonal. Otherwise, the process \defn{ends in row insertion}.
 
\item If $y \neq \widetilde y$  then the primes on these entries are interchanged
and $x+1$ is inserted into the next column (if $y$ is on the diagonal) or the next row (otherwise).
 If $y=\widetilde y$ is off the diagonal then $x$ replaces $y$ and  
$y$ is inserted into
the next row.
 If $y=\widetilde y$ is on the diagonal then $\lceil x \rceil $ replaces $y$ and
 $y - (\lceil x \rceil  - x)$ is inserted into the next column.
\een
Define $\PO(a) := T_m$ and construct
$\QO(a)$ as the shifted tableau with the same shape as $\PO(a)$
that contains $j$ (respectively, $j'$) in the box added to $T_{i-1}$ to form $T_i$
if $w_i$ is in the $j$th factor of $a$ and the insertion process ends in row (respectively, column) insertion.
\end{definition}



%
 
\begin{example}\label{oeg-ex1}
If $a = (4,1'35,\emptyset,4',\emptyset, 2)$ then $\PO(a)$ and $\QO(a)$ are computed as follows:
\[
\ba 
\emptyset \leadsto \ytab{4} \leadsto  \ytab{1 & 4'}   \leadsto \ytab{ \none & 4 \\ 1 & 3} \leadsto \ytab{ \none & 4 \\ 1 & 3 & 5}
\leadsto  \ytab{ \none & 4 & 5 \\ 1 & 3 & 4'} \leadsto \ytab{ \none & 3 & 5' \\ 1 & 2 & 4 & 5} &= \PO(a), \\
\emptyset \leadsto \ytab{1} \leadsto \ytab{1 & 2'}   \leadsto  \ytab{ \none & 2' \\ 1 & 2'} \leadsto  \ytab{  \none & 2' \\ 1 & 2' & 2}
\leadsto  \ytab{   \none & 2' & 4 \\ 1 & 2' & 2} \leadsto  \ytab{  \none & 2' & 4 \\ 1 & 2' & 2 & 6'} &= \QO(a).
\ea
\]
\end{example}



The map $a \mapsto (\PO(a), \QO(a))$ is called \defn{orthogonal Edelman--Greene insertion} in \cite{M2021a}.
It is a shifted version of the \defn{Edelman--Greene correspondence} from \cite{EG},
and the  counterpart
to a ``symplectic'' insertion algorithm
studied in \cite{Hiroshima,Marberg2019a,Marberg2019b}.
Restricted to  $\Incr_n(\iR(z)) \subsetneq \Incr_n(\iR^+(z))$,
the map  is
a special case of \defn{shifted Hecke insertion} from 
\cite{PatPyl2014}. 


The \defn{row reading word}   of a shifted tableau is the word given by concatenating the rows of
in reverse order (so, starting with the last row).
%A shifted tableau is \defn{increasing} if its rows and columns are strictly increasing.
The map
$a \mapsto (\PO(a), \QO(a))$
is a bijection from
$ \Incr_n^+(z)$ to the set of
pairs $(P,Q)$ of shifted tableaux of the same shape, in which $Q$ is semistandard with all entries at most $n$, 
and $P$ has strictly increasing rows and columns, 
 no primes on the diagonal, row reading word in
$  \iR^+(z)$ \cite[Corollary~3.9]{M2021a}.

We prove some further results connecting $a \mapsto (\PO(a), \QO(a))$ to crystals. To start:

\begin{theorem}
The subsets on which $a \mapsto \PO(a)$ is constant are
\begin{itemize}
\item the full $\q_n$-subcrystals of $\Incr_n(\iR(z)) $ and
\item the full $\qq_n$-subcrystals of $\Incr_n(\iR^+(z)) $.
\end{itemize}
\end{theorem}

A \defn{quasi-isomorphism} between $\gl_n$-, $\q_n$-, or $\qq_n$-crystals, respectively,
is
a map $\psi \colon \cB \to \cC$  
such that 
 for each full subcrystal $\cX \subset \cB$, the image $\cY := \psi(\cX)$ is a full subcrystal of $ \cC$
and the restricted map $\psi\colon \cX \to \cY$ is a crystal isomorphism.

\begin{theorem}
There exists a unique $\qq_n$-crystal structure on $\SShTab_n(\mu)$ 
for each strict partition $\mu \in \NN^n$ such that the map
$a\mapsto \QO(a)$ is quasi-isomorphism of $\qq_n$-crystals
\[
\bigsqcup_{z \in I_\ZZ} \Incr_n(\iR^+(z)) \to \bigsqcup_{\text{strict partitions }\mu \in \NN^n}  \SShTab_n(\mu).
\]
The $\qq_n$-crystal $\SShTab_n(\mu)$  is connected and normal with unique highest weight $\mu$.
For each strict partition $\mu \in \NN^n$, the subset $\ShTab_n(\mu)\subset \SShTab_n(\mu)$
is a full $\q_n$-subcrystal, so inherits a $\q_n$-crystal structure which is also connected and normal
with unique highest weight $\mu$.
\end{theorem}



This result is highly nontrivial, since the preimage of any given $\SShTab_n(\mu)$ under 
$\QO$ will intersect $\Incr_n(\iR^+(z))$ for infinitely many different $z\in I_\ZZ$.
The $\q_n$-crystal  $\ShTab_n(\mu)$  coincides with the one studied in \cite{AssafOguz,HPS,Hiroshima2018}.
The results in \cite{AssafOguz,HPS,Hiroshima2018} give explicit formulas for the relevant $\q_n$-crystal operators. There are similarly explicit, slightly more complicated formulas for the 
$\qq_n$-crystal operators
on $\SShTab_n(\mu)$. This information will appear in the full-length version of this extended abstract.








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