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\documentclass[finalversion]{FPSAC2023}
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\usepackage{tikz}
\usepackage{ytableau}

\def\fkS{\mathfrak{S}}
\def\fkSO{\fkS^{\Or}}
\def\fkSS{\fkS^{\Sp}}

\def\fpf{\mathsf{fpf}}
\def\lambdafpf{\lambda^{\mathsf{ss}}}
\def\ch{\mathrm{ch}}
\DeclareMathOperator{\Dem}{Dem}
\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\q}{\mathfrak{q}}
\newcommand{\xx}{\mathbf{x}}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\key}{\kappa}
\newcommand{\Schubert}{\mathfrak{S}}
\newcommand{\GL}{\mathsf{GL}}
\newcommand{\Sp}{\mathsf{Sp}}
\newcommand{\Or}{\mathsf{O}}
\newcommand{\K}{\mathsf{K}}
\newcommand{\zero}{\mathbf{0}}
\newcommand{\bon}{\overline{1}}
\newcommand{\qquand}{\qquad\text{and}\qquad}
\newcommand{\quand}{\quad\text{and}\quad}

\DeclareMathOperator{\RF}{RF}
\DeclareMathOperator{\BRF}{BRF}
\DeclareMathOperator{\wt}{wt}
\DeclareMathOperator{\spanning}{span}
\DeclareMathOperator{\Stab}{Stab}

\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\mcB}{\mathcal{B}}
\renewcommand{\mid}{:}

\def\qsymbol{\mathsf{Q}}
\def\qkey{\kappa^\qsymbol}
\def\qatom{\overline\kappa^\qsymbol}
\def\qlascoux{L^\qsymbol}

\def\psymbol{\mathsf{P}}
\def\pkey{\kappa^\psymbol}
\def\patom{\overline\kappa^\psymbol}
\def\plascoux{L^\psymbol}
\def\plascouxatom{\overline L{}^\psymbol}


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\title{Shifted key polynomials}

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\author[E.~Marberg and T.~Scrimshaw]{Eric Marberg\thanks{\href{mailto:emarberg@ust.hk}{emarberg@ust.hk}. Partially supported by Hong Kong RGC grant GRF 16306120.}\addressmark{1} \and Travis Scrimshaw\thanks{\href{mailto:tcscrims@gmail.com}{tcscrims@gmail.com}. Partially supported by Grant-in-Aid for JSPS Fellows 21F51028.}\addressmark{2}}

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\address{\addressmark{1}Department of Mathematics, Hong Kong University of Science and Technology \\ \addressmark{2}Department of Mathematics, Hokkaido University}

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\abstract{
We introduce shifted analogues of key (resp.\ atom) polynomials that we call $P$- and $Q$-key (resp.\ atom) polynomials. These families are defined in terms of isobaric divided difference operators applied to dominant symplectic and orthogonal Schubert polynomials. We establish a number of fundamental properties of these functions, formally similar to classical results on key polynomials. For example, we show that our shifted key polynomials are partial versions of Schur $P$- and $Q$-functions in a precise sense. We conjecture that symplectic/orthogonal Schubert polynomials expand positively in terms of $P$/$Q$-key polynomials. As evidence for this conjecture, we also show that shifted key polynomials are the characters of certain shifted analogues of Demazure crystals.
}

\keywords{Schur $P$/$Q$-function, key polynomial, Lie superalgebra, Demazure crystal}

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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}

Fix a positive integer $n$.
Set $G = \GL_n(\CC)$ and write $B\subseteq G$ for the Borel subgroup of upper triangular matrices.
Let $\K$ be either the orthogonal group $\Or_n(\CC)$ or symplectic group $\Sp_{n}(\CC)$ (when $n$ is even).
For brevity, we omit the rank and field.
%
The \defn{(complete) flag variety} $G/B$ decomposes into finitely many $B$-orbits indexed by 
 the symmetric group $S_n$.
 The closures of these orbits give rise to the \defn{Schubert classes} in  $H^*(G/B)$.
The cohomology ring $H^*(G/B)$ is naturally a quotient of $\ZZ[\xx] = \ZZ[x_1,x_2,\dots]$
and the well-known \defn{Schubert polynomials} $\Schubert_w \in \ZZ[\xx]$ for $w \in S_n$
provide representatives for the Schubert classes.

The flag variety $G/B$ also decomposes into finitely many
$\K$-orbits, indexed by  involutions in $S_n$ when $\K=\Or$ and by fixed-point-free involutions in $S_n$ when $\K=\Sp$ and $n$ is even. The closures of the $\K$-orbits give rise to cohomology classes in $H^*(G/B)$ that are positive sums of Schubert classes.
Polynomial representatives for these classes are provided by the \defn{orthogonal} and \defn{symplectic Schubert polynomials} (which we abbreviate as \defn{$\K$-Schubert polynomials}) characterized in~\cite{WY17}.
A precise expansion of $\K$-Schubert polynomials into usual Schubert polynomials was given in~\cite{Brion98}; see also \cite{CJW16,HMP17}.

It is known~\cite[Ex.~2.2.2]{Manivel01} that $\Schubert_w = \xx^{\lambda}$ whenever $w \in S_n$
has \defn{Rothe diagram} equal to the Young diagram of a partition
 $\lambda = \lambda(w)$. 
Such $w$ are called \defn{dominant} and correspond to $\GL$ dominant weights.
There exists a stable limit $F_w$ of $\Schubert_w$ known as a \defn{Stanley symmetric function}~\cite{Stanley84}, and when $w$ is dominant, it holds that $F_w = s_{\lambda(w)}$ is a Schur function.
Combining the \defn{Billey--Jockusch--Stanley (BJS) formula}~\cite{BJS93} 
for $\Schubert_w$
with the stable limit shows that if $w \in S_n$ is dominant, then $s_{\lambda(w)}$
is the weight-generating function for the set  
 $\RF(w)$ of factorizations of reduced words for $w$ into decreasing subwords. (We refer to elements of $\RF(w)$ as \defn{reduced factorizations}.)
Morse and Schilling showed this directly in~\cite{MS16} by constructing a \defn{(Kashiwara) crystal}~\cite{K91} on the set $\RF(w)$.
 

The BJS formula for $\Schubert_w$ is a sum over a certain class of \defn{bounded reduced factorizations} $\BRF(w) \subset \RF(w)$, and so we can restrict the crystal structure on $\RF(w)$ to this subset and consider the resulting connected components.
These connected components were shown in~\cite{AS18} to be the crystals for $B$-representations called \defn{Demazure modules} that are constructed as ``partial'' versions of highest weight $\GL$-representations.
The characters of these \defn{$\gl_n$-Demazure crystals} are the so-called \defn{key polynomials}
$\key_{u\lambda} $; here $u \in S_n$ and  $\lambda$ is a partition with at most $n$ parts.

 
The precise definition of a key polynomial is $\key_{u\lambda}= \pi_u \Schubert_w$
where $\pi_u$ is an \defn{isobaric divided difference operator} 
and $w$ is a dominant permutation with $\lambda(w)=\lambda$.
For each choice of $\K \in \{\Or,\Sp\}$, there is an analogous notion of a \defn{$\K$-dominant} involution $z$.
These elements index the $\K$-Schubert polynomials that are
products of binomials $x_i+x_j$ indexed by positions in the associated Rothe diagram.
By considering all expressions of the form 
$\pi_u \Schubert_z^{\K}$ where $z$ is a $\K$-dominant involution in $S_n$,
we obtain a new family of objects that we refer to as \defn{$P$- and $Q$-key polynomials}, or collectively as \defn{shifted key polynomials}.


 
Using the fact that each $\Schubert_z^{\K}$ is an $\NN$-linear combination of Schubert polynomials, we can show that each shifted key polynomial is an $\NN$-linear combination of key polynomials (see \cref{positiv-thm}).
This suggests that shifted key polynomials may form a combinatorially interesting family.
Key polynomials are partial versions of Schur functions, since if $w_0 \in S_n$ denotes the reverse permutation then $s_{\lambda}(x_1,x_2,\dots,x_n) = \key_{w_0 \lambda}$.
Similarly, we show that $P$- and $Q$-key polynomials are partial versions of the Schur $P$- and $Q$-functions  related to the projective representation theory of $S_n$ (see \cref{thm:long_element}).


Classical key polynomials form a $\ZZ$-basis for all polynomials, are uniquely indexed by \defn{weak compositions},
and decompose every Schubert polynomial with positive coefficients.
Shifted key polynomials are not so well-behaved: they are not linearly independent over $\ZZ$, nor 
is it clear how to index them uniquely.
In spite of this, we conjecture (\cref{conj:K_Schubert_key}) that $\K$-Schubert polynomials also expand positively for some choice of shifted key polynomials.

In order to classify a good set of linear independent shifted key polynomials, we consider a certain ``truncated'' crystal structure on a set of bounded reduced factorizations associated to an involution $z$, analogous to constructions in ~\cite{AS18,MS16}.
However, in our case, the crystal will not be for $\gl_n$ but for the queer Lie superalgebra $\q_n$ and its extended version $\q_n^+$ recently introduced in~\cite{MT21}.
The full set of reduced factorizations relevant to $\K$-Schubert polynomials were given a $\q_n$/$\q_n^+$ crystal structure in \cite{Marberg22,MT21}.
We show in \cref{thm:dominant} that when this crystal structure is restricted to its bounded elements for a $\K$-dominant involution, we obtain a connected object whose character is a shifted key polynomial.
We conclude by describing a crystal-theoretic generalization of  our conjecture that 
$\K$-Schubert polynomials expand positively into shifted key polynomials.

 

This extended abstract is organized as follows.
\cref{sec:keys} gives some background on  key polynomials.
\cref{sec:shifted} contains our main results on shifted key polynomials.
We have omitted all proofs to save space. Complete arguments can be found in two full-length articles associated to this abstract, this first of which is available as \cite{MarScr}. 

 






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Key polynomials, Schubert calculus, and crystals}
\label{sec:keys}

Throughout, $n$ is a positive integer, $[n] = \{1,2,\dotsc,n\}$, $\NN = \{0,1,2,\dotsc\}$, and $\PP = \{1,2,3,\dotsc\}$.
Let $\xx = (x_1, x_2, \ldots)$ be  commuting indeterminates.

Define $S_\infty = \langle s_1,s_2,s_3,\dots\rangle$ to be the group permutations of $\PP$ fixing all but finitely many elements, with $s_i = (i \; i+1)$ denoting a simple transposition. Set $S_n =\langle s_i : i \in [n-1]\rangle \subset S_\infty$.
A \defn{reduced word} for $w \in S_{\infty}$ a minimal length sequence $i_1i_2 \dotsc i_{\ell}$ such that $w = s_{i_1}s_{i_2}\cdots s_{i_{\ell}}$, where $\ell(w) = \ell$ is the length of $w$.
The group $S_{\infty}$ acts on the polynomial ring $\ZZ[\xx]$ by permuting variables.
The \defn{Rothe diagram} of $w \in S_{\infty}$ is $D(w) = \{ (i, w(j)) \mid i < j \text{ and } w(i) > w(j) \} \subset \PP \times \PP$.

A \defn{word} is a possibly empty sequence of positive integers.
For $w \in S_{\infty}$, let $\RF(w)$ denote the set of sequences $a= (a^1,a^2,a^3, \dotsm) $ 
where each $a^i$ is a strictly decreasing word such that the concatenation $a^1a^2a^3\cdots$ is a reduced word for $w$. We refer to elements of this set as \defn{reduced factorizations}
and define $\RF_n(w)$ to be the set of such $a$ with $a^i$ empty for all $i>n$.
In examples we express elements of $\RF_n(w)$ as $n$-tuples rather than as infinite sequences. 
Let $\BRF_n(w)$ denote the set of reduced factorizations in $\RF_n(w)$ that are \defn{bounded} in the sense that $i \leq \min(a^i)$ for all nonempty $a^i$.
Set $\BRF(w) := \bigsqcup_{n=1}^{\infty} \BRF_n(w)$. %, which is a finite set.

A \defn{weak composition} is a nonnegative integer sequence $\alpha = (\alpha_i \in \NN)_{i=1}^{\infty}$ with finite sum $\abs{\alpha} := \sum_{i=1}^{\infty} \alpha_i$, and a \defn{partition} is a weakly decreasing weak composition.
We frequently omit the trailing $0$'s when writing weak compositions in examples.
Given a weak composition $\alpha$, let $\lambda(\alpha)$ be the partition sorting $\alpha$
and define $\xx^{\alpha} = x_1^{\alpha_1} x_2^{\alpha_2} \cdots$.
There is a unique $u(\alpha) \in S_{\infty}$ such that $u(\alpha) \alpha$ is a partition $\lambda(\alpha)$, where the permutation acts on $\alpha$ by permuting indices;
\textit{e.g.}, if $\alpha = 1021$, then $\lambda(\alpha) = 2110$ and $u(\alpha) = 3142 = s_2 s_1 s_3$.
 
For $i \in \PP$, let $\partial_i $ be the \defn{divided difference operator} on $f \in \ZZ[\xx]$ defined by
$ 
\partial_i f = (f - s_if) / (x_i-x_{i+1}).
$
The \defn{isobaric divided difference operators} are then given by $\pi_i f := \partial_i( x_i f)$ and $\overline{\pi}_i := \pi_i - 1$.
For $w \in S_{\infty}$ with reduced word $i_1 \dotsm i_{\ell}$, define $\pi_w = \pi_{i_1} \cdots \pi_{i_{\ell}}$ and $\overline{\pi}_w = \overline{\pi}_{i_1} \dotsm \overline{\pi}_{i_{\ell}}$; these formulas do not depend on the choice of reduced word.
The \defn{key polynomial} of a weak composition $\alpha$ is then $\key_{\alpha} := \pi_{u(\alpha)} \xx^{\lambda(\alpha)}$
while the \defn{atom polynomial} of $\alpha$ is $\overline{\key}_{\alpha} = \overline{\pi}_{u(\alpha)} \xx^{\lambda(\alpha)}$.
It is well-known that $\{\key_{\alpha} : \text{weak compositions }\alpha\}$ is a basis for $\ZZ[\xx]$ and that key polynomials are unitriangular 
with $\key_{\alpha} = \xx^{\alpha} + (\text{lower order terms})$ with respect to lexicographic order \cite[Cor.~7]{ReinerShimozono}.
Key polynomials are related to atom polynomials
by the identity $\key_{\alpha} = \sum_{\beta \leq \alpha} \overline{\key}_{\beta}$, where 
we write $\alpha \leq \beta$ if $\lambda(\alpha) = \lambda(\beta)$ and $u(\alpha) \leq u(\beta)$ in Bruhat order.
For more background on key polynomials, see  \cite{ReinerShimozono}.

A permutation $w \in S_{\infty}$ is \defn{dominant} if its Rothe diagram $D(w)$ is the Young diagram $D_\lambda =\{ (i,j) \in \PP\times \PP \mid j \leq \lambda_i\}$ of a partition $\lambda = \lambda(w)$. This occurs precisely when $w$ is $132$-avoiding~\cite[Ex.~2.2.2]{Manivel01}.
The \defn{Schubert polynomial} of $w \in S_\infty$ is defined recursively by setting $\Schubert_w = \xx^{\lambda(w)}$ when $w$ is dominant and 
requiring that $\Schubert_{w s_i} = \partial_i \Schubert_w$ for $w(i) > w(i+1)$~\cite{LS82}.
For any $w \in S_{\infty}$ the Billey--Jockusch--Stanley formula~\cite{BJS93} asserts that
\begin{equation}
\label{eq:BJS}
\Schubert_w = \sum_{a \in \BRF(w)} \xx^{\wt(a)}, \qquad \text{ where } \wt(a) := (\ell(a^1), \ell(a^2), \dotsc).
\end{equation}

Key polynomials can be defined in terms of Schubert polynomials,
since if $w \in S_\infty$ is dominant of shape $\lambda(\alpha)$ then $\key_{\alpha} = \pi_{u(\alpha)} \Schubert_w$.
On the other hand, every Schubert polynomial expands as a positive linear combination of key polynomials with an explicit combinatorial description~\cite[Thm.~4]{ReinerShimozono}.
 

 
The BJS formula \eqref{eq:BJS} can be interpreted as a character formula for certain \defn{Demazure crystals} which we describe below.
A \defn{crystal}~\cite{K91} for $\gl_n$ is a set $\mcB$ with \defn{crystal operators} $e_i, f_i \colon \mcB \to \mcB \sqcup \{\zero\}$ for $i \in [n-1]$ and a \defn{weight function} $\wt \colon \mcB \to \ZZ^n$ that satisfy certain conditions.
We can encode this data as a weighted directed graph called a \defn{crystal graph} with vertices $\mcB$ and edges $b \xrightarrow[\hspace{15pt}]{i}f_ib$
whenever $b \in \mcB$ and $ f_i b\neq 0$.
For each $w \in S_\infty$, the set $\RF_n(w)$ already has a weight function as used in \eqref{eq:BJS}.
Morse and Schilling~\cite{MS16} identified a natural $\gl_n$-crystal structure on $\RF_n(w)$,
using a certain bracketing rule 
 to describe the crystal operators.
See \cite[\S10]{BS17} for more information on these crystals.
 

Suppose $w \in S_\infty$ is dominant of shape $\lambda=\lambda(w)$. Assume $\lambda$ has at most $n$ nonzero parts.
Then  $\RF_n(w)$ contains a single bounded reduced factorization $b_\lambda \in \BRF_n(w)$.
This element has weight $\lambda$ and is \defn{highest weight} 
in the sense that $e_i b_\lambda =0$ for all $i \in [n-1]$.   
If $\alpha$ is a weak composition with $\lambda = \lambda(\alpha)$ and $u(\alpha) \in S_n$ then we define
\begin{equation}
\label{eq:Demazure_crystal}
\Dem_n(\alpha) := \left\{ a \in \RF_n(w) \mid \begin{aligned} 
& e_{i_1}^{m_1}e_{i_2}^{m_2} \dotsm e_{i_{\ell}}^{m_{\ell}} a = b_{\lambda} 
\text{ for some reduced word} \\ 
&\text{ $i_1i_2\cdots i_\ell$ of $u(\alpha)$ and some $m_1,m_2,\dotsc,m_\ell \in \NN$}
\end{aligned}
 \right\}.
\end{equation}
We refer to this subset as a \defn{$\gl_n$-Demazure crystal}.
We identify it with the (connected) subgraph induced from the crystal graph of $\RF_n(w)$.
%
See \cref{fig:factorization_crystal} for an example.
The \defn{character} of any finite subset $\mathcal{X}$ of a crystal $\mcB$ is $\ch(\mathcal{X}) := \sum_{b \in \mathcal{X}} \xx^{\wt(b)} \in \ZZ[\xx]$.
\begin{thm}[See \cite{BS17}]
\label{thm:Schubert_Demazure_crystal0}
If $\alpha$ is a weak composition with $u(\alpha) \in S_n$ 
such that $\lambda(\alpha)$ has at most $n$ parts
then $\ch(\Dem_n(\alpha) ) = \kappa_\alpha$.
\end{thm}

On the other hand, Assaf and Schilling \cite{AS18} have shown the following:

\begin{thm}[{\cite{AS18}}]
\label{thm:Schubert_Demazure_crystal}
For any $w \in S_\infty$, the set $\BRF_n(w)$ is a disjoint union of $\gl_n$-Demazure crystals, 
in the sense that there is a weight-preserving isomorphism from each connected component of the subgraph of the crystal graph of $\RF_n(w)$ induced on $\BRF_n(w)$
 to $\Dem_n(\alpha)$ for some $\alpha$.
\end{thm}

Since $\ch(\BRF_n(w)) = \Schubert_w$ if $n$ is sufficiently large,
taking characters in this theorem recovers the nontrivial fact noted above that every Schubert polynomial expands as a positive linear combination of key polynomials~\cite[Thm.~4]{ReinerShimozono}.
 
\begin{figure}
\[
\begin{tikzpicture}[xscale=3,yscale=1,>=latex,baseline=0,every node/.style={scale=0.8}]
\pgfsetlinewidth{1bp}
\definecolor{fillcol}{rgb}{0.95,0.95,0.95};
\node[draw=black,line width=.5pt] (hw) at (0, 0) {$21/3/\cdot$}; %{$(s_2 s_1)(s_3)()$};
\node[draw=black,line width=.5pt] (1) at (1, 1) {$21/\cdot/3$}; %{$(s_2 s_1)()(s_3)$};
\node[fill=fillcol] (2) at (1, -1) {$2/31/\cdot$}; %{$(s_2)(s_3 s_1)()$};
\node[fill=fillcol] (12) at (2, 1) {$2/1/3$}; %{$(s_2)(s_1)(s_3)$};
\node[fill=fillcol] (21) at (2, -1) {$2/3/1$}; %{$(s_2)(s_3)(s_1)$};
\node[fill=fillcol] (211) at (3, -1) {$2/\cdot/31$}; %{$(s_2)()(s_3 s_1)$};
\node[fill=fillcol] (122) at (3, 1) {$\cdot/21/3$}; %{$()(s_2 s_1)(s_3)$};
\node[fill=fillcol] (lw) at (4, 0) {$\cdot/2/31$}; %{$()(s_2)(s_3 s_1)$};
\draw[->,color=blue,solid] (hw) -- node[above] {\scriptsize $2$} (1);
\draw[->,color=red,solid] (hw) -- node[below] {\scriptsize $1$} (2);
\draw[->,color=blue,solid] (2) -- node[below] {\scriptsize $2$} (21);
\draw[->,color=red,solid] (1) -- node[above] {\scriptsize $1$} (12);
\draw[->,color=blue,solid] (21) -- node[below] {\scriptsize $2$} (211);
\draw[->,color=red,solid] (12) -- node[above] {\scriptsize $1$} (122);
\draw[->,color=blue,solid] (122) -- node[above] {\scriptsize $2$} (lw);
\draw[->,color=red,solid] (211) -- node[below] {\scriptsize $1$} (lw);
\end{tikzpicture}
\]
\caption{For the dominant $w = 3142 = s_2 s_1 s_3 \in S_{\infty}$ of shape  $\lambda(w) = (2, 1, 0)$, the $\gl_3$-crystal $\RF_3(w)$.
The unique reduced factorization is $b_{\lambda(w)} = (21, 3, \varnothing)$. The elements in the $\gl_n$-Demazure crystal $\Dem_3(\alpha)$ for $\alpha=(2,0,1)$, which are all in $\BRF(w)$, are boxed.}
\label{fig:factorization_crystal}
\end{figure}









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Shifted key polynomials}
\label{sec:shifted}

In this section, we introduce two shifted analogues of key and atom polynomials.  
We then present our main results about these polynomials and state a number of conjectures.

A partition $\lambda$ is \defn{strict} if its nonzero parts are all distinct; alternatively, if $\lambda = (\lambda_1 > \cdots > \lambda_{\ell} > 0)$.
We say $\lambda$ is \defn{symmetric} if $\lambda^\top = \lambda$, where $\lambda^\top$ is the conjugate shape.
A partition $\lambda$ is \defn{skew-symmetric} if $\lambda^\top = \lambda$ and if $i$ maximal such that $(i, i) \in D_\lambda $, then we cannot add or remove the box $(i, i+1)$ from $D_\lambda$ and still have the diagram of a partition.
When $\lambda$ is symmetric, we define its \defn{shifted diagram} to be $S(\lambda) = \{(i,j) \in D_\lambda  \mid i \leq j\}$ and  \defn{strict shifted diagram} to be $\widehat{S}(\lambda) = \{(i,j) \in D_\lambda  \mid i < j\}$.

Let $H(\lambda)$ (resp.\ $\widehat{H}(\lambda)$) be the \defn{(strict) half diagram} formed by sliding all boxes to the left of $S(\lambda)$ (resp.~$\widehat{S}(\lambda)$). This is the diagram of the strict partition $\lambda^H$ (resp.~$\lambda^{\widehat{H}}$) whose parts count the number of boxes in the distinct rows of $S(\lambda)$ (resp.~$\widehat{S}(\lambda)$).
The map $\lambda \mapsto \lambda^H$ (resp.~$\lambda \mapsto \lambda^{\widehat{H}}$)
is a bijection from symmetric (resp.~skew-symmetric) partitions to strict partitions.
We say a weak composition $\alpha$ is \defn{(skew-)symmetric} if $\lambda(\alpha)$ is (skew-)symmetric.

\begin{dfn}\label{spo-key-def}
Let $\alpha$ be a symmetric weak composition, and set $\lambda = \lambda(\alpha)$. Define
\[ 
\qkey_{\alpha} = \pi_{u(\alpha)} \left(\prod_{(i,j) \in S(\lambda)} (x_i + x_{j})\right)
\quand
\qatom_{\alpha} = \overline{\pi}_{u(\alpha)} \left(\prod_{(i,j) \in S(\lambda)} (x_i + x_{j})\right).
\]
We refer to these functions as \defn{$Q$-key polynomials} and \defn{$Q$-atom polynomials}.
Similarly, when $\alpha$ is skew-symmetric we define
\[ 
\pkey_{\alpha} = \pi_{u(\alpha)} \left(\prod_{(i,j) \in \widehat{S}(\lambda)} (x_i + x_{j})\right)
\quand
\patom_{\alpha} = \overline{\pi}_{u(\alpha)} \left(\prod_{(i,j) \in \widehat{S}(\lambda)} (x_i + x_{j})\right).
\]
We refer to these functions as \defn{$P$-key polynomials} and \defn{$P$-atom polynomials}.
\end{dfn}

The definitions of $\pkey_{\alpha}$ and $\patom_{\alpha}$ make sense if $\alpha$ is symmetric
but not skew-symmetric, but in this case there is always a skew-symmetric $\beta$
with $\pkey_{\alpha}=\pkey_\beta$ and $\patom_{\alpha}=\patom_\beta$. 

\begin{ex}
\label{ex:shifted_keys}
If $\alpha = 3143$ then $\lambda(\alpha) = 4211$ is skew-symmetric with $\lambda^{\widehat{H}} = 3100$ and $u(\alpha) = 3142 = s_2s_1s_3$,
so we have
\begin{align*}
\pkey_{3143} &= \pi_2\pi_1\pi_3 \bigl( (x_1+x_2)(x_1+x_3)(x_1+x_4)(x_2+x_3) \bigr)
 = \key_{0022} + \key_{0031} + \key_{0112},
\allowdisplaybreaks \\
\patom_{3143} &= \overline{\pi}_2 \overline{\pi}_1 \overline{\pi}_3 \bigl( (x_1+x_2)(x_1+x_3)(x_1+x_4)(x_2+x_3) \bigr) = \overline{\key}_{0022} + \overline{\key}_{0031}.
 \end{align*}
If $\alpha = 2031$ then $\lambda(\alpha) = 3210$ is symmetric with $\lambda(\alpha)^H = 3100$ and  $u(\alpha)   = s_2s_1s_3$,
so  
\begin{align*}
 \qkey_{2031} &= \pi_2\pi_1\pi_3 \bigl( 4x_1x_2(x_1+x_2)(x_1+x_3) \bigr)
 = 4 \key_{103} + 4 \key_{202} + 4 \key_{1021},
\allowdisplaybreaks \\
\qatom_{2031} &= \overline{\pi}_2 \overline{\pi}_1 \overline{\pi}_3 \bigl( 4x_1x_2(x_1+x_2)(x_1+x_3) \bigr) = 0.
 \end{align*}
\end{ex}
 
 We refer to $\pkey_\alpha$ and $\qkey_\alpha$ collectively as \defn{shifted key polynomials},
 and to $\patom_\alpha$ and $\qatom_\alpha$ as \defn{shifted atom polynomials}.
Shifted atom polynomials are related to shifted key polynomials via the \defn{Bruhat order} $\leq$ on $S_\infty$.
Recall that $\beta \leq \alpha$ if $\lambda(\beta) = \lambda(\alpha)$ and $u(\beta) \leq u(\alpha)$.

\begin{prop}
We have
$
\pkey_{\alpha} = \sum_{\beta \leq \alpha} \patom_{\beta}
$
and
$
\qkey_{\alpha} = \sum_{\beta \leq \alpha} \qatom_{\beta}.
$
Moreover, $\qkey_{\alpha}$ and $\qatom_{\alpha}$ are divisible by $2^{\ell}$, where $\ell$ is the length of $\lambda(\alpha)^H$.
\end{prop}

 

Our first substantial result about shifted key and atom polynomials is the following.

\begin{thm}\label{positiv-thm}
Let $\alpha$ be a symmetric composition.
Then $\pkey_{\alpha}$ and  $\qkey_{\alpha}$ (resp.~$\patom_{\alpha}$ and $\qatom_\alpha$) are linear combinations of key (resp.\ atom) polynomials with nonnegative integer coefficients.
Consequently, the polynomials $\pkey_{\alpha}$, $\patom_\alpha$, $\qkey_\alpha$, and $\qatom_\alpha$ are all in $ \NN[\xx]$.
\end{thm}

 

Key polynomials are partial Schur functions in 
the sense that if $\lambda=(\lambda_1\geq \lambda_2\geq \dots \geq 0)$ is a partition with at most $n$ nonzero parts then 
$\kappa_{\alpha} = s_\lambda(x_1,x_2,\dots,x_n)$ for $\alpha = (\lambda_n,\dots,\lambda_2,\lambda_1)$ \cite[\S2]{ReinerShimozono}.
Analogously, we can prove that shifted key polynomials are partial \defn{Schur $P$/$Q$-functions} (see, \cite[\S III.8]{Macdonald15} for background on these functions):
 

\begin{thm}
\label{thm:long_element}
If $\lambda=(\lambda_1\geq \lambda_2\geq \dots \geq 0)$ is a symmetric partition with $\lambda_1\leq n$
then
\[\pkey_\alpha= P_{\mu}(x_1,x_2,\dots,x_n)
\quand
\qkey_\alpha= Q_{\nu}(x_1,x_2,\dots,x_n)\]
 for $\alpha = (\lambda_n,\dots,\lambda_2,\lambda_1,0,0,\dots)$, $\mu= \lambda^{\widehat{H}}$, and $\nu = \lambda^{H}$.
\end{thm}

We have conjectural formulas for the leading terms of $\pkey_\alpha$ and $\qkey_\alpha$.
Assume $\alpha$ is a symmetric composition with $\lambda = \lambda(\alpha)$ and $u=u(\alpha)$.
Define $D(\alpha) = \{ (u(i), u(j)) : (i,j) \in D_\lambda\}$. 
Let 
$\rho(\alpha)=(\rho_1,\rho_2,\dots)$ and $\theta(\alpha)=(\theta_1,\theta_2,\dots)$
where $\rho_i =| \{ (a,b) \in D(\alpha) \mid i = a\geq b\}|$ and $\theta_i = | \{ (a,b) \in D(\alpha) \mid  a\geq b=i\}|$.
Also define
$\widetilde\rho(\alpha)=(\widetilde\rho_1,\widetilde\rho_2,\dots)$ and $\widetilde\theta(\alpha)=(\widetilde\theta_1,\widetilde\theta_2,\dots)$
where $\widetilde\rho_i = |\{ (a,b) \in D(\alpha) \mid i = a> b\}|$ and $\widetilde\theta_j = |\{ (a,b) \in D(\alpha) \mid  a> b=i\}|$. 
These are the row/column counts of  $D(\alpha)$ below the main diagonal.

 

\begin{conj}
Suppose $\alpha$ and $\beta$ are symmetric compositions with $\beta$ skew-symmetric. Then
\[
\qkey_{\alpha} \in 2^{\ell(\lambda(\alpha)^H)} \left( \xx^{\rho(\alpha)} + \xx^{\theta(\alpha)} + \sum_{ \gamma \neq \rho(\alpha)} \NN \xx^{\gamma} \right)
\quand
\pkey_{\beta} \in \xx^{\widetilde\rho(\beta)} + \xx^{\widetilde\theta(\beta)} + \sum_{ \gamma\neq \widetilde\rho(\beta)} \NN \xx^{\gamma}.
\]
Moreover, $\xx^{\rho(\alpha)} $ and $\xx^{\widetilde\rho(\beta)}$ are the leading terms of 
$\qkey_{\alpha} $ and $\pkey_{\beta} $ in lexicographic order.
\end{conj}

We can prove that if $\alpha$ and $\beta$ are as above then
$D(\alpha)$ (hence, also $\alpha$) is uniquely determined by $\rho(\alpha)$ and $\theta(\alpha)$,
while  $D(\beta)$ (hence, also $\beta$) is uniquely determined by $\widetilde\rho(\beta)$ and $\widetilde\theta(\beta)$.
This does not hold for general symmetric subsets of $\PP \times \PP$.
%
Shifted key/atom polynomials are not as well-behaved as their classical analogues in a few other ways:
\begin{itemize}
\item Shifted atom polynomials are zero for some indices $\alpha$. They can also coincide for different indices.
For example, $\qatom_{30023} = \qatom_{21014} \neq 0$ and $\patom_{402402} = \patom_{313501} \neq 0$.

 
\item $P$-key polynomials are not uniquely indexed by skew-symmetric compositions: for example,
$
\pkey_{4313} =  \pkey_{4133} \neq 0.
$

\item However, we have not yet been able to find a pair of distinct symmetric compositions $\alpha \neq \beta$ such that $\qkey_{\alpha} = \qkey_{\beta}$.
It is possible that the $Q$-key polynomials are uniquely indexed by symmetric compositions.

\item Even if this is the case,
the $Q$-key polynomials are still not linearly independent. For example, we have
$
\qkey_{123} + \qkey_{0321} = \qkey_{132} + \qkey_{0231}.
$
\end{itemize}

Shifted key polynomials are closely related to certain ``orthogonal'' and ``symplectic'' versions of type A Schubert polynomials.
Let $I_\infty^\Or = \{ z \in S_\infty : z=z^{-1}\}$ and let $I_\infty^\Sp$ be the $S_\infty$-conjugacy class of
$1_\fpf  = (1\; 2)(3\; 4)(5\; 6)\cdots $.
If $\lambda$ is a symmetric partition, then the unique dominant element of $S_\infty$ of shape $\lambda$ already
belongs to $I_\infty^\Or$.
If $\lambda$ is a skew-symmetric partition,
then there is a unique $z \in I_\infty^\Sp $ 
with $\{ (i,j) \in D(z) : i \neq j\} = \{ (i,j) \in D_\lambda : i\neq j\}$,
which we call
 the \defn{dominant} element of $ I_\infty^\Sp $ with shape $ \lambda$.

Let $\K \in \{\Sp, \Or\}$.
By results in \cite{WY17}, there are unique polynomials $\{\Schubert^\K_z\}_{z \in I^\K_\infty}$  
with
$\Schubert^\K_z = \pkey_\lambda$ when $\K=\Sp$ (resp.~$\Schubert^\K_z = \qkey_\lambda$ when $\K=\Or$)
 and $z \in I^\K_\infty$ is dominant of shape $\lambda$,
and which satisfy 
\[
  \partial_i \fkSS_{z} = \begin{cases} 0&\text{if }z(i) < z(i+1),
\\
0&\text{if } z(i)=i+1, \\ 
  \fkSS_{s_i z s_i} &\text{otherwise,}
  \end{cases}
\quand
  \partial_i \fkSO_{z} = \begin{cases} 0&\text{if }z(i) < z(i+1), \\ 
 2 \fkSO_{zs_i} &\text{if }z(i) =i+1,\\
  \fkSO_{s_i z s_i} &\text{otherwise,}
    \end{cases}
    \]
for all $z \in I^\K_\infty$ and $i \in \PP$.
We refer to the
$\Schubert^\K_z$'s as
 \defn{$\K$-Schubert polynomials}.
 These elements,  called \defn{involution Schubert polynomials} in  \cite{HMP18,HMP19,HMP20},
represent cohomology classes of 
the closures of the $\Sp$- and $\Or$-orbits in the complete flag variety \cite{WY17}.
The following conjecture is one of our primary motivations for studying 
shifted key polynomials:

\begin{conj}
\label{conj:K_Schubert_key}
Each polynomial $\Schubert^\Sp_z$ for $z \in I^\Sp_\infty$
(resp.~$\Schubert^\Or_z$ for $z \in I^\Or_\infty$)
 is an $\NN$-linear combination of $P$-key polynomials
   (resp.~$Q$-key polynomials).
\end{conj}

This conjecture is supported by many computational examples and closely parallels the classical case.
Below, we will  outline a shifted analogue of  \cref{thm:Schubert_Demazure_crystal0} that also provides
some heuristic support for the conjecture.
 

Like ordinary Schubert polynomials, $\K$-Schubert polynomials can be expressed via a BJS-type formula
as $\Schubert^\K_z = \sum_{a \in \BRF^\K(z)} \xx^{\wt(a)}$
for an analogue  $\BRF^\K(z)$ of the set $\BRF(z)$ \cite{HMP18}.
For each choice of $\K$, $\BRF^\K(z)$ consists of the bounded elements in a larger set of \defn{$\K$-reduced factorizations} $\RF^\K(z)$.
If $\K=\Sp$ then $\RF^\K(z)$ is explicitly given as the disjoint union of $\RF(w)$
over all minimal length $w \in S_\infty$ with $z=w^{-1} 1_\fpf w$.
The definition of $\RF^\Or(z)$ for $z \in I^\Or_\infty$ is more involved:
this is formed by taking another disjoint union of sets $\RF(w)$ for certain $w \in S_\infty$,
and then optionally annotating some letters in each reduced factorization by primes; see \cite{MT21} for the precise details.

 Define $\BRF_n^\K(z)\subseteq \RF_n^\K(z)$ to be the respective subsets of $\BRF^\K(z)\subseteq \RF^\K(z)$ 
 consisting of the tuples $a=(a^1,a^2,\dots)$ with $a^i$ empty for $i>n$.
Like $\RF_n(w)$, the sets  $ \RF_n^\K(z)$ have natural crystal structures.
Results in \cite{Marberg22}
identify
 a crystal structure on $\RF_n^{\Sp}(z)$ corresponding to the \defn{queer Lie superalgebra} $\q_n$ (as described axiomatically in \cite{GJKKK14}), extending the $\gl_n$ crystal in \cite{MS16}.
The set $\RF_n^{\Or}(z)$ similarly is the prototypical example of what is called an \defn{extended queer supercrystal}
or \defn{$\q_n^+$-crystal} in~\cite{MT21}.

\begin{figure}
\begin{center}
\input{q3-decr-crystal.tex}
\end{center}
\caption{The $\q_4$-crystal on $\RF^{\Sp}(z)$ corresponding to $\pkey_{\lambda}$
for $z =  s_1s_2s_4 \cdot 1_\fpf\cdot s_4s_2s_1$ and $\lambda= (4,1,1,1)$.
The boxed elements are in $\BRF^{\Sp}(z)$.
Solid blue, red, and green arrows indicate $1$-, $2$-, and $3$-edges, respectively, while dashed blue arrows are $\bon$-edges.
}
\label{fig:Sp_crystal}
\end{figure}

Crystals for $\q_n$ are $\gl_n$ crystals with \defn{odd crystal operators} $e_{\overline{\imath}}, f_{\overline{\imath}}$ for $\overline{\imath} \in [\overline{n-1}] := \{ \bon, \dotsc, \overline{n-1}\}$ satisfying certain axioms.
It is sufficient to define $e_{\bon}, f_{\bon}$ as the other odd crystal operators come from inductively twisting the root system~\cite[Lemma~2.2]{GHPS20}:
\begin{equation}
\label{eq:bar_induction}
e_{\overline{\imath}} = s_{i-1} s_i e_{\overline{\imath-1}} s_i s_{i-1}
\quand
f_{\overline{\imath}} = s_{i-1} s_i f_{\overline{\imath-1}} s_i s_{i-1}.
\end{equation}
Here $s_i$ acts as the crystal operator that reverses each \defn{$i$-string}.
We remark that $e_{\bon}$ is not defined by a usual bracketing rule but by a weight condition.
For $\q_n^+$ crystals, there are additional crystal operators $e_0, f_0$.
Similar to $\gl_n$ crystals, we encode $\q_n$/$\q_n^+$ crystals as crystal graphs.
In view of \eqref{eq:bar_induction}, we omit $\overline{2}, \overline{3}, \ldots$ when drawing the crystal graphs.

 

Now suppose $z \in I^\K_\infty$ is dominant with (skew-)symmetric shape $\lambda$ having $\lambda_1 \leq n$. In general,
the set 
  $\BRF^\K_n(z)$ contains more than one element in $ \RF^\K_n(z)$.
  Nevertheless, there is an interesting shifted analogue of the $\gl_n$-Demazure crystal \eqref{eq:Demazure_crystal}.

Fix a (skew-)symmetric composition $\alpha$ with $\lambda = \lambda(\alpha)$ and $u(\alpha) \in S_n$.  Then define
\[
\Dem^\K_n(\alpha) := \left\{ a \in \RF^\K_n(z) \mid \begin{aligned} 
& e_{i_1}^{m_1}e_{i_2}^{m_2} \dotsm e_{i_{\ell}}^{m_{\ell}} a  \in \BRF^\K_n(z)
\text{ for some reduced word} \\ 
&\text{ $i_1i_2\cdots i_\ell$ for $u(\alpha)$ and some $m_1,m_2,\dots,m_\ell \in \NN$}
\end{aligned}
 \right\}.
\]
We refer to this subset as a \defn{$\K$-Demazure crystal}; see \cref{fig:Sp_crystal,fig:Or_crystal} for examples.
We identify $\Dem^\K_n(\alpha)$ with the subgraph that it induces in 
the \defn{extended $\q_n$/$\q_n^+$-crystal graph} of $\RF^\K_n(z)$ (formed by drawing arrows corresponding to all
even and odd crystal operators).
This directed subgraph is connected, although this is not at all obvious:
 
\begin{figure}
\begin{center}
\input{qq3-decr-crystal.tex}
\end{center}
\caption{The $\q_3^+$-crystal on $\RF^{\Or}(z)$ corresponding to $\qkey_{\lambda}$ for $z=(1 \; 3)(2 \; 4)$ and $\lambda = (3,3,3)$.
The boxed elements are in $\BRF^{\Or}(z)$. Solid blue, solid red, dotted green, and dashed blue arrows are $i$-edges for $i = 1, 2, 0, \bon$, respectively.
}
\label{fig:Or_crystal}
\end{figure}

\begin{thm}
\label{thm:dominant}
Assume $u(\alpha) \in S_n$ 
and $\lambda(\alpha)$ has at most $n$ parts.
Then the $\K$-Demazure crystal $\Dem^\K_n(\alpha) $ is connected with character equal to $\pkey_\alpha$ when $\K=\Sp$ 
 and to $\qkey_\alpha$ when $\K=\Or$. 
\end{thm}


Our second main conjecture is the following analogue of \cref{thm:Schubert_Demazure_crystal}:

\begin{conj}\label{conj-last}
If $z \in I^\K_\infty$ then $\BRF^\K_n(z)$ is a disjoint union of $\K$-Demazure crystals,
in the sense that there is a weight-preserving isomorphism from each connected component of the subgraph of the extended crystal graph of $\RF^\K_n(z)$ induced on $\BRF^\K_n(z)$
 to $\Dem^\K_n(\alpha)$ for some  $\alpha$.
\end{conj}

As $\ch(\BRF^\K_n(z)) = \Schubert^\K_z$ if $n\gg 0$, \cref{conj:K_Schubert_key}
would follow from 
this conjecture on taking characters. This conjecture is again supported by extensive computer calculations.
\Cref{conj:K_Schubert_key,conj-last}
also have more refined versions involving $\K$-reduced factorizations that are bounded by arbitrary \defn{flags},
which would generalize \cite[Thm. 21]{ReinerShimozono}.


 





 
An interesting discrepancy with the classical case is that the subsets $\Dem^\K_n(\alpha)$ are not closed under $e_{\overline{\imath}}$ for all $\overline{\imath} \in [\overline{n-1}]$.
Contrast this with the situation for $\gl_n$-Demazure crystals:
if $b_{\beta}$ is the unique element of weight $\beta \leq \alpha$ in a $\gl_n$-Demazure crystal,
then while we cannot obtain all elements from $b_{\alpha}$ through applying $\{ e_i  \mid  i \in [n-1] \}$, we can obtain everything by applying these operators to the closure of $\{b_{\beta} \mid \beta \leq \alpha\}$.

\begin{ex}
Consider the $\q_4$-crystal $\RF^{\Sp}(z)$  
in \cref{fig:Sp_crystal}.
For  $b = (2/3/4/\cdot) \in \BRF^{\Sp}(z)$, we have
$e_{\overline{2}}(2/3/4/\cdot)  = s_1 s_2 e_{\bon} s_2 s_1(2/3/4/\cdot) = s_1 s_2 e_{\bon} (2/3/4/\cdot) = s_1 s_2 (21/ \cdot/4/\cdot) = s_1 (21/4/\cdot/\cdot) = (2/41/\cdot/\cdot) \notin \BRF^{\Sp}(z).$
\end{ex}
 




\acknowledgements{
The authors thank T.~Matsumura, A.~Schilling, and A.~Yong for useful discussions.
}

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