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\title{Combinatorial Howe duality of symplectic type}

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%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Taehyeok Heo\thanks{\href{mailto:gjxogur123@snu.ac.kr}{gjxogur123@snu.ac.kr}}\addressmark{1} \and Jae-Hoon Kwon\thanks{\href{mailto:jaehoonkw@snu.ac.kr}{jaehoonkw@snu.ac.kr}}\addressmark{1,2}}

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\address{\addressmark{1}Department of Mathematical Sciences, Seoul National University, Seoul, Korea\\
\addressmark{2}Research Institute of Mathematics, Seoul National University, Seoul, Korea}

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\received{\today}

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\abstract{We give a symplectic analog of the RSK algorithm for Howe dual pairs of the form $(\mathfrak{g}, \textrm{Sp}_{2\ell})$, where $\mathfrak{g}$ is a Lie (super)algebra of classical type. We introduce an analog of jeu de taquin sliding for spinor model of irreducible characters of a Lie superalgebra $\mathfrak{g}$ to define $P$-tableau, and then define the associated $Q$-tableau in terms of a symplectic tableau due to King.}

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\keywords{crystal graphs, Howe duality, RSK correspondence, jeu de taquin}

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\begin{document}

\maketitle
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\section{Introduction}

Let $\mathscr{P}$ be the set of partitions or Young diagrams $\lambda=(\lambda_1,\lambda_2,\ldots)$
and, for $n\geq 1$, let $\mathscr{P}_{n}=\{\,\lambda \in \mathscr{P}\,|\,\ell(\lambda)\leq n\,\}$, 
where $\ell(\lambda)$ is the length of $\lambda$. 

Let $\mathcal{A}$ be a $\mathbb{Z}_2$-graded linearly ordered set and let $\mathscr{E}_\mathcal{A}$ be the exterior algebra generated by the superspace with a linear basis indexed by $\mathcal{A}$. 
Then $\mathscr{F}_\mathcal{A}=\mathscr{E}_\mathcal{A}^\ast \otimes \mathscr{E}_\mathcal{A}$ is a semisimple module over a classical Lie (super)algebra $\mathfrak{g}_\mathcal{A}$, the type of which depends on $\mathcal{A}$, 
%
and the $\ell$-fold tensor power $\mathscr{F}^{\otimes \ell}_\mathcal{A}$ ($\ell\geq 1$) is a $(\mathfrak{g}_\mathcal{A}, {\rm Sp}_{2\ell})$-module with the following multiplicity-free decomposition:
\begin{equation}\label{eq:Howe duality type C}
\mathscr{F}^{\otimes \ell}_\mathcal{A} \cong \bigoplus_{(\lambda, \ell) \in \mathscr{P}({\rm Sp})_\mathcal{A}} V_{\mathfrak{g}_\mathcal{A}}(\lambda,\ell)\otimes V_{{\rm Sp}_{2\ell}}(\lambda),
\end{equation}
where the direct sum is over a set $\mathscr{P}({\rm Sp})_\mathcal{A}$ of pairs $(\lambda, \ell) \in \mathscr{P} \times \mathbb{Z}_+$ with $\ell(\lambda)\leq \ell$ (see \cite{CKW,Howe95, Howe89,HTW,LZ,Wang99} for various choices of $\mathcal{A}$).
Here $V_{{\rm Sp}_{2\ell}}(\lambda)$ is the irreducible ${\rm Sp}_{2\ell}$-module corresponding to $\lambda$, and $V_{\mathfrak{g}_\mathcal{A}}(\lambda,\ell)$ is the irreducible highest weight $\mathfrak{g}$-module corresponding to $V_{{\rm Sp}_{2\ell}}(\lambda)$ appearing in $\mathscr{F}^{\otimes \ell}_\mathcal{A}$. This decomposition is obtained from a more general principle called Howe duality \cite{Howe89}.

There exists a combinatorial object called a spinor model of type $C$, which gives the character of $V_{\mathfrak{g}_\mathcal{A}}(\lambda,\ell)$ in \eqref{eq:Howe duality type C} in a uniform way \cite{Kwon15}. As a set, the spinor model $\mathbf{T}_\mathcal{A}(\lambda,\ell)$ consists of sequences of usual semistandard tableaux of two-columned shapes with letters in $\mathcal{A}$, where two adjacent tableaux satisfy certain configuration.
%

In this extended abstract, we give an analog of RSK algorithm for \eqref{eq:Howe duality type C} in terms of spinor model. More precisely, we construct an explicit bijection
\begin{equation}\label{eq:main result}
\xymatrixcolsep{4pc}\xymatrixrowsep{0pc}\xymatrix{
\mathbf{F}_\mathcal{A}^{\ell} 
\ar@{->}[r] & \ 
\displaystyle{\bigsqcup_{\lambda \in \mathscr{P}({\rm Sp})_\mathcal{A}} \mathbf{T}_\mathcal{A}(\lambda,\ell)\times \mathbf{K}(\lambda,\ell)}
},
\end{equation}
where $\mathbf{F}_\mathcal{A}^{\ell}$ is the set of $2\ell$-tuple of $\mathcal{A}$-semistandard tableaux of single-columned shapes with letters in $\mathcal{A}$, and $\mathbf{K}(\lambda, \ell)$ is the set of symplectic tableaux of shape $\lambda$ due to King \cite{King76} giving the character of $V_{{\rm Sp}_{2\ell}}(\lambda)$. The bijection \eqref{eq:main result} yields the Cauchy type identity which follows from the decomposition \eqref{eq:Howe duality type C} for arbitrary $\mathcal{A}$, and it recovers well-known identities \cite{King75,Li77,Weyl46} when $\mathcal{A}$ is a finite set of homogeneous degree.
A full version of this paper including detailed proofs has appeared in \cite{HK}.

%To construct a bijection \eqref{eq:main result}, we first introduce an analog of jeu de taquin sliding for spinor model of a skew shape, which plays a crucial role in this paper. 
%A key observation to generalize the symplectic sliding algorithm to a spinor model with respect to arbitrary $\mathcal{A}$ is that in terms of spinor model, the Sheats' algorithm can be described as a sequence of Kashiwara operators with respect to an $\mathfrak{sl}_{2\ell}$-crystal structure on ${\bf F}_\mathcal{A}^{\ell}$ defined by the jeu de taquin sliding of type $A$ \cite{La03}. 
%This implies that the sliding does {\em not} depend on the choice of $\mathcal{A}$, and enables us to define a symplectic analog of RSK in a uniform way.
%
%Then we show that for ${\bf T}\in {\bf F}_\mathcal{A}^{\ell}$, there exists a unique tableau ${\tt P}({\bf T})\in {\bf T}_\mathcal{A}(\lambda,\ell)$ which can be obtained by applying our jeu de taquin sliding. We also define the recording tableau corresponding to ${\tt P}({\bf T})$, which is a sequence of oscillating tableaux, and then show that it corresponds bijectively to a tableau ${\tt Q}({\bf T})$ in ${\bf K}(\lambda,\ell)$ thanks to a recent result by Lee \cite{Lee19}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Preliminaries}
\subsection{Notations}

Let $\mathbb{Z}_+$ denote the set of non-negative integers.
We denote by $\lambda'=(\lambda'_1,\lambda'_2,\ldots)$ the conjugate of $\lambda$.
Let $\mathcal{A}$ be a linearly ordered set with a $\mathbb{Z}_2$-grading $\mathcal{A}=\mathcal{A}_0 \sqcup \mathcal{A}_1$.
For example, for a positive integer $n$, let 
\[ [n] = \{ 1 < 2 < \cdots < n \}, \quad [\overline{n}] = \{ \overline{n} < \overline{n-1} < \cdots < \overline{1} \}, \]
where we assume that all the entries are assumed to be of degree $0$.
For positive integers $m$ and $n$, let
$\mathbb{I}_{m|n} = \{ 1 < 2 < \cdots < m < 1' < 2' < \cdots < n' \}$
with $(\mathbb{I}_{m|n})_0 = \{ 1 < \cdots < m \}$ and $(\mathbb{I}_{m|n})_1 = \{ 1' < \cdots < n' \}$.



For a skew Young diagram
$\lambda/\mu$, let $SST_\mathcal{A}(\lambda/\mu)$ be the set of semistandard (or $\mathcal{A}$-semistandard) tableaux of shape $\lambda/\mu$, that is, tableaux with entries in $\mathcal{A}$ such that (1) the entries in each row (resp. column) are
weakly increasing from left to right (resp. from top to bottom), (2)
the entries in $\mathcal{A}_0$ (resp. $\mathcal{A}_1$) are strictly increasing in each
column (resp. row).

For $a, b, c \in \mathbb{Z}_+$, let $\lambda(a, b, c) = (2^{b+c}, 1^a)/(1^b)$ be a skew Young diagram with two columns. For an $\mathcal{A}$-semistandard tableau $T$ with two columns, let $T^{\tt L}$ and $T^{\tt R}$ denote the left and right columns of $T$, respectively.

We place a tableau so that its top or bottom edges is parallel with 
or same as a given horizontal line $L$. 
More precisely, let $U_1,\ldots, U_r$ be column tableaux (that is, tableaux of single-columned shapes), which are $\mathcal{A}$-semistandard. 
For $(u_1,\ldots, u_r) \in \mathbb{Z}_+^r$, let 
\begin{equation*}
\llfloor U_1,\ldots,U_r \rrfloor_{(u_1,\ldots,u_r)},\quad 
\llceil U_1,\ldots,U_r \rrceil^{(u_1,\ldots,u_r)}
\end{equation*}
be the tableaux such that the $i$-th column from the left is $U_i$ and the bottom (resp. top) edge of $U_i$ is slid by $u_i$ positions up (resp. down) from $L$
(see Examples \ref{ex:sl2action}, \ref{ex:admissible}, \ref{ex:sliding}, \ref{ex:insertion tableau}).
In particular, we do not record the tuple $(u_1, \ldots, u_r)$ when $(u_1, \ldots, u_r) = (0, \ldots, 0)$.

%\color{red}
%If necessary, we assume that a tableau is placed on the plane $\mathbb{P}_L$ with a horizontal line $L$. 
%Let $U_1,\ldots, U_r$ be column tableaux (that is, tableaux of single-columned shapes), which are $\mathcal{A}$-semistandard. 
%Let 
%$\llfloor U_1,\ldots,U_r \rrfloor$
%denote the tableau in $\mathbb{P}_L$ (not necessarily of partition shape), 
%where the $i$-th column from the left is $U_i$ and its bottom edge lies on $L$. 
%Similarly, let $\llceil U_1,\ldots,U_r \rrceil$ denote the tableau, 
%where the $i$-th column from the left is $U_i$ and its top edge lies on $L$.
%\color{black}


\subsection{Crystal and Sch\"{u}tzenberger's jeu de taquin}
Note that Sch\"{u}tzenberger's jeu de taquin is also available for $\mathcal{A}$-semistandard tableaux.
We use this algorithm in terms of crystal operator $\mathcal{E}$ and $\mathcal{F}$ for $\mathfrak{sl}_2$ (cf.~\cite{La03}).

For $T = \llfloor T^{\tt L}, T^{\tt R} \rrfloor_{(0, a)} \in SST_{\lambda(a, b, c)}$ such that $\llfloor T^{\tt L}, T^{\tt R} \rrfloor_{(0, a-1)}$ is not $\mathcal{A}$-semistandard, we define $\mathcal{E}T$ and $\mathcal{F}T$ to be the tableaux obtained by applying the usual jeu de taquin to the outer and inner corners of $T$, respectively. Here we define $\mathcal{E}T = \mathbf{0}$ and $\mathcal{F}T = \mathbf{0}$ when $a=0$ and $b=0$, respectively, where $\mathbf{0}$ is a formal symbol.
\begin{ex}\label{ex:sl2action}
Suppose that $\mathcal{A} = \mathbb{I}_{4|3}$.

\[ \ytableausetup{boxsize=1.2em}
\begin{tikzpicture}
\node [above right] at (0, 0) { 
\begin{ytableau} 
\tl{$2$} & \tl{$2'$} \\ 
\tl{$1'$} & \tl{$2'$} \\ 
\tl{$1'$} \\ 
\tl{$3'$}
\end{ytableau}};
\node [above right] at (2.5, 0) {
\begin{ytableau} 
\none & \tl{$2$} \\ 
\tl{$1'$} & \tl{$2'$} \\ 
\tl{$1'$} & \tl{$2'$} \\ 
\tl{$3'$} 
\end{ytableau}};
\node [above right] at (5, 0) {
\begin{ytableau} 
\none & \tl{$2$} \\ 
\none & \tl{$2'$} \\ 
\tl{$1'$} & \tl{$2'$} \\ 
\tl{$1'$} & \tl{$3'$} 
\end{ytableau}};


\node at (1.87, 3em) {$\longrightarrow$}; 
\node at (4.4, 3em) {$\longrightarrow$};
\node [above] at (1.87, 3em) {$\mathcal{E}$}; 
\node [above] at (4.4, 3em) {$\mathcal{E}$};
\node at (1.87, 2.6em) {$\longleftarrow$}; 
\node at (4.4, 2.6em) {$\longleftarrow$};
\node [above] at (1.87, 1.35em) {$\mathcal{F}$}; 
\node [above] at (4.4, 1.35em) {$\mathcal{F}$};
\node [left] at (-0.3, 0.362em) {$L$};

\draw [dotted] (-0.3, 0.362em) -- (6.7, 0.362em);
\end{tikzpicture}
\]
\end{ex}

Now, for $(U,V)\in SST_\mathcal{A}((1^u))\times SST_\mathcal{A}((1^v))$ ($u,v\in\mathbb{Z}_+$), we define 
\begin{equation}\label{eq:E and F via jdt}
\mathcal{X}(U,V)=
\begin{cases}
((\mathcal{X}T)^{\tt L},(\mathcal{X}T)^{\tt R}) &\text{if $\mathcal{X}T\neq \mathbf{0}$},\\
\mathbf{0} &\text{if $\mathcal{X}T=\mathbf{0}$},
\end{cases} \quad\quad (\mathcal{X}=\mathcal{E}, \mathcal{F}),
\end{equation}
where $T$ is the unique tableau in $SST_\mathcal{A}(\lambda(u-k,v-k,k))$ for some $0\leq k\leq \min\{u,v\}$ such that $(T^{\tt L},T^{\tt R})=(U,V)$ and $\llfloor T^{\tt L}, T^{\tt R} \rrfloor_{(0, u-k-1)}$ is not $\mathcal{A}$-semistandard. Put $\varphi(U,V)=\max\{\,k\,|\,\mathcal{F}^k(U,V)\neq {\bf 0}\,\}$ and $\mathcal{F}^{\rm max}(U,V)=\mathcal{F}^{\varphi(U,V)}(U,V)$.

For $r\geq 2$, let $\mathbf{E}_\mathcal{A}^r$ be the set of $r$-tuples of single-columned $\mathcal{A}$-semistandard tableaux. For $(U_r,\ldots,U_1)\in \mathbf{E}^r_\mathcal{A}$ and $1\leq i\leq r-1$, we define
\begin{equation}%\label{eq:Ei and Fi via jdt}
\mathcal{X}_i(U_r,\ldots,U_1) = 
\begin{cases}
(U_r,\ldots, \mathcal{X}(U_{i+1},U_i),\ldots,U_1) & \text{if $\mathcal{X}(U_{i+1},U_i)\neq \mathbf{0}$},\\
\mathbf{0} &\text{if $\mathcal{X}(U_{i+1},U_i)=\mathbf{0}$},
\end{cases}
\end{equation} 
where $\mathcal{X}(\, \cdot\, ,\, \cdot \, )$ for $\mathcal{X}=\mathcal{E},\mathcal{F}$ is defined in \eqref{eq:E and F via jdt}. Then we have the following.

\begin{lem}[{\cite[Lemma 2.3]{HK}}] \label{lem:regularity of E} 
$\mathbf{E}^r_\mathcal{A}$ is a regular $\mathfrak{sl}_r$-crystal with respect to $\mathcal{E}_i$ and $\mathcal{F}_i$ for $1\leq i\leq r-1$.
\end{lem}

\subsection{Spinor model}
%Let us recall our main combinatorial object, which is introduced in \cite{Kwon15}.
%
Let 
\begin{equation*}
\mathscr{P}({\rm Sp})=\{\,(\lambda,\ell)\,|\,\ell\geq 1,\ \lambda\in \mathscr{P}_\ell\,\}.
\end{equation*}
For $a\in \mathbb{Z}_+$, let
\[ \mathbf{T}_\mathcal{A}(a)=\bigsqcup_{c\in\mathbb{Z}_+}SST_\mathcal{A}(\lambda(a, 0, c)). \]
For $T \in \mathbf{T}_\mathcal{A}(a)$, we define 
\[ {^{\tt L}}T=(\mathcal{E}^a T)^{\tt L}, \quad   {^{\tt R}}T= (\mathcal{E}^a T)^{\tt R}. \]

\begin{df}[{\cite[Definition 6.7, Definition 6.10]{Kwon15}}]\label{def:spinor tableaux}
\begin{itemize}
\item[(1)] For $a_1, a_2\in \mathbb{Z}_+$ with $a_2\leq a_1$ and $(T_2,T_1)\in \mathbf{T}_\mathcal{A}(a_2)\times \mathbf{T}_\mathcal{A}(a_1)$, we define 
\begin{equation*}
\text{$T_2\prec T_1$\quad if  $\llfloor {^{\tt R}}T_2, T^{\tt L}_1 \rrfloor$ and $\llfloor T_2^{\tt R},{^{\tt L}}T_1 \rrfloor_{(a_2,a_1)}$ are $\mathcal{A}$-semistandard.}
\end{equation*}

\item[(2)] For $(\lambda,\ell)\in \mathscr{P}({\rm Sp})$,
we define
\begin{equation*}
\mathbf{T}_\mathcal{A}(\lambda,\ell)=
\{\,\mathbf{T}=(T_\ell,\ldots,T_1) \ |\ \text{$T_i \in \mathbf{T}_\mathcal{A}(\lambda_i)$ for all $1 \le i \le \ell$ and $T_{\ell}\prec \cdots \prec T_{1}$}\,\}.
\end{equation*}
\end{itemize}
\end{df}

Put $$\mathscr{P}({\rm Sp})_\mathcal{A}=\{\,(\lambda,\ell)\in \mathscr{P}({\rm Sp})\,|\,\mathbf{T}_\mathcal{A}(\lambda,\ell)\neq \emptyset\,\}.$$
%
Let ${\bf x}_\mathcal{A}=\{\,x_a\,|\,a\in \mathcal{A}\,\}$ be the set of formal commuting variables indexed by $\mathcal{A}$. 
For an $\mathcal{A}$-semistandard tableau $T$, let
${\bf x}_\mathcal{A}^T =\prod_{a\in\mathcal{A}}x_a^{m_a}$, where $m_a$ is the number of occurrences of $a$ in $T$. 
Let $t$ be a variable commuting with all $x_a$ ($a\in \mathcal{A}$).
%
For $(\lambda,\ell)\in \mathscr{P}({\rm Sp})_\mathcal{A}$, we define the character of $\mathbf{T}_\mathcal{A}(\lambda,\ell)$ to be 
\begin{equation}\label{eq:character of spinor}
S_{(\lambda,\ell)}({\bf x}_\mathcal{A})=
t^\ell \sum_{(T_\ell,\ldots,T_1)\in {\bf T}_\mathcal{A}(\lambda,\ell)}
{\bf x}_\mathcal{A}^{T_\ell}\cdots {\bf x}_\mathcal{A}^{T_1}.  
\end{equation}
The character $S_{(\lambda,\ell)}({\bf x}_\mathcal{A})$ gives various irreducible characters of Lie (super)algebras under suitable choices of $\mathcal{A}$ (cf.~\cite{Kwon15}). In particular, it is the character of a finite-dimensional irreducible $\mathfrak{sp}_{2n}$-module when $\mathcal{A}=[\overline{n}]$.

\begin{ex} \label{ex:admissible}
Suppose $\mathcal{A} = \mathbb{I}_{4|3}$ and $(\lambda,\ell)=((3, 2, 1), 3)$. Let $\mathbf{T} = (T_3, T_2, T_1)$ be given by
\[ \resizebox{0.3\textwidth}{!}{
\begin{tikzpicture}
\node [above right] at (0, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2$} \\
\tl{$3$} & \tl{$4$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (1.5, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$3$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$2'$}
\end{ytableau}$};
\node [above right] at (3, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2'$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\tl{$3'$}
\end{ytableau}$};

\draw [dotted] (-0.2, 0.362em) -- (4.7, 0.362em);
\node [below] at (0.75, 0) {$T_3$};
\node [below] at (2.25, 0) {$T_2$};
\node [below] at (3.75, 0) {$T_1$};
\end{tikzpicture}
} \]
where the dotted line is the common bottom line. In this case, $T_3 \prec T_2 \prec T_1$ and so $\mathbf{T} \in \mathbf{T}_\mathcal{A}(\lambda, \ell)$.
\end{ex}

%We introduce a spinor model of skew shape.

\begin{df}[{\cite[Definition 5.1]{HK}}] %\label{def:spinor of skew shape}
Let
${\bf T}=(T_\ell,\ldots,T_1)\in {\bf T}_\mathcal{A}(a_\ell)\times\cdots\times{\bf T}_\mathcal{A}(a_1)$ be given for some $a_1,\ldots,a_\ell\in \mathbb{Z}_+$. 
Let $\lambda/\mu$ be a skew diagram with $\lambda,\mu\in\mathscr{P}_\ell$. We say that
\begin{itemize}
\item[(1)] ${\bf T}$ is of shape $\lambda/\mu$ if  
\begin{equation*}
a_i=\lambda_i-\mu_i,\quad {\rm ht}(T^{\tt L}_{i+1})+\mu_{i+1}\leq {\rm ht}(T^{\tt L}_i)+\mu_i\quad (1\leq i\leq \ell),
\end{equation*}
where ${\rm ht}(U)$ denotes the height of a single-columned tableau $U$,

\item[(2)] ${\bf T}$ is $\mathcal{A}$-admissible of shape $\lambda/\mu$ if ${\bf T}$ is of shape $\lambda/\mu$ and
\begin{equation*}
\text{
$\llfloor {^{\tt R}}T_{i+1}, T^{\tt L}_i \rrfloor_{(\mu_{i+1},\mu_i)}$ 
and 
$\llfloor T_{i+1}^{\tt R},{^{\tt L}}T_i \rrfloor_{(\lambda_{i+1},\lambda_i)}$ 
are $\mathcal{A}$-semistandard\quad$(1\leq i\leq \ell-1)$.}
\end{equation*}
\end{itemize}
\end{df}

We denote by ${\bf T}_\mathcal{A}(\lambda/\mu,\ell)$ the set of $\mathcal{A}$-admissible tableaux of shape $\lambda/\mu$. 
When ${\bf T}$ is of shape $\lambda/\mu$, let us often identify ${\bf T}$ with a tableau
\begin{equation*}
\llfloor T_\ell,\ldots,T_1 \rrfloor_{(\mu_\ell,\ldots,\mu_1)} := 
\llfloor T^{\tt L}_\ell,T^{\tt R}_\ell,\ldots,T^{\tt L}_1,T^{\tt R}_1 \rrfloor_{(\mu_\ell,\lambda_\ell,\ldots,\mu_1,\lambda_1)}.
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Symplectic jeu de taquin}

\subsection{Symplectic jeu de taquin for KN tableaux}

Recall that there exists a well-known combinatorial model for the irreducible highest weight $\mathfrak{sp}_{2n}$-module, called Kashiwara--Nakashima tableaux of type $C$ (KN tableaux for short) \cite{KN}. For $\lambda \in \mathscr{P}_n$, denote by $\mathbf{KN}_\lambda$ the set of KN tableaux of shape $\lambda$ with letters in $\{ 1 < \cdots < n < \overline{n} < \cdots < \overline{1} \}$.

Note that a tableau with letters in $\{ 1 < \cdots < n < \overline{n} < \cdots < \overline{1} \}$ is a KN tableau if and only if it satisfies the \textit{admissibility} condition (see \cite[Section 4]{Sh99}). 
With this characterization, one can define the set $\mathbf{KN}_{\lambda/\mu}$ of admissible tableaux of a skew shape $\lambda/\mu$ with $\lambda, \mu \in \mathscr{P}_n$ \cite[Definition 6.1.1]{Le02}.

For $(\lambda,\ell)\in \mathscr{P}({\rm Sp})_{[\overline{n}]}$, put $\rho_n(\lambda, \ell)=(n-\lambda_\ell, n-\lambda_{\ell-1}, \ldots, n-\lambda_1)'$, which is the conjugate of the rectangular complement of $\lambda$ in $(n^\ell)$.

For $U\in SST_{[\bar{n}]}((1^m))$, let $U^c$ be the tableau in $SST_{[n]}((1^{n-m}))$ such that $k$ appears in $U^c$ if and only if $\overline{k}$ does not appear in $U$ for each $k\in [n]$.  
For $T\in {\bf T}_n(a)$, define $T^{\tt ad}$ to be the tableau obtained by putting ${^{\tt L}}T$ below $({^{\tt R}}T)^c$. 
Then the map $T \longmapsto {T}^{\tt ad}$ is a bijection from ${\bf T}_n(a)$ to ${\bf KN}_{(1^{n-a})}$ \cite[Lemma 3.11]{Kwon18-1}. Moreover, we have a bijection  \cite[Corollary 5.6]{HK}
\begin{equation}\label{eq:skewPsi_lambda:C}
\xymatrixcolsep{2pc}\xymatrixrowsep{0.5pc}\xymatrix{
\mathbf{T}_{[\overline{n}]}(\lambda/\mu,\ell)  \ar@{->}[r]  & \ \mathbf{KN}_{\rho_{n+\mu_1}(\lambda,\ell)/\rho_{\mu_1}(\mu,\ell)} \\
\mathbf{T}=\llfloor T_\ell, \ldots, T_1 \rrfloor_{(\mu_\ell, \ldots, \mu_1)}  \ar@{|->}[r] & \mathbf{T}^{\tt ad}:=\llceil T_\ell^{\tt ad},\ldots, T_1^{\tt ad} \rrceil^{\rho_{\mu_1}(\mu,\ell)}}.
\end{equation}

For ${\bf T}\in \mathbf{T}_{[\overline{n}]}(\lambda/\mu,\ell)$ and an inner corner $c$ of $T$ (if exists), one may apply the symplectic analog of jeu de taquin in \cite{Sh99} (see also \cite[Section 6]{Le02}) to obtain another admissible tableau, say $\texttt{jdt}_{KN}(T, c)$.  
It is proved in \cite{Le02} that there exists a unique KN tableau in ${\bf KN}_\nu$ for some $\nu\in \mathscr{P}_n$, which is obtained from $T$ by applying $\texttt{jdt}_{KN}$ successively.

\subsection{Symplectic jeu de taquin for spinor model}

Note that $\mathbf{T} \in \mathbf{T}_\mathcal{A}(\lambda/\mu, \ell)$ does not necessarily correspond to a KN tableau of skew shape for arbitrary $\mathcal{A}$ as in \eqref{eq:skewPsi_lambda:C}. In order to define an analog of jeu de taquin, we first introduce the notion of $n$-conjugate of $\mathbf{T}$. 

Let us assume that $\mathbf{T} = (T_\ell^{\tt L}, T_\ell^{\tt R}, \ldots, T_1^{\tt L}, T_1^{\tt R}) \in \mathbf{E}_\mathcal{A}^{2\ell}$. Then we have a map
\begin{equation*}
\xymatrixcolsep{2pc}\xymatrixrowsep{0.5pc}\xymatrix{
\mathbf{E}_\mathcal{A}^{2\ell}  \ar@{->}[r]  & \ \displaystyle \bigsqcup_{\mu \in \mathscr{P}} SST_\mathcal{A}(\mu) \times SST_{[2\ell]}(\mu') \\
 \mathbf{U}  \ar@{|->}[r] & (P_\mathcal{A}({\bf U}), Q_\mathcal{A}({\bf U})) },
\end{equation*}
where $P_\mathcal{A}({\bf U})$ is obtained by the usual Schensted's column insertion and $Q_\mathcal{A}({\bf U})$ is the associated recording tableau.

\begin{df}
Let ${\bf T}=(T_\ell,\ldots,T_1)\in {\bf T}_\mathcal{A}(\lambda/\mu, \ell)$ be given and set $\nu = {\rm sh}(P_\mathcal{A}(\mathbf{T}))$. 
For $n \geq \ell(\nu)$, we define the $n$-conjugate of ${\bf T}$ to be the unique tableau 
$\overline{\bf T} \in {\bf T}_{[\overline{n}]}(\lambda/\mu, \ell)$ such that
\begin{equation*}
P_{[\overline{n}]}(\overline{\bf T}) = H_\nu \quad\mbox{and}\quad Q_{[\overline{n}]}(\overline{\bf T}) =  Q_\mathcal{A}({\bf T}),
\end{equation*}
where $H_\nu \in SST_{[\overline{n}]}(\nu)$ is such that the $i$-th row from the top is filled with $\overline{n-i+1}$ for $1\leq i\leq n$. 
\end{df}

Now, let us introduce an analog of jeu de taquin for spinor model. 
We first consider the case when $\ell=2$.
Suppose that $\mathbf{T}=(T_2,T_1)\in \mathbf{T}_\mathcal{A}(a_2)\times \mathbf{T}_\mathcal{A}(a_1)$ is given for some $a_1, a_2\in \mathbb{Z}_+$. Let 
\begin{equation*}
d(T_1,T_2)=\min\left\{\,d \in \mathbb{Z}_+\,\Big|\,\text{$\llfloor T_2,T_1 \rrfloor_{(0,d)}$ is $\mathcal{A}$-admissible (of a skew shape)}\,\right\}.
\end{equation*}
We assume that $\mathbf{T}=(T_2^{\tt L},T_2^{\tt R},T_1^{\tt L},T_1^{\tt R})\in \mathbf{E}^4_\mathcal{A}$ and consider the $\mathfrak{sl}_4$-crystal structure on $\mathbf{E}_\mathcal{A}^4$ given in Lemma \ref{lem:regularity of E}.

\begin{df}[{\cite[Section 5.2]{HK}}]\label{def:sjdt spin 2 col}
Under the above hypothesis, suppose that $d=d(T_1,T_2)>0$. Define
\[ {\tt jdt}_{spin}(\mathbf{T}) = \mathbf{T}'=(T'_2,T'_1) \]
to be the pair $(T_2', T_1')$ obtained by applying a sequence of crystal operators as follows: 

\begin{itemize}
\item[] {\em Case 1}. Suppose that $\llfloor {^{\tt R}}T_2, T_1^{\tt L} \rrfloor_{(0,d-1)}$ is not $\mathcal{A}$-semistandard. Then
\begin{equation}\label{eq:sliding-1}
\mathbf{T}':=(U_4,U_3,U_2,U_1)=
\begin{cases}
\mathcal{F}^{a_2-1}_3\mathcal{E}_2\mathcal{E}_3^{a_2}\mathbf{T} & \text{if $\varepsilon_3(\mathcal{E}_2\mathcal{E}_3^{a_2}\mathbf{T})=0$},\\
\mathcal{F}^{a_2}_3\mathcal{E}_2\mathcal{E}_3^{a_2}\mathbf{T} & \text{if $\varepsilon_3(\mathcal{E}_2\mathcal{E}_3^{a_2}\mathbf{T})=1$}.\\
\end{cases}
\end{equation}
%


\item[] {\em Case 2}. Suppose that $\llfloor {^{\tt R}}T_2, T_1^{\tt L} \rrfloor_{(0,d-1)}$ is $\mathcal{A}$-semistandard, but $\llfloor T_2^{\tt R},{^{\tt L}}T_1 \rrfloor_{(0,d-1)}$ is not.
Then 
\begin{equation}\label{eq:sliding-2}
\mathbf{T}':=(U_4,U_3,U_2,U_1)=
\mathcal{F}^{a_1+1}_1\mathcal{F}_2\mathcal{E}_1^{a_1}\mathbf{T}.
\end{equation}
\end{itemize}
\end{df}

\begin{ex}\label{ex:sliding}
Suppose that $\mathcal{A} = \mathbb{I}_{4|3}$.

(1) The following is an example of {\em Case 1} with $\varepsilon_3(\mathcal{E}_2\mathcal{E}_3^{a_2}\mathbf{T}) = 0$.
\[ \resizebox{0.8\textwidth}{!}{
\begin{tikzpicture}
\node [above right] at (0, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$2$} & \tl{$3$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$3'$}
\end{ytableau}$};
\node [above right] at (1.2, 1.2em) {$\begin{ytableau}
\tl{$2$} & \tl{$2'$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$3'$}
\end{ytableau}$};

\node [above right] at (3.2, 0) {$\begin{ytableau}
\none \\
\tl{$1$} \\
\tl{$2$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (3.9, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$3$} \\
\tl{$2'$} \\
\tl{$3'$}
\end{ytableau}$};
\node [above right] at (4.6, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$1'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\none
\end{ytableau}$};
\node [above right] at (5.3, 0) {$\begin{ytableau}
\tl{$2'$} \\
\tl{$2'$} \\
\none \\
\none \\
\none
\end{ytableau}$};

\node [above right] at (6.8, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$2$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (7.5, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$3$} \\
\tl{$2'$}
\end{ytableau}$};
\node [above right] at (8.2, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$1'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\tl{$3'$}
\end{ytableau}$};
\node [above right] at (8.9, 0) {$\begin{ytableau}
\tl{$2'$} \\
\tl{$2'$} \\
\none \\
\none \\
\none
\end{ytableau}$};

\node [above right] at (10.4, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$2$} & \tl{$3$} \\
\tl{$1'$} & \tl{$2'$}
\end{ytableau}$};
\node [above right] at (11.6, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2'$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\tl{$3'$}
\end{ytableau}$};

\draw [dotted] (-0.2, 0.362em) -- (13.3, 0.362em); 
\draw [->] (2.5, 2.4em) -- (3.1, 2.4em);
\draw [->] (6.1, 2.4em) -- (6.7, 2.4em);
\draw [->] (9.7, 2.4em) -- (10.3, 2.4em);
\node [above] at (2.8, 2.4em) {$\mathcal{E}_3$};
\node [above] at (6.4, 2.4em) {$\mathcal{E}_2$};
\node [below] at (3.55, 0.2em) {\textit{4}}; \node [below] at (4.3, 0.2em) {\it 3}; \node [below] at (5.05, 0.2em) {\it 2}; \node [below] at (5.8, 0.2em) {\it 1};
\node [below] at (7.25, 0.2em) {\textit{4}}; \node [below] at (7.95, 0.2em) {\it 3}; \node [below] at (8.65, 0.2em) {\it 2}; \node [below] at (9.35, 0.2em) {\it 1};
\node [below] at (1.6, 0.1) {$\mathbf{T} =\llfloor T_2, T_1 \rrfloor_{(0, 1)}$};
\node [below] at (11.8, 0.1) {$\mathbf{T}' =\llfloor T_2', T_1' \rrfloor$};
\end{tikzpicture}
}
\]

(2) The following is an example of {\em Case 2}.
\[ \resizebox{0.8\textwidth}{!}{
\begin{tikzpicture}
\node [above right] at (0, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$4$} \\
\tl{$3$} & \tl{$2'$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (1.2, 2.4em) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$2$} & \tl{$3$} \\
\tl{$1'$} & \tl{$2'$}
\end{ytableau}$};

\node [above right] at (3.2, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$3$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (3.9, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$4$} \\
\tl{$2'$}
\end{ytableau}$};
\node [above right] at (4.6, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$1'$} \\
\none \\
\none
\end{ytableau}$};
\node [above right] at (5.3, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$3$} \\
\tl{$2'$} \\
\none
\end{ytableau}$};

\node [above right] at (6.8, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$3$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (7.5, 0) {$\begin{ytableau}
\tl{$2$} \\
\tl{$4$} \\
\tl{$2'$}
\end{ytableau}$};
\node [above right] at (8.2, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$3$} \\
\tl{$1'$} \\
\none
\end{ytableau}$};
\node [above right] at (8.9, 0) {$\begin{ytableau}
\tl{$1$} \\
\tl{$2'$} \\
\none \\
\none
\end{ytableau}$};

\node [above right] at (10.4, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2$} \\
\tl{$3$} & \tl{$4$} \\
\tl{$1'$} & \tl{$2'$}
\end{ytableau}$};
\node [above right] at (11.6, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$3$} & \tl{$2'$} \\
\tl{$1'$} \\
\none
\end{ytableau}$};

\draw [dotted] (-0.2, 0.362em) -- (13.3, 0.362em); 
\draw [->] (2.5, 2.4em) -- (3.1, 2.4em);
\draw [->] (6.1, 2.4em) -- (6.7, 2.4em);
\draw [->] (9.7, 2.4em) -- (10.3, 2.4em);
\node [above] at (2.8, 2.4em) {$\mathcal{F}_2$};
\node [above] at (6.4, 2.4em) {$\mathcal{F}_1$};
\node [below] at (3.55, 0.2em) {\textit{4}}; \node [below] at (4.3, 0.2em) {\it 3}; \node [below] at (5.05, 0.2em) {\it 2}; \node [below] at (5.8, 0.2em) {\it 1};
\node [below] at (7.25, 0.2em) {\textit{4}}; \node [below] at (7.95, 0.2em) {\it 3}; \node [below] at (8.65, 0.2em) {\it 2}; \node [below] at (9.35, 0.2em) {\it 1};
\node [below] at (1.6, 0.1) {$\mathbf{T} =\llfloor T_2, T_1 \rrfloor_{(0, 2)}$};
\node [below] at (11.8, 0.1) {$\mathbf{T}' =\llfloor T_2', T_1' \rrfloor_{(0, 1)}$};
\end{tikzpicture}
}
\]
\end{ex}
One can check that ${\tt jdt}_{spin}$ is compatible with the $n$-conjugate for a sufficiently large $n$, that is, $\ov{{\tt jdt}_{spin}({\bf T})}={\tt jdt}_{spin}(\ov{\bf T})$. 
Furthermore, ${\tt jdt}_{spin}$ can be viewed as a generalization of ${\tt jdt}_{KN}$ in the following sense.
\begin{equation}\label{eq:jdt compatibility}
\left({\tt jdt}_{spin}(\ov{\bf T})\right)^{\tt ad}
={\tt jdt}_{KN}\left(\ov{\bf T}^{\tt ad},c\right),
\end{equation}
where $(\,\cdot \, )^{\tt ad}$ is given in \eqref{eq:skewPsi_lambda:C}, and $c$ is the inner corner of $\ov{\bf T}^{\tt ad}$.

Now consider a general case.
Let $\mathbf{T}=(T_\ell,\ldots,T_1)\in \mathbf{T}_\mathcal{A}(\lambda/\mu, \ell)$ be given. 
Let $c$ be the inner corner of $\lambda/\mu$ in the $i$-th column from the right.

Let $b$ be the inner corner of $\overline{\bf T}^\texttt{ad}$ in the $(i+1)$-th column from the right, where $\overline{\bf T}$ is the $n$-conjugate of ${\bf T}$ for a sufficiently large $n$, and consider ${\tt jdt}_{KN}\left(\overline{\bf T}^{\tt ad}, b \right)$. 
By Definition \ref{def:sjdt spin 2 col} and \eqref{eq:jdt compatibility}, there exists 
a composite of operators $\mathcal{E}_i$ and $\mathcal{F}_i$, say $\mathcal{X}$, such that
\begin{equation}
\left( \mathcal{X}\,\overline{\bf T}\right)^{\tt ad} = {\tt jdt}_{KN}\left(\overline{\bf T}^{\tt ad},b\right).
\end{equation}
We remark that the composite $\mathcal{X}$ is independent of the choice of $n$.
\begin{df}[{\cite[Definition 5.14]{HK}}]
Under the above hypothesis, we define
\begin{equation}\label{eq:sjdt spin general}
{\tt jdt}_{spin}({\bf T},c)=\mathcal{X}\,{\bf T}.
\end{equation}
\end{df}

\begin{thm}[{\cite[Theorem 5.15]{HK}}]\label{thm:P tableau for spinor}
Let $\mathbf{T}=(T_\ell,\ldots,T_1)\in {\bf T}_\mathcal{A}(\lambda/\mu, \ell)$ be given. 
There exists a unique ${\tt P}({\bf T})\in {\bf T}_\mathcal{A}(\nu,\ell)$ for some $(\nu,\ell)\in \mathscr{P}({\rm Sp})_\mathcal{A}$, which can be obtained from $\mathbf{T}$ by applying ${\tt jdt}_{spin}(\,\cdot\,,c)$ finitely many times with respect to inner corners $c$.
\end{thm}

\begin{ex} \label{ex:insertion tableau}
Let $\mathbf{T}=\llfloor T_3, T_2, T_1 \rrfloor_{(0, 1, 2)}$ be given as below.
Then ${\tt P}({\bf T})$ can be obtained by
\[ {\tt P}(\mathbf{T}) = {\tt jdt}_{spin}({\tt jdt}_{spin}({\tt jdt}_{spin}(\mathbf{T}, c_1), c_2), c_3) = \mathcal{E}_4\mathcal{F}_3\mathcal{F}_4\mathcal{E}_2\mathcal{E}_3\mathbf{T}. \]
%For detailed computations, see Example \ref{ex:sliding}.
\[ \resizebox{300pt}{!}{
\begin{tikzpicture}
\node [above right] at (0, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$4$} \\
\tl{$3$} & \tl{$2'$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (1.5, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$2$} & \tl{$3$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$3'$} \\
\none[\tl{$c_2$}]
\end{ytableau}$};
\node [above right] at (3, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2'$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\none[\tl{$c_1$}] \\
\none[\tl{$c_3$}]
\end{ytableau}$};

\node [above right] at (6, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2$} \\
\tl{$3$} & \tl{$4$} \\
\tl{$1'$}
\end{ytableau}$};
\node [above right] at (7.5, 0) {$\begin{ytableau}
\tl{$1$} & \tl{$1$} \\
\tl{$3$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$2'$}
\end{ytableau}$};
\node [above right] at (9, 0) {$\begin{ytableau}
\tl{$2$} & \tl{$2'$} \\
\tl{$1'$} & \tl{$2'$} \\
\tl{$1'$} \\
\tl{$3'$} \\
\tl{$3'$}
\end{ytableau}$};

\draw [dotted] (-0.5, 0.362em) -- (10.8, 0.362em); 
\draw [->] (4.5, 4.8em) -- (5.7, 4.8em);
\node [below] at (2.5, 0.1) {$\mathbf{T} =\llfloor T_3, T_2, T_1 \rrfloor_{(0, 1, 2)}$};
\node [below] at (8.4, 0.1) {${\tt P}(\mathbf{T})=\llfloor T_3', T_2', T_1' \rrfloor$};
\end{tikzpicture}
} \]
\end{ex}

\section{Symplectic RSK correspondence}

\subsection{Oscillating tableaux and King tableaux}

 An oscillating tableau is a sequence of partitions $Q=(Q_1,\ldots,Q_s)$ for some $s\geq 1$ such that each pair $(Q_i,Q_{i+1})$ differs by one box for $1\leq i\leq s-1$, i.e., 
$Q_i/Q_{i+1} = \square$ or $Q_{i+1}/Q_i = \square$. 
We say that $Q=(Q_1, \ldots, Q_s)$ is vertical if 
$Q_1\subsetneq  \cdots \subsetneq Q_r \supsetneq \cdots \supsetneq Q_s$ for some $1 \le r \le s$ and $Q_r/Q_1$ and $Q_r/Q_s$ is a skew diagram of vertical strip.
Here $A \subsetneq B \iff A \subseteq B$ and $A \neq B$.
We denote by $|Q|=s$ the length of $Q=(Q_1, \ldots, Q_s)$.

\begin{df}[{\cite[Section 6.1]{HK}}]
Let $(\lambda,\ell)\in \mathscr{P}({\rm Sp})$ be given.
For $n\geq \lambda_1$, define ${\bf O}(\lambda,\ell;n)$ to be a set of sequences of oscillating tableaux $Q=(Q^{(1)}:\cdots :Q^{(\ell)})$ such that
\begin{itemize}
\item[-] $Q$ is itself an oscillating tableau,
\item[-] $Q^{(i)}=(Q_{i,1},\ldots,Q_{i,s_i})$ is a \textit{vertical} oscillating tableau for $1\leq i\leq \ell$,

\item[-] $\ell(Q_{i, j}) \le n$ for $1 \le i \le \ell$ and $1 \le j \le s_i$,

\item[-] $Q_{1,1}=\square$ and $Q_{\ell,s_\ell}=\rho_n(\lambda,\ell)$.
\end{itemize} 
\end{df}

For $Q = (Q^{(1)} : \cdots : Q^{(\ell)}) \in \mathbf{O}(\lambda, \ell; n)$ with $|Q^{(i)}| = s_i$, let $\sigma(Q)=(\widehat{Q}^{(1)}:\cdots:\widehat{Q}^{(\ell)})$ be a sequence of oscillating tableaux $\widehat{Q}^{(i)}$ given as follows;
\begin{equation*}
\widehat{Q}^{(i)} =((i) \cup Q_{i-1, s_{i-1}},(i)\cup Q_{i,1},\ldots,(i)\cup Q_{i,s_i}) \quad (1\leq i\leq \ell).
\end{equation*}
Here we denote by $(i)\cup Q_{i,k}$ the partition obtained by adding $i$ to $Q_{i,k}$ as its first part. 
Then we show $\sigma(Q) \in {\bf O}(\lambda,\ell;n+1)$. 
Indeed, $\sigma : {\bf O}(\lambda,\ell;n) \longrightarrow {\bf O}(\lambda,\ell;n+1)$ is a bijection for $n\geq \lambda_1$, 
and induces an equivalence relation on $\bigsqcup_{n\geq \lambda_1}{\bf O}(\lambda,\ell;n)\times \{n\}$, where 
$(Q',m)\sim (Q,n)$ if and only if $\sigma^{m-n}(Q)=Q'$
for $Q'\in {\bf O}(\lambda,\ell;m)$ and $Q \in {\bf O}(\lambda,\ell;n)$ with $m\geq n$.
We define 
\begin{equation*}
\mathbf{O}(\lambda,\ell) =\{\,[Q,n]\,|\, Q\in \mathbf{O}(\lambda,\ell;n)\ (n\geq \lambda_1) \,\},
\end{equation*}
where $[Q,n]$ is the equivalence class of $Q\in \mathbf{O}(\lambda,\ell;n)$ with respect to $\sim$.
We call $[Q,n]$ an oscillating tableau of shape $(\lambda,\ell)$.

For $(\lambda,\ell)\in \mathscr{P}({\rm Sp})$, let $\mathbf{K}(\lambda,\ell)$ be the set of tableaux of shape $\lambda$ with letters in $\{ 1 < \overline{1} < \cdots < \ell < \overline{\ell} \}$, whose letters are of degree $0$, such that all entries in the $i$th row are larger than or equal to $i$. It is known as the set of King tableaux of shape $\lambda$ \cite{King76}. 
Based on \cite{Lee19}, we construct a bijection between King tableaux and oscillating tableaux.

\begin{thm}[{\cite[Theorem 2.7]{Lee19}, \cite[Corollary 6.8]{HK}}] 
For $\lambda \subseteq (n^\ell)$, we have an explicit bijection
\begin{equation}\label{eq:bijection 4} 
\mathbf{K}(\lambda, \ell) \ \longrightarrow \ \mathbf{O}(\lambda, \ell).
\end{equation}
\end{thm}

\subsection{Symplectic RSK correspondence}
For $\mathbf{a} = (a_1, \ldots, a_\ell) \in \mathbb{Z}_+^\ell$, let $\mathbf{T}_\mathcal{A}(\mathbf{a}) = \mathbf{T}_\mathcal{A}(a_\ell) \times \cdots \times \mathbf{T}_\mathcal{A}(a_1)$, and $\mathbb{Z}/(\mathbf{a+1})\mathbb{Z} = \mathbb{Z}/(a_\ell+1)\mathbb{Z} \times \cdots \times \mathbb{Z}/(a_1+1)\mathbb{Z}$, 
where $\mathbb{Z}/(a+1)\mathbb{Z}$ is understood as the set $\{ 0, 1, \ldots, a \}$ for $a \in \mathbb{Z}_+$, and let $\mathbf{F}_\mathcal{A}^{\ell} = \mathbf{E}_\mathcal{A}^{2\ell}$.
 
\begin{lem}[{\cite[Corollary 7.5]{HK}}]\label{lem:sl-crystal}
We have a bijection
\begin{equation}\label{eq:bijection 1}
\xymatrixcolsep{4pc}\xymatrixrowsep{0.5pc}\xymatrix{
{\bf F}_\mathcal{A}^\ell \ar@{->}[r]  & 
\ \displaystyle{\bigsqcup_{\bf a}{\bf T}_\mathcal{A}({\bf a})
%\left({\bf T}_\A(a_\ell)\times\cdots\times{\bf T}_\A(a_1)\right)
\times \mathbb{Z}/({\bf a}+{\bf 1})\mathbb{Z}}\\
{\bf T} \ar@{|->}[r] & (\mathcal{F}^{\rm max}{\bf T}, \varphi({\bf T}))},
\end{equation}
where  
\begin{equation*}
\begin{split}
\mathcal{F}^{\rm max}{\bf T}&=(\mathcal{F}^{\rm max}(U_{2\ell},U_{2\ell-1}),\ldots,\mathcal{F}^{\rm max}(U_{2},U_{1})),\\
\varphi({\bf T}) &=(\varphi(U_{2\ell},U_{2\ell-1}),\ldots,\varphi(U_{2},U_{1})),
\end{split}
\end{equation*}
for ${\bf T}=(U_{2\ell},\ldots,U_1)\in \mathbf{F}_\mathcal{A}^\ell$ and the union is over $\mathbf{a} \in \mathbb{Z}_+^\ell$ such that ${\bf T}_\mathcal{A}({\bf a}) \ne \emptyset$.
\end{lem}

Let $\mathbf{T}=(T_\ell,\ldots,T_1)\in \mathbf{T}_\mathcal{A}(\mathbf{a})$ be given. 
%
Let us define a recording tableau for ${\tt P}({\bf T})$ (see Theorem \ref{thm:P tableau for spinor}). 
We assume that $\mathbf{T}\in {\bf T}_\mathcal{A}(\lambda/\mu, \ell)$ for some skew shape $\lambda/\mu$. 
Let $\overline{\bf T}$ be the $n$-conjugate of ${\bf T}$ for a sufficiently large $n$. 
Then $\overline{\bf T}^{\tt ad}$ is a KN tableau of shape $\rho_{n+\mu_1}(\lambda, \ell)/\rho_{\mu_1}(\mu, \ell)$. 

By using an analog of Robinson-Schensted correspondence for KN tableaux of type $C$ \cite[Theorem 5.2.2]{Le02} and given $(c_1, \ldots, c_\ell) \in \mathbb{Z}/(\mathbf{a+1})\mathbb{Z}$, we obtain an oscillating tableaux $Q(\mathbf{T}; n)$ from $\overline{\bf T}^{\tt ad}$ such that $Q(\mathbf{T};n) \in \mathbf{O}(\lambda, \ell; n)$ and $\sigma(Q(\mathbf{T};n)) = Q(\mathbf{T};n+1)$ \cite[Lemma 6.2]{HK}. 
Then we define $$Q(\mathbf{T}) = [Q(\mathbf{T};n), n] \in \mathbf{O}(\lambda, \ell).$$

\begin{thm}[{\cite[Proposition 6.4, Theorem 7.2]{HK}}]
We have a bijection
\begin{equation}\label{eq:bijection 2}
\xymatrixcolsep{2pc}\xymatrixrowsep{0.5pc}\xymatrix{
\displaystyle \bigsqcup_{\mathbf{a} \in \mathbb{Z}_+^\ell} \mathbf{T}_\mathcal{A}(\mathbf{a}) \times \mathbb{Z}/(\mathbf{a+1})\mathbb{Z} \ar@{->}[r]  & 
\ \displaystyle{\bigsqcup_{(\lambda,\ell)\in \mathscr{P}({\rm Sp})_\mathcal{A}} \mathbf{T}_\mathcal{A}(\lambda,\ell)\times \mathbf{O}(\lambda,\ell)}\\
({\bf T}, (c_1, \ldots, c_\ell)) \ar@{|->}[r]  & ({\tt P}({\bf T}), Q({\bf T}) )}.
\end{equation}
\end{thm}

Now we consider the composition of the following sequence of bijections \eqref{eq:bijection 1}, \eqref{eq:bijection 2}, and \eqref{eq:bijection 4}
\begin{align*}
\mathbf{F}_\mathcal{A}^\ell 
& \stackrel{\eqref{eq:bijection 1}}{\longrightarrow}
\ \bigsqcup_\mathbf{a} \mathbf{T}_\mathcal{A}(\mathbf{a}) \times \mathbb{Z}/(\mathbf{a+1})\mathbb{Z} \\
& \stackrel{\eqref{eq:bijection 2}}{\longrightarrow}
\ \bigsqcup_{(\lambda, \ell)} \mathbf{T}_\mathcal{A}(\lambda, \ell) \times \mathbf{O}(\lambda, \ell) \quad
\stackrel{\eqref{eq:bijection 4}}{\longrightarrow} \ \bigsqcup_{(\lambda, \ell)} \mathbf{T}_\mathcal{A}(\lambda, \ell) \times \mathbf{K}(\lambda, \ell)
\end{align*}
and denote by $(\texttt{P}(\mathbf{T}), \texttt{Q}(\mathbf{T}))$ the image of $\mathbf{T} \in \mathbf{F}_\mathcal{A}^\ell$ under the above composition.

\begin{thm}[{\cite[Theorem 7.7]{HK}}]\label{thm:RSK}
The map $\mathbf{T} \mapsto ({\tt P}(\mathbf{T}), {\tt Q}(\mathbf{T}))$ gives a bijection
\begin{equation*}
\xymatrixcolsep{4pc}\xymatrixrowsep{0.5pc}\xymatrix{
{\bf F}_\mathcal{A}^\ell
\ar@{->}[r] & \ 
\displaystyle{\bigsqcup_{(\lambda,\ell)\in \mathscr{P}({\rm Sp})_\mathcal{A}}{\bf T}_\mathcal{A}(\lambda,\ell)\times {\bf K}(\lambda,\ell).}} %\\
%{\rm \mathbf{T}}
%\ar@{->}[r] & \
%({\tt P}(\mathbf{T}), {\tt Q}(\mathbf{T}))
%}.
\end{equation*}
\end{thm}

\subsection{Cauchy identity}
Let $\mathbf{z} = \{ z_1, \ldots, z_\ell \}$ be formal commuting variables, which commutes with $\mathbf{x} = \mathbf{x}_\mathcal{A}$. 
%
Let $(\lambda,\ell)\in \mathscr{P}({\rm Sp})$ be given. 
For $K\in {\bf K}(\lambda,\ell)$, let ${\bf z}^K =\prod_{i=1}^\ell z_i^{m_i-m_{\overline{i}}}$, where $m_i$ (resp. $m_{\overline{i}}$) is the number of occurrences of $i$ (resp. $\overline{i}$) in $K$. 
Then put
\begin{equation*}
sp_\lambda({\bf z})= \sum_{K\in {\bf K}(\lambda,\ell)}{\bf z}^K.
\end{equation*}
It is well-known that $sp_\lambda({\bf z})$ is the character of the irreducible highest weight module of ${\rm Sp}_{2\ell}$ with highest weight corresponding to $\lambda$ \cite{King76}.

Let ${\bf U}=(U_{2\ell},\ldots,U_1)\in {\bf F}_\mathcal{A}^\ell$ be given with the ${\rm ht}(U_i) = u_i$. 
Let ${\bf x}^{{\bf U}}=\prod_{i=1}^{2\ell}{\bf x}^{U_i}$ and
${\bf z}^{{\bf U}}=\prod_{i=1}^\ell z_i^{u_{2i}-u_{2i-1}}$.
Then we have
\begin{equation*}
{\rm ch} {\bf F}_\mathcal{A}^\ell
:= \sum_{\bf U} {\bf x}^{{\bf U}}{\bf z}^{{\bf U}}
=
\prod_{j=1}^\ell \frac{\prod_{a \in \mathcal{A}_0} (1+x_az_j)(1+x_az_j^{-1})}{ \prod_{a \in \mathcal{A}_1} (1-x_az_j)(1-x_az_j^{-1})}.
\end{equation*}  


\begin{thm}[{\cite[Theorem 7.9]{HK}}]\label{thm:Cauchy identity}
We have the following identity
\begin{equation}\label{eq:cauchy}
t^{\ell}\prod_{j=1}^\ell \frac{\prod_{a \in \mathcal{A}_0} (1+x_az_j)(1+x_az_j^{-1})}{ \prod_{a \in \mathcal{A}_1} (1-x_az_j)(1-x_az_j^{-1})} = 
\sum_{(\lambda, \ell) \in \mathscr{P}({\rm Sp})_\mathcal{A}}
S_{(\lambda,\ell)}({\bf x}_\mathcal{A})sp_\lambda({\bf z}).
\end{equation}
\end{thm}

We recover well-known identities when $\mathcal{A}$ is homogeneous. 
If $\mathcal{A} = \mathcal{A}_0$, then \eqref{eq:cauchy} is the identity \cite[ (6.19)]{King75}. 
If $\mathcal{A} = \mathcal{A}_1$ and $\ell\ge n$, then \eqref{eq:cauchy} and the stability of ${\bf T}_\mathcal{A}(\lambda,\ell)$ \cite[Theorem 6.5]{Kwon18-2} imply the following identity \cite{Li77, Weyl46} 
\begin{equation}\label{eq:Littlewood}
\frac{1}{\prod_{i=1}^n \prod_{j=1}^\ell (1-x_iz_j)(1-x_iz_j^{-1})} = \sum_{\ell(\lambda)\leq n} sp_\lambda(\mathbf{z}) s_\lambda(\mathbf{x}) \prod_{1 \le i < j \le n}{(1-x_ix_j)^{-1}}, \\
% &=& \sum_{\ell(\la)\leq n} sp_\lambda(\mathbf{z}) s_\lambda(\mathbf{x}) \sum_{\beta':\mbox{\scriptsize even}} s_\beta(\mathbf{x}),
\end{equation}
where $s_\lambda({\bf x})$ is the Schur polynomial in $x_1,\ldots,x_n$, and $\ell(\lambda)$ is the length of $\lambda$. 


\acknowledgements{This work was supported by National Research Foundation of Korea (NRF) grants funded by the Korean government (MSIT) (Grant numbers 2019R1A2C108483311 and 2020R1A5A1016126).}

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