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\title{Quantum nilpotent subalgebras of classical quantum groups and affine crystals}

\author{Il-Seung Jang\thanks{\href{mailto:ignatioschoi@snu.ac.kr}{is\_jang@snu.ac.kr}}\addressmark{1} \and 
Jae-Hoon Kwon\thanks{\href{mailto:jaehoonkw@snu.ac.kr}{jaehoonkw@snu.ac.kr}}\addressmark{2}}

\address{\addressmark{1}Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea \\ \addressmark{2}Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826, Korea}

\received{\today}

\abstract{We give a new interpretation of RSK correspondence of type $D$ in terms of affine crystals. We show that the crystal of quantum nilpotent subalgebra of $U_q(D_n)$ associated to a maximal Levi subalgebra of type $A_{n-1}$ has an affine $D_n^{(1)}$-crystal structure, and it is isomorphic to a direct limit of perfect Kirillov-Reshetikhin crystal $B^{n,s}$ for $s\geq 1$. An analogue of RSK correspondence for type $D$ due to Burge is naturally defined on this crystal and shown to be an isomorphism of affine crystals. We further obtain a generalization of Greene's formula for type $D$ and as a byproduct a new polytope realization of $B^{n,s}$.
}

\keywords{quantum groups, quantum nilpotent subalgebra, crystal graphs}

%\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle

\let\thefootnote\relax\footnotetext{This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1501-01.}

\section{Introduction}

Let $\mf g$ be a classical Lie algebra and let $\mf l$ be its proper maximal Levi subalgebra of type $A$ (or a sum of type $A$). Let $\mf u^-$ be the negative  nilradical of the parabolic subalgebra $\mf p=\mf l + \mf b$, where $\mf b$ is a Borel subalgebra of $\mf g$.
The enveloping algebra $U(\mf u^-)$ is an integrable $\mf l$-module, which has a multiplicity-free decomposition \cite{H}, and the expansion of its character
\begin{equation}
{\rm ch} \ U(\mf u^-) = {\prod_{\alpha\in \Phi({\mf u}^-)}(1-e^\alpha)^{-1}}
\end{equation}
into irreducible $\mf l$-characters (that is, Schur polynomials or a product of Schur polynomials) gives the celebrated Cauchy identity when $\mf g$ is of type $A$, and Littlewood identities when $\mf g$ is of type $B, C, D$, where $\Phi(\mf u^-)$ is the set of roots of $\mf u^-$.


The decomposition of $U(\mf u^-)$ into $\mf l$-modules has a purely combinatorial interpretation by RSK 
correspondence and its variations, say $\kappa$ (cf. \cite{B,Ful}). In \cite{La}, Lascoux showed that $\kappa$ is an isomorphism of $\mf l$-crystals, which immediately implies the same result for type $B$ and $C$ \cite{K09} by using similarity of crystals %\cite{Kas96}
. Furthermore, it is shown in \cite{K13} that the RSK correspondence $\kappa$ can be extended to an isomorphism of affine crystals of type $A_{n}^{(1)}$ when $\mf g$ is of type $A_{n}$, and of type $D_{n+1}^{(2)}$ and $C_n^{(1)}$ when $\mf g$ is of type $B_n$ and $C_n$, respectively.

In this paper, we establish an analogue of the above result when $\g$ is of type $D$. First, we consider the crystal $\B_{\bi_0}$ of $\bi_0$-Lusztig data, where $\bi_0$ is  a reduced expression associated to a specific convex order on the set of positive roots of $\mf g$. The subcrystal $B(U_q(\mf u^-))$ of $\B_{\bi_0}$ consisting of Lusztig data on $\Phi(\mf u^-)$ has a nice combinatorial realization, and naturally admits an affine crystal structure of type $D_n^{(1)}$ isomorphic to a direct limit of KR crystals $B^{n,s}$ for $s\geq 1$. %where $B^{n,1}$ is the crystal of the spin representation as a classical crystal of type $D_n$. 
We give an explicit description of $B^{n,s}\subset B(U_q(\mf u^-))$ in terms of double paths on $\Phi(\mf u^-)$, which yields a polytope realization of $B^{n,s}$ (\cref{thm:main-1}). 
Next, we consider an analogue of RSK correspondence for type $D$ due to Burge \cite{B}. %As a main result of this paper, 
We prove that it is an isomorphism of affine crystals of type $D_n^{(1)}$, where a suitable affine crystal structure is defined on the side of tableaux  (\cref{thm:isomorphism theorem}). Furthermore, we present an interesting formula for the shape of a semistandard tableau corresponding to a Lusztig datum on $\Phi(\mf u^-)$ in terms of non-intersecting double paths on $\Phi(\mf u^-)$ (\cref{thm:shape}). %This formula can be viewed as an analogue of Greene's formula for the shape of a tableau corresponding to a biword under RSK given in terms of disjoint weakly decreasing subwords \cite{G}.
A full version of this paper including detailed proofs has appeared in \cite{JK}.

\section{PBW crystals}

\subsection{PBW  basis and crystals}

We refer the reader to \cite{Kas95,Kas91} for definitions of crystal base and crystal.
Suppose that $\g$ is of finite type and $U_q(\mathfrak{g})$ is the associated quantized enveloping algebra (see \cite{HK, Lu93}). Let us briefly recall the notion of PBW basis and the crystal of Lusztig data which is isomorphic to $B(\infty)$ (see \cite{Lu90, S94}).
Let $W$ be the Weyl group of $\g$ generated by the simple reflections $s_i$ for $i\in I$.
Let $w_0$ be the longest element in $W$ of length $N$, and let $R(w_0)=\{\, \bi=(i_1,\ldots,i_N)\,|\,w_0=s_{i_1}\ldots s_{i_N}\,\}$ be the set of reduced expressions of $w_0$.  For $i\in I$, let $T_i$ be the $\mathbb{Q}(q)$-algebra automorphism of $U_q(\mathfrak{g})$, which is given as {$T''_{i,1}$} in \cite{Lu93}.

For $\bi\in R(w_0)$,
\begin{equation}\label{eq:beta_k}
\Phi^+=\{\beta_1:=\alpha_{i_1}, \beta_2:=s_{i_1}(\alpha_{i_2}),  \ldots ,\beta_N:=s_{i_1}\cdots s_{i_{N-1}}(\alpha_{i_N})\}
\end{equation}
is the set of positive roots of $\g$. For ${\bf c}=(c_{\beta_1},\ldots,c_{\beta_N})\in\Z_+^N$.  For $1\leq k\leq N$, put 
$f_{\beta_k}:=T_{i_{1}}T_{i_2}\cdots T_{i_{k-1}}(f_{i_{k}})$ and let
%\begin{equation*}
%\begin{split}
$
b _{\bi}({\bf c})=
f_{\beta_1}^{(c_{\beta_1})}f_{\beta_2}^{(c_{\beta_2})}\cdots f_{\beta_N}^{(c_{\beta_N})}.
$
%\end{split}
%\end{equation*}
Then the set $B_{\bf i}:=\{\,b _{\bi}({\bf c})\,|\,{\bf c}\in\Z_+^{N}\,\}$ is a $\mathbb{Q}(q)$-basis of $U^-_q(\g)$ called a {\it PBW basis}.

Let $A_0$ be the subring of $\mathbb{Q}(q)$ consisting of rational functions regular at $q=0$. The $A_0$-lattice $L(\infty)$ of $U^-_q(\g)$ generated by $B_{\bf i}$ is independent of the choice of $\bi$ and invariant under $\te_i$, $\tf_i$, and the induced crystal $\pi(B_{\bf i})$ under a canonical projection $\pi : L(\infty) \rightarrow L(\infty)/q L(\infty)$ is isomorphic to $B(\infty)$. We identify ${\bf B}_{\bi}:=\Z_+^N$ with a crystal $\pi(B_{\bi})$ under the map ${\bf c}\mapsto b_{\bi}({\bf c})$, and call ${\bf c}\in \B_{\bi}$ an {\it $\bi$-Lusztig datum}.

Let $w\in W$ be given with length $r$. There exists $i = (i_1, \cdots , i_N ) \in R(w_0)$ such that $w = s_{i_1} \cdots s_{i_r}$ by the properties of the Bruhat order.
%One may assume that there exists $\bi =(i_1,\ldots,i_N) \in R(w_0)$ such that $w=s_{i_1}\cdots s_{i_r}$. 
The $\Q(q)$-subspace of $U^-_q(\mf g)$ spanned by $b_{\bi}({\bf c})$ for ${\bf c}\in \B_{\bi}$ with $c_k=0$ for $r+1\leq k\leq N$ is a subalgebra called the {\em quantum nilpotent subalgebra associated to $w\in W$} and denoted by $U_q^-(w)$ (see for example, \cite{Ki} and references therein). 
%is the $\Q(q)$-subalgebra of $U^-_q(\mf g)$ generated by $f_{\beta_k}$ for $1\leq k\leq r$. 
%It is independent of the choice of the reduced expression of $w$.
%We call this subalgebra the {\em quantum nilpotent subalgebra associated to $w\in W$} and denote it by $U_q^-(w)$ (see for example, \cite{Ki} and references therein). 

\subsection{Description of $\tilde{f}_i$} \label{subssec:tensor product rule on B}
%Suppose that $\mf g$ is of finite type. 
Let $\bi\in R(w_0)$ be given. For $\beta\in \Phi^+$, we denote by ${\bf 1}_{\beta}$ the element in $\B_{\bi}$ where $c_\beta=1$ and $c_\gamma=0$ for $\gamma\in \Phi^+\setminus \{\beta\}$. The Kashiwara operators $\tf_i$ or $\tf_i^\ast$ on $\B_{\bi}$ for $i\in I$ is not easy to describe in general except 
\begin{equation} \label{eq:f}
\begin{split}
\tf_{i}{\bf c} &= (c_{\beta_1}+1,c_{\beta_2},\ldots,c_{\beta_N})= {\bf c}+{\bf 1}_{\alpha_i},\quad 
\text{when $\beta_{1}=\alpha_i$},\\
\tf^\ast_{i}{\bf c} &= (c_{\beta_1},\ldots,c_{\beta_{N-1}},c_{\beta_N}+1)= {\bf c}+{\bf 1}_{\alpha_i},\quad \text{when $\beta_{N}=\alpha_i$},\\
\end{split}
\end{equation}
for ${\bf c}\in \B_{\bi}$ \cite{Lu93}.

Let us review the results in \cite{SST}, which plays an important role in our paper. For simplicity, let us assume that $\mf g$ is simply laced.
Let $\sigma=(\sigma_{1},\sigma_2,\ldots,\sigma_s)$ be a sequence with $\sigma_{u}\in \{\,+\,,\,-\, , \ \cdot\ \}$. We replace a pair $(\sigma_{u},\sigma_{u'})=(+,-)$, where $u<u'$ and $\sigma_{u''}=\,\cdot\,$
for $u<u''<u'$, with $(\,\cdot\,,\,\cdot\,)$, and repeat this process as far as possible until we get a sequence with no $-$ placed to the right of $+$. We denote the resulting sequence by ${\sigma}^{\rm red}$. For another sequence $\tau=(\tau_1,\ldots,\tau_{t})$, we denote by $\sigma\cdot\tau$ the concatenation of $\sigma$ and $\tau$.


Recall that a total order $\prec$ on $\Phi^+$ is called {\em convex} 
if either $\gamma \prec \gamma' \prec \gamma''$ or $\gamma'' \prec \gamma' \prec \gamma$ whenever $\gamma'=\gamma+\gamma''$ for $\gamma, \gamma', \gamma''\in \Phi^+$. It is well-known that there exists a one-to-one correspondence between $R(w_0)$ and the set of convex orders on $\Phi^+$, where the convex order $\prec$ 
associated to $\bi=(i_1,\ldots,i_N)\in R(w_0)$ is given by 
\begin{equation} \label{eq:convex}
\beta_1\prec \beta_2\prec \ \ldots \prec \beta_N,
\end{equation} 
where $\beta_k$ is as in \eqref{eq:beta_k}.
%
Recall that there exists a reduced expression $\bi'$ obtained from $\bi$ by a 3-term braid move 
$(i_k,i_{k+1},i_{k+2}) \rightarrow (i_{k+1},i_{k},i_{k+1})$ with $i_k=i_{k+2}$ if and only if 
$\{\,\beta_{k},\beta_{k+1},\beta_{k+2}\,\}$ forms the positive roots of  type $A_2$,
where the corresponding convex order $\prec'$ is given by replacing $\beta_k\prec \beta_{k+1}\prec \beta_{k+2}$ with $\beta_{k+2}\prec' \beta_{k+1}\prec' \beta_{k}$. 
%
Also there exists a reduced expression $\bi'$ obtained from $\bi$ by a 2-term braid move $(i_k,i_{k+1})\rightarrow (i_{k+1},i_{k})$ if and only if 
$\beta_{k}$ and $\beta_{k+1}$ are orthogonal, where the associated convex ordering $\prec'$ is given by replacing $\beta_k\prec \beta_{k+1}$ with 
$\beta_{k+1}\prec'\beta_{k}$.

Given $i\in I$, suppose that $\bi$ is {\em simply braided for $i\in I$}, that is, if one can obtain $\bi'=(i'_1,\ldots ,i'_N)\in R(w_0)$ with $i'_1=i$ by applying a sequence of braid moves consisting of either a 2-term move or 3-term braid move  
$(\gamma,\gamma',\gamma'')\rightarrow (\gamma'',\gamma',\gamma)$ with $\gamma''=\alpha_i$.
%
Suppose that 
\begin{equation}\label{eq:triple in 3-term}
\Pi_s=\{\gamma_s,\gamma'_s,\gamma''_s\}
\end{equation}
is the triple of positive roots of type $A_2$ with 
$\gamma'_s=\gamma_s+\gamma''_s$ and $\gamma''_s=\alpha_i$ corresponding to the $s$-th 3-term braid move for $1\leq s\leq t$.

For ${\bf c}\in \B_{\bi}$, let
%\begin{equation*}
%\sigma^+_s:=c_{\alpha^{(s)}}, \quad \sigma^-_s:=c_{\beta^{(s)}} \quad (1\leq s\leq t).
%\end{equation*}
%and
%\begin{equation}\label{eq:sigma_i}
$
\sigma_i({\bf c})= 
(\underbrace{-\cdots -}_{c_{\gamma'_1}}\,\underbrace{+\cdots +}_{c_{\gamma_1}}
\ \cdots \
\underbrace{-\cdots -}_{c_{\gamma'_t}}\,\underbrace{+\cdots +}_{c_{\gamma_t}}
).
$
%\end{equation}
%Then we have the following description of $\tf_i$ on $\B_{\bi}$ \cite[Theorem 4.6]{SST}.
\begin{thm}\cite[Theorem 4.6]{SST} \label{thm:signature rule}  
Let $\bi\in R(w_0)$ and $i\in I$. 
Suppose that $\bi$ is simply braided for $i$. 
Let ${\bf c}\in \B_{\bi}$ be given.
\begin{itemize}
\item[(1)] If there exists $+$ in $\sigma_i({\bf c})^{\rm red}$ and the leftmost $+$ appears in $c_{\gamma_s}$, then 
$\tf_i{\bf c} = {\bf c} - {\bf 1}_{\gamma_s} + {\bf 1}_{\gamma'_s}.$
\item[(2)] If there exists no $+$ in $\sigma_i({\bf c})^{\rm red}$, then 
$\tf_i{\bf c} = {\bf c} + {\bf 1}_{\alpha_i}.$
\end{itemize}
\end{thm}

\section{Crystal of quantum nilpotent subalgebra}
\subsection{Crystal $\B_{{\bf i}_0}$}
From now on, we assume that $\g$ is of type $D_n$ ($n\geq 4$). 
We assume that the weight lattice is $P=\bigoplus_{i=1}^n\Z\epsilon_i$, where $\{\,\epsilon_i\,|\,1\leq i\leq n\,\}$ is an orthonormal basis with respect to a symmetric bilinear form $(\, ,\,)$. 
The set of positive roots is $\Phi^+=\{\,\ep_i\pm\ep_j\,|\,1\leq i<j\leq n\,\}$ and let $\alpha_i$ be the $i$-th simple root given by $\alpha_i = \ep_i - \ep_{i+1}$ for $i \neq n$, and $\alpha_n = \ep_{n-1}+\ep_n$.
Recall that $W$ acts faithfully on $P$ by $s_i(\ep_i)=\ep_{i+1}$, $s_i(\ep_k)=\ep_k$ for $1\leq i\leq n-1$ and $k\neq i, i+1$, and $s_n(\ep_{n-1})=-\ep_n$ and $s_n(\ep_k)=\ep_k$ for $k\neq n-1, n$.
The fundamental weights are $\varpi_i=\sum_{k=1}^i\epsilon_k$ for $i=1,\ldots, n-2$, $\varpi_{n-1}=(\epsilon_1+\cdots+\epsilon_{n-1}-\epsilon_n)/2$ and $\varpi_{n}=(\epsilon_1+\cdots+\epsilon_{n-1}+\epsilon_n)/2$. 
Put $J=I\setminus \{n\}$. 
Let $\mf l$ be the Levi subalgebra of $\mf g$ associated to $\{\,\alpha_i\,|\, i\in J\,\}$ of type $A_{n-1}$. Then 
\begin{equation*}
\Phi^+=\Phi^+(J)\cup \Phi^+_J,
\end{equation*}
where $\Phi^+_J=\{\,\ep_i-\ep_j\,|\,1\leq i<j\leq n\,\}$ is the set of positive roots of $\mf l$ and $\Phi^+(J)=\{\,\ep_i+\ep_j\,|\,1\leq i<j\leq n\,\}$ is the set of roots of the nilradical $\mf u$ of the parabolic subalgebra associated to ${\mf l}$.

Throughout this paper, we consider a specific $\bi_0\in R(w_0)$,
whose associated convex order on $\Phi^+$ is given by
\begin{equation}\label{eq:convex order i_0}
\begin{split}
&\ep_i+\ep_j \prec \ep_k-\ep_l,\\
\ep_i+\ep_j \prec \ep_k+& \ep_l \quad \Longleftrightarrow \quad 
\text{$(j>l)$ or $(j=l$, $i>k)$}, \\
\ep_i-\ep_j \prec \ep_k-& \ep_l \quad \Longleftrightarrow \quad
\text{$(i<k)$ or $(i=k$, $j<l)$},
\end{split}
\end{equation}
for $1\leq i<j\leq n$ and $1\leq k<l\leq n$.
An explicit form of $\bi_0$ is as follows.
%
For $1\leq k\leq n-1$, put
{\allowdisplaybreaks
\begin{align*}
%\begin{split}
\bi_k & = 
\begin{cases}
(n,n-2,\ldots,k+1,k), & \text{if $k$ is odd},\\
(n-1,n-2,\ldots,k+1,k), & \text{if $k$ is even},\\
(n), & \text{if $n$ is even and $k=n-1$},
\end{cases}
\\
\bi'_k & =
\begin{cases}
(n-1,n-2,\ldots, k+1,k), & \text{if $n$ is even and $1\leq k\leq n-1$},\\
(n,n-2,\ldots, k+1,k), & \text{if $n$ is odd and $1\leq k\leq n-2$},\\
(n), & \text{if $n$ is odd and $k=n-1$}.
\end{cases}
%\end{split}
\end{align*}}
Let 
$\bi^J = \bi_1 \cdots \cdots \cdot \bi_{n-1}$ and 
$\bi_J = \bi'_1 \cdots \cdots \cdot \bi'_{n-1}$. Then
%\begin{equation} \label{eq:rx}
$
\bi_0
$ as the concatenation ${\bf i}^J \cdot {\bf i}_J$.
%\end{equation}
%where $\bi\cdot \bj$ denotes the concatenation of $\bi\in I^r$ and $\bj \in I^s$.
We write $\bi_0=(i_1,\ldots,i_N)$,
where $i_1=n$, and put $\bi^J=(i_1,\ldots,i_M)$, $\bi_J=(i_{M+1},\ldots,i_N)$ with $N=n^2-n$ and  $M=N/2$. 
%\red{
%\begin{ex}\label{ex:i_0}
%{\rm 
%We have 
%\begin{equation*}
%\begin{split}
%&\bi^J=(4,2,1,3,2,4),\quad\quad\quad\quad\ \ \bi_J=(3,2,1,3,2,3),\quad\quad\quad\quad\quad  \text{when $n=4$},\\
%&\bi^J=(5,3,2,1,4,3,2,5,3,4),\quad \bi_J=(5,3,2,1,5,3,2,5,3,5),\, \quad \text{when $n=5$}.
%\end{split}
%\end{equation*}
%}
%\end{ex}
%}
Throughout the paper, we set
$
\B:=\B_{\bi_0}.
$
For ${\bf c}=(c_\beta)\in \B$, we also write 
\begin{equation*}
c_{\beta_k} = 
\begin{cases}
c_{\ov{j}\ov{i}}, & \text{if $\beta_k=\ep_i+\ep_j$ for $1\leq i<j\leq n$}, \\
c_{j\ov{i}} , & \text{if $\beta_k=\ep_i-\ep_j$ for $1\leq i<j\leq n$}.
\end{cases}
\end{equation*}

\begin{prop}\label{prop:signature for type D}
\mbox{}
\begin{itemize}
\item[(1)] The reduced word $\bi_0$ is simply braided for any $i\in I$.

\item[(2)] For $i\in I\setminus\{n\}$ and ${\bf c}\in \B$,  we have
%\begin{equation*}
%\begin{split}
$
\sigma_i({\bf c})=
\sigma_{i,1}({\bf c})\cdot \sigma_{i,2}({\bf c})\cdot \sigma_{i,3}({\bf c}),
$
%\end{split}
%\end{equation*}
where
\begin{equation} \label{eq:sigma}
\begin{split}
\sigma_{i,1}({\bf c})=&
(
\underbrace{-\cdots -}_{c_{\ov{n}\,\ov{i}}}\,\underbrace{+\cdots +}_{c_{\ov{n}\,\ov{i+1}}}
\underbrace{-\cdots -}_{c_{\ov{n-1}\,\ov{i}}}\,\underbrace{+\cdots +}_{c_{\ov{n-1}\,\ov{i+1}}}
\ \cdots \
\underbrace{-\cdots -}_{c_{\ov{i+2}\,\ov{i}}}\,\underbrace{+\cdots +}_{c_{\ov{i+2}\,\ov{i}}}), \\
\sigma_{i,2}({\bf c})=& (\underbrace{-\cdots -}_{c_{\ov{i}\,\ov{i-1}}}\,\underbrace{+\cdots +}_{c_{\ov{i+1}\,\ov{i-1}}}
\underbrace{-\cdots -}_{c_{\ov{i}\,\ov{i-2}}}\,\underbrace{+\cdots +}_{c_{\ov{i+1}\,\ov{i-2}}}
\ \cdots \
\underbrace{-\cdots -}_{c_{\ov{i}\,\ov{1}}}\,\underbrace{+\cdots +}_{c_{\ov{i+1}\,\ov{1}}}
), \\
\sigma_{i,3}({\bf c})=& (\underbrace{-\cdots -}_{c_{{i+1}\,\ov{1}}}\,\underbrace{+\cdots +}_{c_{{i}\,\ov{1}}} \underbrace{-\cdots -}_{c_{{i+1}\,\ov{2}}}\,\underbrace{+\cdots +}_{c_{{i}\,\ov{2}}}\ \cdots \ \underbrace{-\cdots -}_{c_{{i+1}\,\ov{i-1}}}\,\underbrace{+\cdots +}_{c_{{i}\,\ov{i-1}}} \underbrace{-\cdots -}_{c_{{i+1}\,\ov{i}}}).
\end{split}
\end{equation}
Here $c_{ab}$ is assumed to be zero when it is not defined.
\end{itemize}
\end{prop}

\begin{rem}
The crystal operator $\widetilde{f}_i$ on ${\bf B}$ can be described in the same way as in Theorem \ref{thm:signature rule} with $\sigma_i({\bf c})$  in Proposition \ref{prop:signature for type D} (see \cite[Remark 3.3]{JK} for more details).
%Although we need some $2$-term moves on ${\bf i}^J$ on showing (1) in Proposition \ref{eq:sigma} (see \cite[Example 3.3]{JK}), we can apply \cite[Theorem 4.6]{SST} for ${\bf i}_0$ because the bracketing rule is local.
\end{rem}

Set
\begin{equation}
\begin{split}
\B^J &=\left\{\,{\bf c}=(c_{\beta})\in \B\,\big\vert \,c_{\beta}=0 \text{ unless $\beta\in \Phi^{+}(J)$}\,\right\},\\
\B_J &=\left\{\,{\bf c}=(c_{\beta})\in \B\,\big\vert \,c_{\beta}=0 \text{ unless $\beta\in \Phi_J$}\,\right\}. 
\end{split}
\end{equation}
which we regard them as subcrystals of $\B$, where we assume that 
$\te_n{\bf c}=\tf_n{\bf c}={\bf 0}$ with  
$\varepsilon_n({\bf c})=\varphi_n({\bf c})=-\infty$ for ${\bf c}\in \B_J$.
The subcrystal $\B^J$ is the crystal of the quantum nilpotent subalgebra $U^-_q(w^J)$, where $w^J=s_{i_1}\cdots s_{i_M}$ with $\bi^J=(i_1,\ldots,i_M)$, which can be viewed as a $q$-deformation of $U(\mf u^-)$ by definition.


\begin{cor} \label{tensor_decomposition}
\mbox{}
The map $\B \longrightarrow \B^J \otimes \B_J$ sending ${\bc}$ to ${\bc}^J \otimes {\bc}_J$
%\begin{equation*}
%\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
%\B \ar@{->}[r]  & \ \B^J \,\otimes \B_J \\
%{\bf c}  \ar@{|->}[r] &   {\bf c}^J\otimes{\bf c}_{J}}
%\end{equation*}
is an  isomorphism of $\mf g$-crystals.
\end{cor}

\subsection{Subcrystal $\B^J$} \label{section:Delta_n}
Let us consider the subcrystal $\B^J$ in more details.  
Let $\Delta_n$ be the arrangements of dots in the plane to represent the $(n-1)$-th triangular number. 
We identify $\Delta_n$ with $\Phi^+(J)$ in such a way that $\ep_{k+1}+\ep_{l+1}$, $\ep_{k+1}+\ep_{l}$ and $\ep_{k}+\ep_{l}$ for $1\leq k,l\leq n-1$ are the vertices of a triangle of minimal shape in $\Delta_n$ as follows:
\begin{equation}\label{eq:small triangle}
\xymatrixcolsep{-1.8pc}\xymatrixrowsep{0.1pc}\xymatrix{
& &    & & \stackrel{{\ep_k+\ep_{l+1}}}{\bullet} & & &  & \\
&  & & \stackrel{{\ep_{k+1}+\ep_{l+1}}}{\bullet} & &  \ {}\stackrel{{\ep_k+\ep_l}}{\bullet}  }
\end{equation}
%
We also identify ${\bf c}\in \B^J$ with an array of $c_{\beta}$'s in ${\bf c}$ with $c_\beta$ at the corresponding dot in $\Delta_n$.

%\begin{ex}\label{ex:Delta_n}
%{\rm For $n=5$ and $\bc\in \B^J$, we have
%\begin{equation*} 
%\xymatrixcolsep{-0.9pc}\xymatrixrowsep{0.0pc}
%\xymatrix{
%& & & & \stackrel{{}_{\ep_1+\ep_5}}{\bullet} & & &  & \\
%\Delta_5=\quad & & & \stackrel{{}_{\ep_2+\ep_5}}{\bullet} & &  {}\stackrel{{}_{\ep_1+\ep_4}}{\bullet} & & \\
%& & \stackrel{{}_{\ep_3+\ep_5}}{\bullet}  & & \stackrel{{}_{\ep_2+\ep_4}}{\bullet} &  & \stackrel{{}_{\ep_1+\ep_3}}{\bullet}  & \\  
%& \stackrel{{}_{\ep_4+\ep_5}}{\bullet} & & \stackrel{{}_{\ep_3+\ep_4}}{\bullet} & & \stackrel{{}_{\ep_2+\ep_3}}{\bullet} & & \stackrel{{}_{\ep_1+\ep_2}}{\bullet}}
%\quad\quad
%\xymatrixcolsep{-0.2pc}\xymatrixrowsep{0.3pc}\xymatrix{
%& & & & c_{\ov{5}\ov{1}} & & & & \\
%\bc= & & & c_{\ov{5}\ov{2}} & & c_{\ov{4}\ov{1}} & & \\
%& & c_{\ov{5}\ov{3}} & & c_{\ov{4}\ov{2}} & & c_{\ov{3}\ov{1}} & \\  
%& c_{\ov{5}\ov{4}} & & c_{\ov{4}\ov{3}} & & c_{\ov{3}\ov{2}} & & c_{\ov{2}\ov{1}}}
%\end{equation*}\vskip 3mm
%}
%\end{ex}

\begin{lem}\label{lem:B^J}
We have
$\B^J=\{\,{\bf c}\,|\,\varepsilon_i^*({\bf c})=0\ (i\in J)\,\}$.
\end{lem} 

For $s\geq 1$, let 
\begin{equation}\label{eq:B^J_s-1}
\B^{J,s}:=\{\,{\bf c}\in \B^J\,|\,\varepsilon_n^*({\bf c})\leq s\,\},
\end{equation} 
which is a subcrystal of $\B^J$.
By \cref{lem:B^J} and \cite[Proposition 8.2]{Kas95} %(cf. \cite{Kas93})
, we have 
\begin{equation}\label{eq:B^J_s-2}
B(s\varpi_n)\cong \B^{J,s}\otimes T_{s\varpi_n},\quad\quad \bigcup_{s\geq 1}\B^{J,s}=\B^J,
\end{equation}
as $\mf g$-crystals.
By \eqref{eq:B^J_s-2}, $\B^J$ is a regular $\mf l$-crystal, that is, any connected component with respect to $\te_i$ and $\tf_i$ for $i\in J$ is isomorphic to the crystal of an integrable highest weight $U_q(\mf l)$-module, say $B_J(\la)$ for some $\la =\sum_{i=1}^n\la_i\ep_i\in P$ with $(\la,\alpha_i)\in \Z_+$ for $i\in J$.
%
Then \cref{prop:signature for type D} enables us to decompose $\B^J$ into $\mf l$-crystals directly as follows, and hence the decomposition of $U_q(w^J)$ into irreducible $U_q(\mf l)$-modules.
\begin{prop}\label{prop:decomp of B^J}
As an $\mf l$-crystal, we have
%\begin{equation*}
$
\B^J \cong \bigsqcup_{\la}B_J(\la),
$
%\end{equation*}
where the union is over $\la =\sum_{i=1}^n\la_i\ep_i\in P$ such that $0\ge \la_1=\la_2\ge \la_3=\la_4\ge \cdots$. 
\end{prop}

\subsection{Combinatorial description of $\varepsilon_n^*$} \label{section:bipath}
Let us give an explicit combinatorial description of $\varepsilon_n^*$ on $\B^J$, whose proof is obtained by using the formula of Berenstein-Zelevinsky \cite{BZ-3}.

\begin{df}{\rm

 A {\em path in $\Delta_n$} is a sequence $p=(\gamma_1,\ldots,\gamma_s)$ in $\Phi^+(J)$ for some $s\geq 1$ such that
\begin{itemize} 
\item[(1)] $\gamma_1,\ldots,\gamma_s\in \Phi^+(J)$,
\item[(2)] if $\gamma_{i} = \ep_{k}+\ep_{l+1}$ for some $k<l$, then $\gamma_{i+1}=\ep_{k+1}+\ep_{l+1}$ or $\ep_k+\ep_{l}$ (see \eqref{eq:small triangle}), 
\item[(3)] $\gamma_s = \ep_k+\ep_{k+1}$ for some $k$.
\end{itemize}
For $\beta\in \Phi^+(J)$, a {\em double path at $\beta$ in $\Delta_n$} is a pair of paths ${\bf p}=(p_1,p_2)$ in $\Delta_n$ of the same length with $p_1=(\gamma_1,\ldots,\gamma_s)$ and $p_2=(\delta_1,\ldots,\delta_s)$ such that 
\begin{itemize} 
\item[(1)] $\gamma_1=\delta_1=\beta$,   
\item[(2)] $\gamma_i$ is located to the strictly left of $\delta_i$ for $2\leq i\leq s$, 
\item[(3)] $\gamma_s=\ep_{k+1}+\ep_{k+2}$, $\delta_s=\ep_k+\ep_{k+1}$ for some $k\geq 1$.
\end{itemize}}
\end{df}

\begin{ex}\label{ex:double path}
{\rm The followings are some examples of double paths ${\bf p}$ at $\theta = \ep_1+\ep_5$ in $\Delta_5$.  
{\allowdisplaybreaks\begin{align*}
%\begin{split}
&\xymatrixcolsep{0.1pc}\xymatrixrowsep{0.1pc}
\xymatrix{
 &  & &  {\underset{\theta}{\bullet}}\ar@{->}[ld]\ar@{->}[rd] & & &  & \\
  & &  {\bullet}\ar@{->}[ld]  & &  {\bullet}\ar@{->}[ld] & & \\
   &  {\bullet}\ar@{->}[ld] & &  {\bullet}\ar@{->}[ld] &  &  {\bullet}  & \\  
     {\bullet} & &  {\bullet} & &  {\bullet} & &  {\bullet}
     } \ \ \ \
\xymatrix{
 &  & &  {\underset{\theta}{\bullet}}\ar@{->}[ld]\ar@{->}[rd] & & &  & \\
  & &  {\bullet}\ar@{->}[ld] & &  {\bullet}\ar@{->}[ld]  & & \\
   &  {\bullet}\ar@{->}[rd] & &  {\bullet}\ar@{->}[rd] &  &  {\bullet} & \\  
     {\bullet} & &  {\bullet} & &  {\bullet} & &  {\bullet}
     }\ \ \ \
\xymatrix{
 &  & &  {\underset{\theta}{\bullet}}\ar@{->}[ld]\ar@{->}[rd] & & &  & \\
  & &  {\bullet}\ar@{->}[ld] & &  {\bullet}\ar@{->}[rd] & & \\
   &  {\bullet}\ar@{->}[rd] & &  {\bullet}  &  &  {\bullet}\ar@{->}[ld] & \\  
     {\bullet} & &  {\bullet} & &  {\bullet} & &  {\bullet}
     }
%\end{split}
\end{align*}}
}
\end{ex}
For ${\bf c}\in \B^J$ and a double path ${\bf p}$, let 
%\begin{equation}
$||{\bf c}||_{\bf p} = \sum_{\text{$\beta$ lying on ${\bf p}$}}c_\beta$.
%\end{equation}

\begin{thm}\label{thm:epsilon^*_n}
For ${\bf c}\in \B^J$, $
\varepsilon_n^*({\bf c}) = \max\{\,||{\bf c}||_{\bf p}\,|\,\text{$\bf p$ is a double path at $\theta$ in $\Delta_n$}\,\},$ where $\theta = \ep_1+\ep_n$.
\end{thm}

%\begin{rem}{\rm
%Let $\theta=\ep_1+\ep_n$ is the longest root in $\Phi^+$. Since $\theta$ is located at the top of $\Delta_n$, the formula in Theorem \ref{thm:epsilon^*_n} is equivalent to 
%\begin{equation*}
%\varepsilon_n^*({\bf c}) = \max\{\,||{\bf c}||_{\bf p}\,|\,\text{$\bf p$ is a double path at $\theta$ in $\Delta_n$}\,\}.
%\end{equation*}
%}
%\end{rem}


\subsection{Realization of KR crystals $B^{n,s}$} 
Let $\hat{\mf g}$ be an affine Kac-Moody algebra of type $D_n^{(1)}$ with
$\hat{I}=\{\,0,1,\ldots,n\,\}$ the index set for the simple roots.

\begin{center}
\setlength{\unitlength}{0.16in}
\begin{picture}(16,4.5)
\put(2.8,0.7){\makebox(0,0)[c]{$\bigcirc$}}
\put(2.8,3.3){\makebox(0,0)[c]{$\bigcirc$}}
\put(5.6,2){\makebox(0,0)[c]{$\bigcirc$}}
\put(10.4,2){\makebox(0,0)[c]{$\bigcirc$}}
\put(13.1,3.3){\makebox(0,0)[c]{$\bigcirc$}}
\put(13.1,0.7){\makebox(0,0)[c]{$\bigcirc$}}
%
\put(5.3,1.8){\line(-2,-1){2.1}}
\put(5.3,2.2){\line(-2,1){2.1}}
%
\put(6,2){\line(1,0){1.3}} 
\put(8.7,2){\line(1,0){1.3}} 
%
\put(10.7,2.2){\line(2,1){2}}
\put(10.7,1.8){\line(2,-1){2}}

\put(8,1.95){\makebox(0,0)[c]{$\cdots$}}
%
\put(2.8,2.5){\makebox(0,0)[c]{\tiny ${\alpha}_0$}}
\put(2.8,0.0){\makebox(0,0)[c]{\tiny ${\alpha}_1$}}
\put(5.6,1){\makebox(0,0)[c]{\tiny ${\alpha}_2$}}
\put(10.4,1){\makebox(0,0)[c]{\tiny ${\alpha}_{n-2}$}}
\put(13.1,2.5){\makebox(0,0)[c]{\tiny ${\alpha}_{n-1}$}}
\put(13.1,0.0){\makebox(0,0)[c]{\tiny ${\alpha}_{n}$}}

\end{picture}
\end{center}

For $r\in \{0,n\}$, let $\hat{\mf g}_r$ be the subalgebra of $\hat{\mf g}$ corresponding to $\{\,\alpha_i\,|\,i\in \hat{I} \setminus\{r\}\,\}$. Then $\hat{\mf g}_0=\mf g$, and $\hat{\mf g}_0\cap \hat{\mf g}_n=\mf l$. 
%Let $\hat{\mf h}$ be the Cartan subalgebra of $\hat{\mf g}$.
Let $\hat{P}=\bigoplus_{i\in \hat{I}}\Z\La_i\oplus\Z\delta$ be the weight lattice of $\hat{\g}$, where $\delta$ is the positive imaginary null root and $\La_i$ is the $i$-th fundamental weight %(see \cite{K})
. We regard $P=\bigoplus_{i=1}^n\Z\epsilon_i$ as a sublattice of $\hat{P}/\Z\delta$ by putting 
$\epsilon_1=\Lambda_1-\Lambda_0$, $\epsilon_2=\Lambda_2-\Lambda_1-\Lambda_0$, $\epsilon_k=\Lambda_k-\Lambda_{k-1}$ for $k=3,\ldots, n-2$, $\epsilon_{n-1}=\Lambda_{n-1}+\Lambda_n-\Lambda_{n-2}$ and $\epsilon_{n}=\Lambda_{n}-\Lambda_{n-1}$.
In particular, we have $\alpha_0 = -\ep_1-\ep_2$ in $P$. 
If $\varpi'_i$ are the fundamental weights for $\hat{\mf g}_{n}$ for $i\in \hat{I}\setminus\{n\}$, then $\varpi'_i=\varpi_i$ for $i\in \hat{I}\setminus\{0,n\}$ and $\varpi'_0=-\varpi_n$.

For ${\bf c}\in \B^J$, define
\begin{equation}\label{eq:e_0 and f_0 on B^J}
\begin{split}
&\te_0{\bf c} = {\bf c} + {\bf 1}_{\ep_1+\ep_2},\quad
\tf_0{\bf c} = 
\begin{cases}
{\bf c} - {\bf 1}_{\ep_1+\ep_2},& \text{if $c_{\ep_1+\ep_2}>0$},\\
{\bf 0}, & \text{otherwise},
\end{cases}\\
&\varphi_0({\bf c})=\max\{\,k \,|\, \tf_0^k {\bf c}\neq {\bf 0}\,\}, \ \ 
\varepsilon_0({\bf c})=\varphi_0({\bf c})- ({\rm wt}({\bf c}),\alpha_0).
\end{split}
\end{equation}

\begin{lem}\label{lem:affine crystal B^J}
The set $\B^J$ is a $\hat{\mf g}$-crystal with respect to ${\rm wt}$, $\varepsilon_i$, $\varphi_i$, $\te_i$, $\tf_i$ for $i\in \hat{I}$, where ${\rm wt}$ is the restriction of ${\rm wt} : \B \longrightarrow P$ to $\B^J$.
\end{lem}
%\red{
%\begin{rem}{\rm
%The inclusions  
%	$\B^{J,s}\hookrightarrow\B^{J,t}\hookrightarrow\B^J$
%	for $s \le t$ are embeddings of $\widehat{\g}$-crystals (cf. \eqref{eq:B^J_s-2}), and hence $\B^J$ is a direct limit of $\left\{ \B^{J,s} \,|\, s \in \mathbb{Z}_+ \right\}$.
%	}
%\end{rem}
%}


\begin{thm}\label{thm:main-1}
For $s\geq 1$, $\B^{J,s}\otimes T_{s\varpi_n}$ is a regular $\hat{\mf g}$-crystal and $\B^{J,s}\otimes T_{s\varpi_n} \cong B^{n,s},$ 
where $B^{n,s}$ is the Kirillov-Reshetikhin crystal of type $D_n^{(1)}$ associated to $s\varpi_n$ (cf.\cite{FOS}).
\end{thm}

\begin{rem}\label{rem:polytope and limit}
By \cref{thm:epsilon^*_n}, we have 
	%\begin{equation*}
	$
	\B^{J,s} = \underset{\text{${\bf p}$}}{\bigcap}\left\{ {\bf c} \in \B^J \,|\, ||{\bf c}||_{\bf p} \le s \right\},
	$
	%\end{equation*}
	where $\bf p$ runs over the double paths in $\Delta_n$. This gives a polytope realization of the KR crystal $B^{n,s}$.
	Also, we note that by construction, the crystal $\B^J$ is the direct limit of $\{ \B^{J,s} : s \in \mathbb{Z}_{+} \}$. By \cite{FOS2}, $\{ B^{n,s} \}$ is a family of perfect KR crystals. It is conjectured that $\{ B^{n,s} \}$ has the limit in the sense of \cite{KNO}, that is, $\{ B^{n,s} \}$ is a coherent family.
\end{rem}

\section{RSK correspondence for type $D$ and affine crystals}\label{sec:burge}
%
%Let us recall some necessary notions. %following \cite{Ful}. 

\subsection{Burge correspondence}
Let $\mathscr{P}$ be the set of partitions $\lambda=(\lambda_i)_{i\geq 1}$, 
which are often identified with Young diagrams. 
%We identify a partition $\lambda=(\lambda_i)_{i\geq 1}$ with a {Young diagram}. 
%or a subset $\{\,(i,j)\,|\,1\leq j\leq
%\lambda_i\,\}$ of $\mathbb{N}\times\mathbb{N}$. 
Let $\lambda'=(\lambda'_i)_{i\geq 1}$ be the conjugate of $\lambda$, and 
let $\lambda^\pi$ be the skew Young diagram obtained by $180^{\circ}$-rotation of $\lambda$. Let $\ell(\lambda)$ denote the {length of $\lambda$}, and let $\cP_n=\{\,\la\,|\,\ell(\la)\leq n\,\}$.
Let $[\ov{n}]:=\{\,\ov{n}<\cdots< \ov{1}\,\}$ be a linearly ordered set. 
Let $\W$ be the set of finite words in $[\ov{n}]$. For a skew Young diagram $\lambda^{\pi}$, let $SST_{[\ov{n}]}(\lambda^{\pi})$ or simply $SST(\la^{\pi})$ denote the set of semistandard tableaux of shape $\lambda^{\pi}$ with entries in $[\ov{n}]$.
For $T\in SST(\lambda^{\pi})$, let $w(T)$ be a word in $\W$ obtained by reading the entries of $T$ row by row from top to bottom, and from right to left in each row, and let ${\rm sh}(T)$ denote the shape of $T$.
Note that we use English convention for partitions and tableaux.
Let $T^{\nw}$ be the unique semistandard tableau such that ${\rm sh}(T^{\nw})\in \cP$ and $w(T^{\nw})$ is Knuth equivalent to $w(T)$. We define $T^{\se}$ in a similar way such that ${\rm sh}(T^{\se})\in \cP^\pi$. Note that if ${\rm sh}(T^{\nw})=\nu$, then ${\rm sh}(T^{\se})=\nu^\pi$.
For $a\in [\ov{n}]$ and $U\in SST(\lambda)$ with $\la\in\cP_n$, let $a\rightarrow U$ be the tableau obtained by applying the Schensted's column insertion of $a$ into $U$. 
Similarly, for $V\in SST(\lambda^\pi)$ and $b\in [\ov{n}]$, let $V\leftarrow b$ be the tableau obtained by applying the Schensted's column insertion of $b$ into $V$ in a reverse way starting from the rightmost column. 
For $w=w_1\ldots w_r\in\W$, we define $P(w)=( w_r\rightarrow(\cdots(w_2\rightarrow w_1)\cdots))$. Note that $P(w)^{\se} = ( (w_r\leftarrow w_{r-1}) \leftarrow \cdots \leftarrow w_1)$.
%

Let us recall a variation of RSK correspondence for type $D$ \cite{B}. Set 
\begin{equation}
\begin{split}
{\mc T}^{\se}:= \bigsqcup_{\substack{\la\in \cP_n \\ \la' \text{:even}}} SST(\la^\pi),
\quad\quad
{\mc T}^{\nw}:= \bigsqcup_{\substack{\la\in \cP_n \\ \la' \text{:even}}} SST(\la),
\end{split}
\end{equation}
where we say that $\la'$ is even if each part of $\la'$ is even
Let $\Omega$ be the set of biwords $(\ba,\bb)\in
\W\times \W$ such that
\begin{itemize}
\item[(1)] $\ba=a_1\cdots a_r$ and $\bb=b_1\cdots b_r$ for some $r\geq 0$,

\item[(2)] $a_i < b_i$ for $1\leq i\leq r$,

\item[(3)] $(a_1,b_1)\leq \cdots \leq (a_r,b_r)$, 
\end{itemize}
where $(a,b)< (c,d)$ if and only if $(a<c)$ or ($a=c$ and $b>d$) for $(a,b)$, $(c,d)\in \W \times \W$. 
We denote by ${\bf c}(\ba,\bb)$ the unique element in $\B^J$ corresponding to $(\ba,\bb) \in \Omega$ such that $c_{ab}=\left|\{\,k\,|\,(a_k,b_k)=(a,b) \,\}\right|$.

For $(\ba,\bb)\in \Omega$ with $\ba=a_1\cdots a_r$ and $\bb=b_1\cdots b_r$, we define a sequence of tableaux $P_r, P_{r-1}, \ldots, P_1$ inductively as follows:
\begin{itemize}
\item[(1)] let $P_1$ be a vertical domino  
{\tiny 
${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{.1ex}{$#1$}}}\raisebox{-2.5ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{a_r}\\
\cline{1-1}
\lr{b_r}\\
\cline{1-1}
\end{array}$}
}$
},

\item[(2)] if $P_{k+1}$ is given for $1\leq k\leq r-1$, then define $P_{k}$ to be the tableau obtained by first applying the column insertion to get $P_{k+1}\leftarrow b_{k}$, and then adding $\boxed{a_{k}}$ at the conner of $P_{k+1}\leftarrow b_{k}$ located above the box ${\rm sh}(P_{k+1}\leftarrow b_{k})/{\rm sh}(P_{k+1})$.
\end{itemize}
We put $P^{\se}(\ba,\bb):=P_1$.
It is not difficult to see from the definition that $P^{\se}(\ba,\bb) \in SST(\la^\pi)$ for some $\la\in \cP$ such that $\la'$ is even.

For ${\bf c}\in \B^J$, let $P^{\se}({\bf c})=P^{\se}(\ba,\bb)$ where ${\bf c}={\bf c}(\ba,\bb)$.
Since the map $(\ba,\bb) \mapsto P^{\se}(\ba,\bb)$ is a bijection from $\Omega$ to $\mc T^{\se}$ \cite{B}, we have a bijection 
\begin{equation}\label{eq:kappa_se}
\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
\kappa^{\se} : \B^J  \ \ar@{->}[r] & \ {\mc T}^{\se}  %\\
%\quad {\bf c} \ar@{|->}[r] & P^{\se}({\bf c})
},
\end{equation} where $\kappa^{\se}(\bc) = P^{\se}(\bc)$.
Similarly, let $\Omega'$ be the set of biwords $(\ba,\bb)\in
\W \times \W$ satisfying the same conditions as in $\Omega$ except that $<$ is replaced by $<'$, where $(a,b)<' (c,d)$ if and only if $(b<d)$ or ($b=d$ and $a<c$) for $(a,b)$ and $(c,d)\in \W \times \W$. We define ${\bf c}'(\ba,\bb)$ in the same way as in ${\bf c}(\ba,\bb)$. Given $(\ba,\bb)\in \Omega'$ with $\ba=a_1\cdots a_r$ and $\bb=b_1\cdots b_r$, define a sequence of tableaux $P_1, P_2, \ldots, P_r$ inductively as follows:
\begin{itemize}
\item[(1)] let $P_1$ be a vertical domino  
{\tiny 
${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{.1ex}{$#1$}}}\raisebox{-2.5ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{a_1}\\
\cline{1-1}
\lr{b_1}\\
\cline{1-1}
\end{array}$}
}$
},

\item[(2)] if $P_{k-1}$ is given for $2\leq k\leq r$, then define $P_{k}$ to be the tableau obtained by first applying the column insertion to get $a_k\rightarrow P_{k-1}$, and then adding $\boxed{b_{k}}$ at the conner of $a_k\rightarrow P_{k-1}$ located below the box ${\rm sh}(a_k\rightarrow P_{k-1})/{\rm sh}(P_{k-1})$,
\end{itemize}
and put $P^{\nw}(\ba,\bb):=P_r$.
For ${\bf c}\in \B^J$, let $P^{\nw}({\bf c})=P^{\nw}(\ba,\bb)$ where ${\bf c}={\bf c}'(\ba,\bb)$. Then we also have a bijection 
\begin{equation}\label{eq:kappa_nw}
\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
\kappa^{\nw} : \B^J  \ \ar@{->}[r] & \ {\mc T}^{\nw}  %\\
%\quad {\bf c} \ar@{|->}[r] & P^{\nw}({\bf c})
},
\end{equation} where $\kappa^{\nw}(\bc) = P^{\nw}(\bc)$

\subsection{Isomorphism of affine crystals}

We regard
$[\ov{n}]=\{\,\ov{n}<\cdots< \ov{1}\,\}$ as the crystal of dual natural representation of $\mf l$ with ${\rm wt}(\ov{k})=-\epsilon_k$. 
Then $\W$ is a regular ${\mf l}$-crystal, where $w=w_1\ldots w_r$ is identified with $w_1\otimes \cdots \otimes w_r$.
For $\la\in \cP_n$, $SST(\la)$ is a regular $\mf l$-crystal with lowest weight $-\sum_{i=1}^n\la_i\ep_i$, where $T$ is identified with $w(T)$. %\cite{KN}. 
In particular $\mc T^{\se}$ and ${\mc T}^{\nw}$ are regular ${\mf l}$-crystals.

Let us recall the $\hat{\mf g}_0$-crystal structure on $\mc T^{\se}$ \cite[Section 5.2]{K13}.
Let $T\in {\mc T}^{\se}$ be given. For $k\geq 1$, let $t_k$  be the entry
in the top of the $k$-th column of $T$ (enumerated from the right). Consider  $\sigma=(\sigma_1,\sigma_2,\ldots)$, where
\begin{equation*}
\sigma_k=
\begin{cases}
+ \ , & \text{if $t_k> \ov{n-1}$ or the $k$-th column is empty}, \\
- \ ,& \text{if  the $k$-th column has both $ \ov{n-1}$ and $ \ov{n}$ as its entries}, \\
\,\cdot\ \ , & \text{otherwise}.
\end{cases}
\end{equation*}
Then $\te_n T$ is
obtained from $T$ by removing  
%
\raisebox{-.6ex}{{\tiny ${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}\raisebox{-.6ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{ \ov{n}}\\
\cline{1-1}
\lr{ \!\ov{n\!-\!1}\!}\\
\cline{1-1}
\end{array}$}}$}}
%
in the column  corresponding to the right-most $-$ in ${\sigma}^{\rm red}$ (see \cref{subssec:tensor product rule on B} for $\sigma^{\rm red}$).  If there is no such $-$ sign, then we define $\te_n T={\bf 0}$, and 
$\tf_n T$ is obtained from $T$ by adding  
%
\raisebox{-.6ex}{{\tiny ${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}\raisebox{-.6ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{ \ov{n}}\\
\cline{1-1}
\lr{\!\ov{n\!-\!1}\!}\\
\cline{1-1}
\end{array}$}}$}}
%
column  corresponding to the left-most $+$ in ${\sigma}^{\rm red}$. 
%
Hence $\mc T^{\se}$ is a $\hat{\mf g}_0$-crystal with respect to ${\rm wt}$, 
$\varepsilon_i$, $\varphi_i$,
$\te_i$, $\tf_i$ $(i\in \hat{I} \setminus\{0\})$, where 
$\varepsilon_n(T)=\max\{\,k \,|\, \te_n^k T\neq {\bf 0}\,\}$ and
$\varphi_n(T)=\varepsilon_n(T)+ ({\rm wt}(T), \alpha_n)$.

Similarly, we have a $\hat{\mf g}_n$-crystal structure on $\mc T^{\nw}$ \cite[Section 5.2]{K13}. 
Let  $T\in {\mc T}^{\nw}$ be given.
For $k\geq 1$, let $t_k$  be the entry
in the bottom of the $k$-th column of $T$ (enumerated from the left). Consider  $\sigma=(\ldots,\sigma_2,\sigma_1)$, where
\begin{equation*}
\sigma_k=
\begin{cases}
- \ , & \text{if $t_k< \ov{2}$ or the $k$-th column is empty}, \\
+ \ ,& \text{if  the $k$-th column has both $\ov{1}$ and $\ov{2}$ as its entries,}\\
\, \cdot \ \ , & \text{otherwise}.
\end{cases}
\end{equation*}
Then $\te_0 T$ is given by adding
%
\raisebox{-.6ex}{{\tiny ${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}\raisebox{-.6ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{ \ov{2}}\\
\cline{1-1}
\lr{ \ov{1}}\\
\cline{1-1}
\end{array}$}}$}}
%
to the bottom of the column  corresponding to the right-most $-$ in ${\sigma}^{\rm red}$,
and $\tf_0 T$ is obtained from $T$ by removing
%
\raisebox{-.6ex}{{\tiny ${\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}}}\raisebox{-.6ex}
{$\begin{array}[b]{c}
\cline{1-1}
\lr{ \ov{2}}\\
\cline{1-1}
\lr{ \ov{1}}\\
\cline{1-1}
\end{array}$}}$}}
%
in the
column  corresponding to the left-most $+$ in ${\sigma}^{\rm red}$. 
If there is no such $+$ sign, then we define $\tf_0 T={\bf 0}$.
Hence ${\mc T}^{\nw}$ is a $\hat{\mf g}_n$-crystal with respect to ${\rm wt}$, $\varepsilon_i$, $\varphi_i$, $\te_i$, $\tf_i$ $(i\in \hat{I}\setminus\{n\})$, where 
$\varphi_0(T)=\max\{\,k\ \,|\, \tf_0^k T\neq {\bf 0}\,\}$ and
$\varepsilon_0(T)=\varphi_0(T)- ({\rm wt}(T), \alpha_0)$.

\begin{thm}\label{thm:isomorphism theorem}
The bijection $\kappa^{\se}$ in \eqref{eq:kappa_se} is an isomorphism of $\hat{\mf g}_0$-crystals, and  
the bijection $\kappa^{\nw}$ in \eqref{eq:kappa_nw} is an isomorphism of $\hat{\mf g}_n$-crystals.
\end{thm}
\begin{rem}
It would be interesting to compare \cref{thm:isomorphism theorem} with the result in \cite{SST16}.
\end{rem}

For a semistandard tableau $T$ of skew shape, let $[T]$ denote the equivalence class of $T$ with respect to Knuth equivalence. Let 
${\mc T}=\{\,[T]\,|\,T\in {\mc T}^{\se}\,\}=\{\,[T]\,|\,T\in {\mc T}^{\nw}\,\}.$

If we define $\td{x}_i[T]=[\td{x}_0 T^{\nw}]$ when $i=0$, $\td{x}_i[T] =[\td{x}_n T^{\se}]$ when $i=n$, and $\td{x}_i[T] =[\td{x}_iT]$ otherwise
%\begin{equation*}
%\begin{split}
%\td{x}_i[T]=
%\begin{cases}
%[\td{x}_0 T^{\nw}], & \text{if $i=0$}, \\
%[\td{x}_n T^{\se}], & \text{if $i=n$}, \\
%[\td{x}_iT], & \text{otherwise},
%\end{cases}
%\end{split}
%\end{equation*}
for $i\in \hat{I}$ and $x=e,f$ (we assume that $[{\bf 0}]={\bf 0}$), then the set ${\mc T}$
is a $\hat{\g}$-crystal with respect to $\te_i$, $\tf_i$ $(i\in I)$, 
where ${\rm wt}$, $\varepsilon_i$, and $\varphi_i$ are induced from either ${\mc T}^{\se}$ or ${\mc T}^{\nw}$ \cite{K13}.
\begin{cor}
The map $\kappa : \B^J \longrightarrow {\mc T}$ sending ${\bc}$ to $[P^{\nw}({\bf c})]=[P^{\se}({\bf c})]$
%\begin{equation*}
%\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
%\kappa : \B^J  \ \ar@{->}[r] & \ {\mc T} \quad\quad\quad\quad\quad \\
%\quad {\bf c} \ar@{|->}[r] & [P^{\nw}({\bf c})]=[P^{\se}({\bf c})]}
%\end{equation*}
is an isomorphism of $\hat{\mf g}$-crystals.
\end{cor}

\subsection{Shape formula} 

For ${\bf c}\in \B^J$, let 
%\begin{equation}\label{eq:la(c)}
$
\la({\bf c})=(\la_1({\bf c})\geq \ldots \geq  \la_{\ell}({\bf c}))
$
%\end{equation}
be the partition corresponding to the regular ${\mf l}$-subcrystal of $\B^J$ including ${\bf c}$, that is, $\la({\bf c})={\rm sh}(\kappa^{\nw}({\bf c}))$ by \cref{thm:isomorphism theorem}. 
Note that $\ell=2[\frac{n}{2}]$ and $\la_{2i-1}(\bc)=\la_{2i}(\bc)$ for $1\leq i\leq [\frac{n}{2}]$.
%We have by Theorem \ref{thm:epsilon^*_n}
%\begin{equation}\label{eq:formula for la_1}
%\la_1({\bf c}) = \max\{\,||{\bf c}||_{\bf p}\,|\,\text{$\bf p$ is a double path at $\theta$}\,\},
%\end{equation} 

%We can further characterize the whole partition $\la(\bc)$ in terms of double paths on $\Delta_n$ generalizing \eqref{eq:formula for la_1} as follows. 
\begin{thm} \label{thm:shape}
For ${\bf c}\in \B^J$ and $1\leq l\leq [\frac{n}{2}]$, we have
\begin{equation*}
\la_1({\bf c})+\la_3({\bf c})+\cdots+\la_{2l-1}({\bf c}) = 
\max_{{\bf p}_1,\ldots , {\bf p}_{l}}
\{\,||{\bf c}||_{{\bf p}_1}+\cdots+||{\bf c}||_{{\bf p}_l}\,\},
\end{equation*}
where ${\bf p}_1,\ldots, {\bf p}_l$ are mutually non-intersecting double paths in $\Delta_n$ and each ${\bf p}_i$ starts at the $(2i-1)$-th row of $\Delta_n$ for $1\leq i\leq l$. 
\end{thm} 

This formula can be viewed as an analogue of Greene's formula for the shape of a tableau corresponding to a biword under usual RSK given in terms of disjoint weakly decreasing subwords \cite{G}.

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\end{document}
