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\title[Crystals and Schur $P$-positive expansions (extended
abstract)]{Crystals and Schur $P$-positive expansions\\ (extended
abstract)}

\author{Seung-Il Choi\thanks{\href{mailto:ignatioschoi@snu.ac.kr}{ignatioschoi@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}Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea \\
\addressmark{2}Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea
}

\received{\today}

\abstract{
We give a new characterization of Littlewood--Richardson--Stembridge tableaux for Schur $P$-functions by using the theory of $\mf{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.
}

\keywords{Schur P-function, crystals, Littlewood--Richardson rule}

\usepackage[backend=bibtex]{biblatex}
\addbibresource{62-Choi-Kwon.bib}

\begin{document}

\maketitle

\section{Introduction}
Let $\cP^+$ be the set of strict partitions and let $P_\la$ be the Schur $P$-function corresponding to $\la\in \cP^+$. The set of Schur $P$-functions is an important class of symmetric functions, which is closely related with representation theory and algebraic geometry. For example, the Schur $P$-polynomial $P_\la(x_1,\ldots,x_n)$ in $n$ variables is the character of a finite-dimensional irreducible representation $V_n(\la)$ of the queer Lie superalgebra $\mf{q}(n)$ with highest weight $\la$ up to a power of $2$ when the length $\ell(\la)$ of $\la$ is no more than $n$ \cite{Srgv85}. 

The set of Schur $P$-functions forms a basis of a subring of the ring of symmetric functions, and the structure constants with respect to this basis are nonnegative integers, that is, given $\mu, \nu \in \cP^+$,
\begin{equation*}
P_{\mu}  P_{\nu} 
=\underset{\la \in \cP^+}{\sum}f^{\la}_{\mu \nu}P_{\la},
\end{equation*}
for some nonnegative integers $f^{\la}_{\mu \nu}$. 
The first and the most well-known result on a combinatorial description of $f^{\la}_{\mu \nu}$ was obtained by Stembridge \cite{Ste} using shifted Young tableaux, which is a combinatorial model for Schur $P$- or $Q$-functions \cite{Sa}. 
It is shown that $f^\la_{\mu\nu}$ is equal to the number of semistandard tableaux with entries in a $\Z_2$-graded set ${\mc N}=\{\,1'<1<2'<2<\cdots \,\}$ of shifted skew shape $\la/\mu$ and weight $\nu$ such that (i) for each integer $k\geq 1$ the southwesternmost entry with value $k$ is unprimed or of even degree and (ii) the reading words satisfy the {\em lattice property}. 
Here we say that the value $|x|$ is $k$
when $x$ is either $k$ or $k'$ in a tableau.
Let us call these tableaux the {\em Littlewood--Richardson--Stembridge (LRS) tableaux} (see Definitions \ref{df:St lattice property} and \ref{Def:Stembridge's description}).

Recently, two more descriptions of $f^\la_{\mu\nu}$ were obtained in terms of semistandard decomposition tableaux, which is another combinatorial model for Schur $P$-functions introduced by Serrano~\cite{Se}.  
It is shown by Cho that $f^\la_{\mu\nu}$ is given by the number of semistandard decomposition tableaux of shifted shape $\mu$ and weight $w_0(\la-\nu)$ whose reading words satisfy the {\em $\la$-good property} (see~\cite[Corollary 5.14]{Cho}).
Here we assume that $\ell(\la), \ell(\mu), \ell(\nu)\leq n$, and  $w_0$ denotes the longest element in the symmetric group $\mf S_n$.
%
Another description is given by  Grantcharov, Jung, Kang, Kashiwara, and Kim~\cite{GJKKK14}
based on their crystal base theory for the quantized enveloping algebra of $\mf{q}(n)$ \cite{GJKKK15}. 
They realize the crystal $\B_n(\la)$ associated to $V_n(\la)$ as the set of semistandard decomposition tableaux of shape $\la$ with entries in  $\{\,1 < \cdots < n\,\}$,
and describe $f^\la_{\mu\nu}$ by characterizing the lowest weight vectors of weight $w_0\la$ in the tensor product $\B_n(\mu)\otimes \B_n(\nu)$.
%
We also remark that bijections between the above mentioned combinatorial models for $f^\la_{\mu\nu}$ are studied in \cite{CNO14} using insertion schemes for semistandard decomposition tableaux.

The main result in this paper is to give another description of $f^\la_{\mu\nu}$ using the theory of $\mf{q}(n)$-crystals, and show that it is indeed equivalent to that of Stembridge. 
More precisely, we show that $f^\la_{\mu\nu}$ is equal to the number of semistandard tableaux with entries in $\mc N$ of shifted skew shape $\la/\mu$ and weight $\nu$
such that (i) for each integer $k\geq 1$ the southwesternmost entry with value $k$ is unprimed or of even degree  and 
(ii) the reading words satisfy the {\em ``lattice property''} 
(see Definitions \ref{Def:new lattice property},~\ref{Def:new LR tab} and
Theorem \ref{Thm:our-characterization}). It is obtained by
semistandardizing the standard tableaux which parametrize the lowest
weight vectors counting $f^\la_{\mu\nu}$ in \cite{GJKKK14}, where the
``lattice property'' naturally arises from the configuration of entries in semistandard decomposition tableaux. We show that these tableaux for $f^{\la}_{\mu\nu}$ are equal to LRS tableaux 
(Theorem \ref{thm:F=LRS}), and hence obtain a new characterization of LRS tableaux.
For more details, see Section~\ref{section3}.

We next consider the Schur $P$-positive expansion of a skew Schur function
\begin{center}
$\displaystyle s_{\lambda/\delta_r} = \sum_{\nu\in\cP^+} a_{\lambda/\delta_r\,\nu} \hskip 0.5mm P_\nu$
\end{center}
for a skew diagram $\lambda/\delta_r$ contained in a rectangle $((r+1)^{r+1})$, where $\delta_r=(r,r-1,\ldots,1)$ (cf. \cite{AS, Dew}). We give a combinatorial description of $a_{\lambda/\delta_r\,\nu}$ (Theorem \ref{Thm:Main-sect5}) by considering a $\mf{q}(n)$-crystal structure on the set of usual semistandard tableaux of shape $\la/\delta_r$ and characterizing the lowest weight vectors corresponding to each $\nu\in \cP^+$.
We refer to Section~\ref{section4}.


Finally, we consider the Schur expansion of a Schur $P$-function
\begin{center}
$\displaystyle P_\lambda = \sum_{\mu} g_{\lambda \mu} s_\mu,$
\end{center}
for $\la\in\cP^+$.
We give a simple and alternate proof of Stembridge's description for $g_{\la\mu}$ \cite{Ste} (Theorem \ref{thm-coeff-g}) by characterizing the type $A$ lowest weight vectors of weight $w_0\mu$ in the $\mf{q}(n)$-crystal $\B_n(\la)$ when $\ell(\la), \ell(\mu)\leq n$
(see Section~\ref{section5}).

A full version of this paper including detailed proofs has appeared in~\cite{CK17}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystals for queer Lie superalgebras}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%55
\subsection{Notation and terminology}  
Let $\Z_+$ be the set of nonnegative integers.
We fix a positive integer $n\geq 2$ throughout this paper.
Let 
\begin{equation*}
\begin{split}
&\mathscr{P}=\{\,\la=(\la_i)_{i\geq 1}\,|\,\la_i\in \Z_+,\, \la_i\geq \la_{i+1}\, (i\geq 1),\, |\lambda|:=\sum \la_i<\infty \,\},\\
&\cP^+=\{\,\la=(\la_i)_{i\geq 1}\,|\,\la\in \cP,\, \la_i=\la_{i+1} \Rightarrow \la_i=0 \ (i\geq 1) \,\}.
\end{split}
\end{equation*}
Let $\cP_n=\{\,\la \,|\,\ell(\la)\leq n\,\}\subseteq \cP$, where $\ell(\la)$ is the length of $\la$,
and $\cP^+_n=\cP^+\cap \cP_n$.  

The (unshifted) diagram of $\lambda\in\cP$ and the shifted diagram of $\lambda\in\cP^+$ are  defined to be 
\begin{equation*}
\begin{split}
&D_\lambda = \{\, (i,j) \in \mathbb{N}^2 \,:\, 1 \leq j \leq \lambda_i, \ 1 \leq i \leq \ell(\lambda)\, \},\\
&D^+_\lambda = \{\, (i,j)  \in \mathbb{N}^2 \,:\, i \leq j \leq \lambda_i+i-1, \ 1 \leq i \leq \ell(\lambda) \, \},
\end{split}
\end{equation*}
respectively.
We identify $D_\la$ and $D^+_{\la}$ with diagrams where a box is placed at the $i$-th row from the top and the $j$-th column from the left for each $(i,j)\in D_{\la}$ and $D^+_{\la}$, respectively.

Let $\mc{A}$ be a linearly ordered set, and $\W_\A$ be the set of words of finite length with letters in $\A$. For $w\in \W_\A$ and $a\in \A$, let $c_a(w)$ be the number of occurrences of $a$ in $w$.

For $\la, \mu\in\cP$ with $D_\mu \subseteq D_\lambda$, 
a {\em tableau of shape $\lambda / \mu$}  means a filling on the skew diagram $D_\lambda \setminus D_\mu$ with entries in  $\mc{A}$.
% 
For $\la, \mu \in \cP^+$ with $D^+_\mu \subseteq D^+_\la$, 
a {\em tableau of shifted  shape $\la/\mu$} is defined in a similar way. For a tableau $T$ of (shifted) shape $\la/\mu$, let $w(T)$ be the word given by reading the entries of $T$ row by row from top to bottom, and from right to left in each row. 
We denote by $T_{i,j}$ the $j$-th entry (from the left) of the $i$-th row of $T$ from the top. 
For $1\leq i\leq \ell(\la)$, let $T^{(i)}=T_{i,{\la_i}}\cdots T_{i,1}$ be the subword of $w(T)$ corresponding to the $i$-th row of $T$. 
Then we have $w(T)=T^{(1)}\cdots T^{(\ell(\la))}$. We denote by $w_{\rev}(T)$ the reverse word of $w(T)$. 
For $a\in\A$, let $c_a(T)=c_a(w(T))$ be the number of occurrences of $a$ in $T$.

Suppose that $\mc{A}$ is a linearly ordered  set with a $\mathbb{Z}_2$-grading $\mc{A}=\mc{A}_0\sqcup\mc{A}_1$. 
For $\la, \mu\in\cP$ with $D_\mu \subseteq D_\lambda$, let $SST_\A(\lambda/\mu)$ be the set of tableaux of shape $\lambda/\mu$ with entries in $\A$ which is semistandard, that is, (a) the entries in each row (resp. column) are
weakly increasing from left to right (resp. from top to bottom), 
(b) the entries in $\mc{A}_0$ (resp. $\mc{A}_1$) are strictly increasing in each
column (resp. row). 
Similarly, for $\la, \mu\in \cP^+$ with $D^+_\mu \subseteq D^+_\la$, 
we define $SST^+_{\A}(\la/\mu)$ to be the set of semistandard tableaux of shifted shape $\lambda/\mu$  with entries in $\A$. 
 
Let ${\mc N}=\{\,1' < 1 < 2' < 2 <  \cdots \,\}$ be a linearly ordered set with a $\Z_2$-grading ${\mc N}_0=\N$ and ${\mc N}_1=\N'=\{1',2',\cdots\}$. Put $[n]=\{\,1,\ldots,n\,\}$ and $[n]'=\{\,1',\ldots,n'\,\}$, where the $\Z_2$-grading and linear ordering are induced from ${\mc N}$. 
For $a\in {\mc N}$, we write $|a|=k$ when $a$ is either $k$ or $k'$.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Semistandard decomposition tableaux and Schur $P$-function}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Semistandard decomposition tableaux and Schur $P$-functions}

Let us recall the notion of semistandard decomposition tableaux \cite{GJKKK14, Se}, which is our main combinatorial object.
A word $u=u_1 \cdots u_s$ in $\W_{\N}$ is called a {\em hook word}
if it satisfies  
$u_1 \geq u_2 \geq \cdots \geq u_k < u_{k+1} < \cdots < u_s$ for some $1 \leq k \leq s$. 
In this case, let $u\!\!\downarrow = u_1 \cdots u_k$ be the weakly decreasing subword of maximal length
and $u\!\!\uparrow = u_{k+1} \cdots u_s$ the remaining strictly increasing subword in $u$.

\begin{df}\label{SSDT}\mbox{}
For $\la\in \cP^+$,
let $T$ be a tableau of shifted shape $\la$ with entries in $\N$. 
Then $T$ is called a semistandard decomposition tableau of shape $\la$ if  
(a) $T^{(i)}$ is a hook word of length $\lambda_i$ for $1 \leq i \leq \ell(\lambda)$,  
(b) $T^{(i)}$ is a hook subword of maximal length in $T^{(i+1)}T^{(i)}$, the concatenation of $T^{(i+1)}$ and $T^{(i)}$, for $1 \leq i < \ell(\lambda)$.
\end{df}

For $\la\in\cP^+$, let $SSDT(\la)$ be the set of semistandard decomposition tableaux of shape $\la$.
Let $x=\{x_1,x_2,\ldots\}$ be a set of formal commuting variables, and let $P_\la=P_\la(x)$ be the Schur $P$-function in $x$ corresponding to $\la\in \cP^+$. 
It is shown in~\cite{Se} that $P_\la$ is given by the weight generating function of $SSDT(\la)$: $P_\lambda = \sum x^T,$
where the sum runs over all $T\in SSDT(\la)$, and 
$x^T = \prod_{i\geq 1} x_i^{c_i(T)}.$

\begin{rem}
Recall that the Schur $P$-function $P_\la$ can be realized as the character of tableaux $T\in SST^+_{\mc N}(\la)$ with no primed entry or entry of odd degree on the main diagonal (cf. \cite{Sa}).  
The notion of semistandard decomposition tableaux was introduced in~\cite{Se} to give a plactic monoid model for Schur $P$-functions. 
In this paper, we follow its modified version (Definition \ref{SSDT}) introduced in \cite{GJKKK14}, by which it is easier to describe $\mf q(n)$-crystals \cite[Remark 2.6]{GJKKK14}. We also refer the reader to~\cite{CNO14} for more details on relation between the combinatorics of these two models.
\end{rem}

For $\la\in \cP^+$, let $SSDT_n(\la)$ be the set of tableaux $T\in SSDT(\la)$ with entries in $[n]$. 
By~\cite[Proposition 2.3]{GJKKK14}, we see that $SSDT_n(\la)\neq \emptyset$ if and only if $\la\in \cP^+_n$. We denote by $P_\la(x_1,\ldots,x_n)$ the Schur $P$-polynomial in $x_1,\ldots,x_n$ given by specializing $P_\la$ at $x_{n+1}=x_{n+2}=\cdots =0$. Then we have $P_\la(x_1,\ldots,x_n)=\sum_{T\in SSDT_n(\la)} x^T$.

For $\la\in\cP^+_n$,
let $H_n^\lambda$ be the element in $SSDT_n(\la)$ where the subtableau with entry  $\ell(\la)-i+1$ is a connected border strip of size $\lambda_{\ell(\lambda)-i+1}$ starting at $(i,i)\in D^+_\la$ for each $i=1,\ldots, \ell(\la)$, and 
let $L_n^\lambda$ be the one where the subtableau with entry $n-i+1$ is a connected horizontal strip of size $\lambda_{i}$ starting at $(i,i)\in D^+_\la$ for each $i=1,\ldots,\ell(\la)$.
Indeed, $H_n^\la$ and $L_n^\la$ are the unique tableaux in $SSDT_n(\la)$ such that 
$(c_1(H^\la_n),\ldots, c_n(H^\la_n))=\la$ and $(c_1(L^\la_n),\ldots,c_n(L^\la_n))=w_0\la.$
Here we assume that $\cP^+_n\subset \Z_+^n$ and the symmetric group $\mf{S}_n$ acts on $\Z_+^n$ by permutation, where $w_0$ is the longest element in $\mf S_n$.

\subsection{Crystals} 

Let us briefly review the crystals for the general linear Lie algebra $\gl(n)$ in \cite{KashNaka}.
Let $P^\vee=\bigoplus_{i=1}^n\Z e_{i}$ be the dual weight lattice and $P={\rm Hom}_\Z(P^\vee,\Z)=\bigoplus_{i=1}^n\Z\epsilon_i$ the weight lattice with $\langle \epsilon_i,e_{j} \rangle =\delta_{ij}$ for $1\leq i,j\leq n$. Define a symmetric bilinear form $(\, \cdot \,|\, \cdot\,)$ on $P$ by $(\epsilon_i |\epsilon_j)=\delta_{ij}$ for $1\leq i,j\leq n$. 
Let $\{\,\alpha_i=\epsilon_i-\epsilon_{i+1}\ (i=1,\ldots,n-1)\,\}$ be the set of simple roots, and $\{\,h_i=e_{i}-e_{i+1}\ (i=1,\ldots,n-1)\,\}$ the set of simple coroots of $\gl(n)$.
Let $P^+=\{\,\la\,|\,\la\in P, \ \langle \la,h_i\rangle\geq 0 \ (i=1,\ldots,n-1) \,\}$ be the set of dominant integral weights.

A {\em $\gl(n)$-crystal} is a set
$B$ together with the maps ${\rm wt} : B \rightarrow P$,
$\varepsilon_i, \varphi_i: B \rightarrow \mathbb{Z}\cup\{-\infty\}$ and
$\te_i, \tf_i: B \rightarrow B\cup\{{\bf 0}\}$ for $i=1,\ldots,n-1$ satisfying the following conditions:  for $b\in B$,\\
\hspace*{5mm} (a) $\varphi_i(b) =\langle {\rm wt}(b),h_i \rangle +
\varepsilon_i(b)$, \\
\hspace*{5mm} (b) $\varepsilon_i(\te_i b) = \varepsilon_i(b) - 1,\ \varphi_i(\te_i b) =
\varphi_i(b) + 1,\ {\rm wt}(\te_ib)={\rm wt}(b)+\alpha_i$ if $\te_i b \in B$, \\
\hspace*{5mm} (c) $\varepsilon_i(\tf_i b) = \varepsilon_i(b) + 1,\ \varphi_i(\tf_i b) =
\varphi_i(b) - 1,\ {\rm wt}({\tf_i}b)={\rm wt}(b)-\alpha_i$ if $\tf_i b \in B$, \\
\hspace*{5mm} (d) $\tf_i b = b'$ if and only if $b = \te_i b'$ for $b' \in B$, \\
\hspace*{5mm} (e) $\te_ib=\tf_ib={\bf 0}$ when $\varphi_i(b)=-\infty$. \\
Here ${\bf 0}$ is a formal symbol and $-\infty$ is the smallest
element in $\Z\cup\{-\infty\}$ such that $-\infty+n=-\infty$
for all $n\in\Z$. 
For $\mu\in P$, let
$B_\mu=\{\, b\in B\,  | \, {\rm wt}(b)=\mu\,\}$.
When $B_\mu$ is finite for all $\mu$, we define the character of $B$ by ${\rm ch}B=\sum_{\mu\in P}|B_\mu|e^\mu$,
where $e^\mu$ is a basis element of the group algebra $\mathbb{Q}[P]$.

Let $B_1$ and $B_2$ be $\gl(n)$-crystals.
A  tensor product $B_1\otimes B_2$
is a $\gl(n)$-crystal, which is defined to be $B_1\times B_2$  as a set with elements  denoted by
$b_1\otimes b_2$, where  \\
\hspace*{5mm} (a) ${\rm wt}(b_1\otimes b_2)={\rm wt}(b_1)+{\rm wt}(b_2),$ \\
\hspace*{5mm} (b) $\varepsilon_i(b_1\otimes b_2)= {\rm
max}\{\varepsilon_i(b_1),\varepsilon_i(b_2)-\langle {\rm
wt}(b_1),h_i\rangle\},$\\
\hspace*{5mm} (c) $\varphi_i(b_1\otimes b_2)= {\rm max}\{\varphi_i(b_1)+\langle {\rm
wt}(b_2),h_i\rangle,\varphi_i(b_2)\},$\\
\hspace*{5mm} (d) ${\te}_i(b_1\otimes b_2)= {\te}_i b_1 \otimes b_2,$ if $\varphi_i(b_1)\geq \varepsilon_i(b_2)$, and $b_1\otimes {\te}_i b_2$ otherwise,\\
\hspace*{5mm} (e) ${\tf}_i(b_1\otimes b_2)=
{\tf}_i b_1 \otimes b_2,$ if $\varphi_i(b_1)>\varepsilon_i(b_2)$,
and $b_1\otimes {\tf}_i b_2$ otherwise,\\
for $i=1,\ldots,n-1$. Here we assume that ${\bf 0}\otimes b_2=b_1\otimes {\bf 0}={\bf 0}$.

For $\la\in \cP_n$, let $B_n(\la)$ be the crystal associated to an irreducible $\gl(n)$-module with highest weight $\la$, where we regard $\la$ as $\sum_{i=1}^n\la_i\epsilon_i\in P^+$.
We may regard $[n]$ as $B_n(\epsilon_1)$, where ${\rm wt}(k)=\epsilon_k$ for $k\in[n]$, and hence $\W_{[n]}$ as a $\gl(n)$-crystal
where we identify $w=w_1\ldots w_r$ with $w_1\otimes \cdots \otimes w_r\in B_n(\epsilon_1)^{\otimes r}$. 
The crystal structure on $\W_{[n]}$ is easily described by the so-called the {signature rule} (cf. \cite[Section 2.1]{KashNaka}).
For $\la\in \cP_n$, the set $SST_{[n]}(\la)$ becomes a $\gl(n)$-crystal under the identification of $T$ with $w(T)\in \W_{[n]}$, and it is isomorphic to $B_n(\la)$ \cite{KashNaka}. In general, one can define a $\gl(n)$-crystal structure on $SST_{[n]}(\la/\mu)$ for a skew diagram $\la/\mu$. By abuse of notation, we set $B_n(\la/\mu):=SST_{[n]}(\la/\mu)$.

Next, let us review the notion of crystals associated to polynomial representations of the queer Lie superalgebra ${\mf q}(n)$ developed in \cite{GJKKK14,GJKKK15}.  

\begin{df}
A {\em $\mathfrak{q}(n)$-crystal} is a set $B$ together with the maps ${\rm wt} : B \rightarrow P$,
$\varepsilon_i, \varphi_i: B \rightarrow \mathbb{Z}\cup\{-\infty\}$ and
$\te_i, \tf_i: B \rightarrow B\cup\{{\bf 0}\}$ for $i\in I:=\{\,1,\ldots,n-1, \ov{1}\,\}$ satisfying the following conditions: 

(a) $B$ is a $\gl(n)$-crystal with respect to ${\rm wt}$, $\varepsilon_i$, $\varphi_i$, $\te_i, \tf_i$ for $i=1,\ldots, n-1$,

(b) $\wt(b) \in \bigoplus_{i\in [n]}\Z_+\epsilon_i$ for $b\in B$, 

(c) $\wt(\widetilde{e}_{\overline{1}}b) = \wt(b) + \alpha_1$, $\wt(\widetilde{f}_{\overline{1}}b)=\wt(b)-\alpha_1$ for $b\in B$,

(d) $\widetilde{f}_{\overline{1}}b = b'$ if and only if 
$b = \widetilde{e}_{\overline{1}}b'$ for all $b, b' \in B$,

(e) for $3 \leq i \leq n-1$, we have

\hspace*{3mm}
(i) the operators $\widetilde{e}_{\overline{1}}$ and $\widetilde{f}_{\overline{1}}$ commute with $\widetilde{e}_{i}$, $\widetilde{f}_{i}$,

\hspace*{3mm}
(ii) if $\widetilde{e}_{\overline{1}}b \in B$, then $\varepsilon_i(\widetilde{e}_{\overline{1}}b)=\varepsilon_i(b)$ and 
$\varphi_i(\widetilde{e}_{\overline{1}}b)=\varphi_i(b)$.
\end{df}

Let $\B_n$ be a $\q(n)$-crystal which is the $\gl(n)$-crystal $B_n(\epsilon_1)$ together with $\tf_{\ov 1} \ \boxed{1}=\boxed{2}$ (in dashed arrow):
\begin{tikzpicture}[baseline=1.5mm]
\def \hhh{4.4mm}
\def \hhhh{18mm}
\def \cdotssize{5mm}
\def \vvv{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*0+\hhhh*1,\vvv*0) rectangle (\hhh*1+\hhhh*1,\vvv*1);
\draw[-] (\hhh*0+\hhhh*2,\vvv*0) rectangle (\hhh*1+\hhhh*2,\vvv*1);
%\draw[-] (\hhh*0+\hhhh*3,\vvv*0) rectangle (\hhh*1+\hhhh*3,\vvv*1);
\draw[-] (\cdotssize+\hhh*0+\hhhh*4,\vvv*0) rectangle (\cdotssize+\hhh*1+\hhhh*4,\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {$1$};
\node at (\hhh*0.5+\hhhh,\vvv*0.5) {$2$};
\node at (\hhh*0.5+\hhhh*2,\vvv*0.5) {$3$};
%\node at (\hhh*0.5+\hhhh*3,\vvv*0.5) {$4$};
\node at (\cdotssize*0.5+\hhh*0.5+\hhhh*3,\vvv*0.5) {$\cdots$};
%\node at (\hhh*0.5+\hhhh*4,\vvv*0.5) {$n-1$};
\node at (\cdotssize+\hhh*0.5+\hhhh*4,\vvv*0.5) {$n$};
%%%%%%%%%%%%%%%%%%% arrow
\draw[->] (\hhh*1.3,\vvv*0.7) -- (-\hhh*0.3+\hhhh*1,\vvv*0.7) node[midway,above] () {\tiny $1$};
\draw[->,densely dashed] (\hhh*1.3,\vvv*0.3) -- (-\hhh*0.3+\hhhh*1,\vvv*0.3) node[midway,below] () {\tiny $\overline{1}$};
\draw[->] (\hhh*1.3+\hhhh*1,\vvv*0.5) -- (-\hhh*0.3+\hhhh*2,\vvv*0.5) node[midway,above] () {\tiny $2$};
\draw[->] (\hhh*1.3+\hhhh*2,\vvv*0.5) -- (-\hhh*0.3+\hhhh*3,\vvv*0.5) node[midway,above] () {\tiny $3$};
\draw[->] (\cdotssize+\hhh*1.1+\hhhh*3,\vvv*0.5) -- (\cdotssize-\hhh*0.3+\hhhh*4,\vvv*0.5) node[midway,above] () {\tiny $n-1$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\cdotssize+\hhh*1.3+\hhhh*4,\vvv*0.1) {.};
\end{tikzpicture}
Here we write $b\,\stackrel{i}{\longrightarrow}\,b'$ if $\tf_i b= b'$ for $b,b'\in B$ and $i\in I\setminus\{\ov{1}\}$ as usual, and %$b\ \stackrel[\ov 1]{\dashrightarrow}\ b'$ 
\begin{tikzpicture}[baseline=-1.3mm]
\def \hhh{13mm}
\def \vvv{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0,\vvv*0) (B) {$b$};
\node at (\hhh*1,\vvv*0) (BP) {$b'$};
%%%%%%%%%%%%%%%%%%% arrow
\draw[->,densely dashed] (B) -- (BP) node[midway,below] () {\tiny $\overline{1}$};
\end{tikzpicture}
if $\tf_{\ov{1}} b= b'$.


For $\mathfrak{q}(n)$-crystals $B_1$ and $B_2$,
the tensor product $B_1 \otimes B_2$ is the $\gl(n)$-crystal $B_1 \otimes B_2$
where the actions of
$\widetilde{e}_{\overline{1}}$ and $\widetilde{f}_{\overline{1}}$
are given by 
\begin{equation}\label{eq:tensor product of qn-crystals}
\begin{split}
{\te}_{\ov1}(b_1\otimes b_2) = {\te}_{\ov1}b_1 \otimes b_2 \text{ if } \langle \epsilon_1, \wt(b_2)\rangle = \langle \epsilon_2,\wt(b_2)\rangle=0, \text{ and } b_1\otimes {\te}_{\ov1}b_2 \text{ otherwise,} \\
{\tf}_{\ov1}(b_1\otimes b_2) = {\tf}_{\ov1} b_1 \otimes b_2 \text{ if } \langle \epsilon_1, \wt(b_2)\rangle = \langle \epsilon_2, \wt(b_2)\rangle=0, \text{ and } b_1\otimes {\tf}_{\ov1}b_2 \text{ otherwise.}
\end{split}
\end{equation}
\noindent Then it is easy to see that $B_1\otimes B_2$ is a $\mathfrak{q}(n)$-crystal, and $\W_{[n]}$ is also a $\q(n)$-crystal.

Let $B$ be a $\q(n)$-crystal. Suppose that $B$ is a regular $\gl(n)$-crystal, that is, each connected component in $B$ is isomorphic to $B_n(\la)$ for some $\la\in\cP_n$. Let $W=\mf S_n$ be the Weyl group of $\gl(n)$ which is generated by the simple reflection $r_i$ corresponding to $\alpha_i$ for $i=1,\ldots,n-1$. We have a group action of $W$ on $B$ 
denoted by $S$ such that 
$S_{r_i}(b)= 
\tf_i^{\langle {\rm wt}(b),h_i\rangle}b$ if $\langle {\rm wt}(b),h_i\rangle\geq 0$, and 
$\te_i^{\,-\langle {\rm wt}(b),h_i\rangle}b$ otherwise,
for $b\in B$ and $i=1,\ldots,n-1$. 
For $2\leq i\leq n-1$, let $w_i\in W$ be such that $w_i(\alpha_{i})=\alpha_1$, and 
let  
$\te_{\ov{i}}=S_{w_i^{-1}}\te_{\ov{1}}S_{w_i}$ and 
$\tf_{\ov{i}}=S_{w_i^{-1}}\tf_{\ov{1}}S_{w_i}$.
For $b\in B$, we say that $b$ is a {\em $\q(n)$-highest weight vector} if 
$\te_i b=\te_{\ov i}b={\bf 0}$ for $1\leq i \leq n-1$, and 
$b$ is a {\em $\q(n)$-lowest weight vector} if $S_{w_0}b$ is a $\q(n)$-highest weight vector.

For $\la\in \cP^+$, let $\B_n(\la)=SSDT_n(\la)$, and consider an injective map
\begin{equation}\label{eq:embedding of SSDT}
\xymatrixcolsep{2pc}\xymatrixrowsep{0pc}
\xymatrix{
\B_n(\la)\ \  \ar@{^{(}->}[r]  & \ \W_{[n]}, \hskip 3mm
\ \ \ T \ \  \ \   \ar@{|->}[r]   &  \  \makebox[2em][l]{$w_\rev(T)$.}}
\end{equation}

\begin{thm}[{\cite[Theorem 2.5]{GJKKK14}}]
Let $\lambda \in \cP^+_n$ be given. 

{\rm (a)} The image of $\B_n(\lambda)$ in \eqref{eq:embedding of SSDT} together with $\{{\bf 0}\}$ is invariant under the action of $\te_i$ and $\tf_i$ for $i\in I$, and hence $\B_n(\la)$ is a $\q(n)$-crystal.

{\rm (b)} The $\q(n)$-crystal $\B_n(\la)$ is connected where $H^\lambda_n$ is a unique $\q(n)$-highest weight vector and $L^\lambda_n$ is a unique $\q(n)$-lowest weight vector.
\end{thm}

Let $B_1$ and $B_2$ be $\q(n)$-crystals. For $b_1\in B_1$ and $b_2\in B_2$, let us say that $b_1$ and $b_2$ are equivalent and write $b_1\equiv b_2$ if there exists an isomorphism of $\q(n)$-crystals $\psi : C(b_1) \longrightarrow C(b_2)$ such that $\psi(b_1)=b_2$ where $C(b_i)$ denotes the connected component of $b_i\in B_i$ $(i=1,2)$ as a $\q(n)$-crystal. 

By \cite[Theorem 4.6]{GJKKK15}, each connected component in $\B_n^{\otimes N}$ $(N\geq 1)$ is isomorphic to $\B_n(\la)$ for some $\la\in \cP^+_n$ with $|\la|=N$.
Indeed, for $b=b_1 \otimes   \cdots \otimes b_N\in \B_n^{\otimes N}$, there exists a unique $\la\in\cP_n^+$ and $T\in \B_n(\la)$ such that $b\equiv T$. In particular, $b$ is a $\q(n)$-lowest (resp. $\q(n)$-highest) weight vector if and only if $b\equiv L^\la_n$ (resp. $H^\la_n$).
The following lemma plays a crucial role in characterization of $\q(n)$-lowest weight vectors in $\B_n^{\otimes N}$ and hence describing the decompositions of $\B_n^{\otimes N}$ and $\B_n(\mu)\otimes\B_n(\nu)$ ($\mu,\nu\in\cP_n^+$) into connected components in \cite{GJKKK14}.

\begin{lem}[{\cite[Lemma 1.15, Corollary 1.16]{GJKKK14}}]
\label{lem:q(n)-lattice property}
For $b=b_1 \otimes \cdots \otimes b_N\in \B_n^{\otimes N}$, the following are equivalent:

{\rm (a)} $b$ is a $\q(n)$-lowest weight vector,

{\rm (b)} $b'=b_2 \otimes \cdots \otimes b_N$ is a $\q(n)$-lowest weight vector and $\epsilon_{b_1}+{\rm wt}(b')\in w_0\cP_n^+$,

{\rm (c)} $\wt(b_M \otimes \cdots \otimes b_N) \in w_0 \cP^+_n$ for all $1\leq M\leq N$.
\end{lem} 

Hence, we have the following immediately by Lemma \ref{lem:q(n)-lattice property}.

\begin{cor}\label{cor:lattice property in general}
For $\la^{(1)},\ldots,\la^{(s)}\in\cP_n^+$ and $T_1 \otimes \cdots \otimes T_s\in\B_n(\la^{(1)})\otimes \cdots \otimes \B_n(\la^{(s)})$, the following are equivalent: 

{\rm (a)} $T_1 \otimes \cdots \otimes T_s$ is a $\q(n)$-lowest weight vector, 

{\rm (b)} $T_r\otimes \cdots\otimes T_s\in \B_n(\la^{(s)})\otimes \cdots \otimes \B_n(\la^{(r)})$ is a $\q(n)$-lowest weight vector for $1\leq r\leq s$.
\end{cor}

\begin{rem}\label{rem:stability}
Let $m\geq n$ be a positive integer, and put $t=m-n$. For $N\geq 1$, let $\psi_t : \B_n^{\otimes N} \longrightarrow \B_m^{\otimes N}$ be the map given by 
$\psi_t(u_1\otimes \cdots \otimes u_N)=(u_1+t)\otimes \cdots \otimes (u_N+t)$. 
Then  for $\la\in\cP_n^+$ and $u\in \B_n^{\otimes N}$ we have $u\equiv L_n^\la$ if and only if $\psi_t(u)\equiv L_m^\la$. This implies that the multiplicity of $\B_n(\la)$ in $\B_n^{\otimes N}$ is equal to that of $\B_m(\la)$ in $\B_m^{\otimes N}$ for $\la\in \cP_n^+$.
\end{rem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Littlewood--Richardson rule for Schur $P$-functions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Littlewood--Richardson rule for Schur $P$-functions}\label{section3}

\subsection{Shifted Littlewood--Richardson rule}

Let $w=w_1\cdots w_N$ be a word in $\W_{\mc N}$. Let $m_k=c_k(w)+c_{k'}(w)$ for $k\geq 1$. We define $w^*=w^*_1\cdots w^*_N$ to be the word obtained from $w$ as follows: for each $k\geq 1$,

{\rm (1)} consider the letters $w_i$'s with $|w_i|=k$. Label them with $1,2,\ldots, m_k$ (as subscripts), first enumerating the $w_p$'s  with $w_p=k$ from left to right, and then the $w_q$'s with $w_q=k'$ from right to left.

{\rm (2)} After the step (1), remove all \ $'$\ \ in each labeled letter $k'_j$, that is, replace any $k'_j$ by $k_j$ for $c_{k}(w)< j\leq m_k$.

\begin{df}\label{Def:new lattice property}
Let $w=w_1\cdots w_N\in \W_{\mc N}$ be given.
We say that $w$ satisfies the {\em ``lattice property''} if 
the word $w^*=w^*_1\cdots w^*_N$ associated to $w$ satisfies the following: for $k\geq 1$

\noindent\hspace*{0.8mm}
{\rm (L1)} if $w^*_i=k_1$, then no $k+1_j$ for $j\geq 1$ occurs in $w^*_1\cdots w^*_{i-1}$,

\noindent\hspace*{0.8mm}
{\rm (L2)} if $(w^*_s,w^*_t)=(k+1_i,k_{i+1})$ for $s<t$ and $i \geq 1$, 
then no $k+1_j$ $(i<j)$ occurs in $w^*_s\cdots w^*_t$, 

\noindent\hspace*{0.8mm}
{\rm (L3)} if $(w^*_s,w^*_t)=(k_{j+1},k+1_{j})$ for some $s<t$ and $j\geq 1$, 
then no $k_i$ $(i\leq j)$ occurs in $w^*_s\cdots w^*_t$.
\end{df}

\begin{df}\label{Def:new LR tab}
For $\la, \mu, \nu \in \cP^+$,
let ${\tt F}^\la_{\mu\nu}$ be the set of tableaux $Q$ such that  

{\rm (a)} $Q\in SST^+_{\mc N}(\la/\mu)$ with $c_{k}(Q)+c_{k'}(Q)=\nu_k$ for $k\geq 1$,

{\rm (b)} for $k\geq 1$, if $x$ is the rightmost letter in $w(Q)$ with $|x|=k$, then $x=k$,

{\rm (c)} $w(Q)$ satisfies the ``lattice property'' in Definition \ref{Def:new lattice property}. 
\end{df}
Then we have the following characterization of $f^\la_{\mu\nu}$.

\begin{thm}[{\cite[Theorem 3.5]{CK17}}]
\label{Thm:our-characterization}
For $\la, \mu, \nu \in \cP^+$, we have  \
$f^\la_{\mu\nu} = \left| {\tt F}^\la_{\mu\nu}\right|.$
\end{thm}

Choose $n$ such that $\la, \mu, \nu\in \cP_n^+$. Put
${\tt L}^\la_{\mu\nu}
=\{\,T\,|\, T \in \B_n(\nu),\ T \otimes L^\mu_n \equiv L^\la_n\,\}.$
By Corollary \ref{cor:lattice property in general},
we have $\B_n(\nu)\otimes \B_n(\mu) \cong \bigsqcup_{\la\in\cP_n^+}\B_n(\la)^{\oplus |{\tt L}^\la_{\mu\nu}|}.$
Hence we have $|{\tt L}^\la_{\mu\nu}|=f^\la_{\mu\nu}=f^\la_{\nu\mu}$.
The key in the proof of this theorem is to construct a bijection 
\begin{equation}\label{eq:LRS recording}
\xymatrixcolsep{2pc}\xymatrixrowsep{2mm}\xymatrix@1{
{\tt L}^\la_{\mu\nu} \   \    \ar[r] \ar@{->}[r]  & \  \ {\tt F}^\la_{\mu\nu}, \hskip 2mm
 \  T  \ \ \  \ar[r] \ar@{|->}[r] &  \ \   Q_T.}
\end{equation}
such that $w(Q_T)$ satisfies the ``lattice property''.
We briefly explain this construction now.
Let $T \in {\tt L}^\la_{\mu\nu}$ be given.
Assume that $w_\rev(T)=u_1\cdots u_N$ where $N=|\nu|$. 
By Lemma \ref{lem:q(n)-lattice property}, there exists $\mu^{(m)}\in \cP_n^+$ for $1\leq m\leq N$  such that 
(i) $(u_{N-m+1} \cdots u_N) \otimes L^\mu_n \equiv L^{\mu^{(m)}}_n$ and $\mu^{(N)}=\la$,
and 
(ii) $\mu^{(m)}$ is obtained by adding a box in the $(n-u_m+1)$-st row of $\mu^{(m-1)}$.
Here we assume that $\mu^{(0)}=\mu$.
Recall that $w_\rev(T)=T^{(\ell(\nu))}\cdots T^{(1)}$,
where $T^{(k)}=T_{k,1}\cdots T_{k,\la_k}$ is a hook word for $1\leq k\leq \ell(\nu)$.
Define $Q_T$ to be a tableau of shifted shape $\la/\mu$ with entries in $\mc N$, where $\mu^{(m)}/\mu^{(m-1)}$ is filled with
\begin{equation}\label{eq:labeling of Q_T}
\begin{cases}
k', & \text{if $u_m$ belongs to $T^{(k)}\!\!\uparrow$},\\
k, & \text{if $u_m$ belongs to $T^{(k)}\!\!\downarrow$},\\
\end{cases}
\end{equation}
for some $1\leq k\leq \ell(\nu)$. 
In other words, the boxes in $Q_T$ corresponding to $T^{(k)}\!\!\uparrow$ are filled with $k'$ from right to left as a vertical strip 
and then those corresponding to $T^{(k)}\!\!\downarrow$ are filled with $k$ from left to right as a horizontal strip.
(cf. the proof of~\cite[Theorem 3.5]{CK17}). 

\begin{rem}
For $T\in {\tt L}^\la_{\mu\nu}$, let $\widehat{Q}_T$ be the tableau of shifted shape $\la/\mu$, which is defined in the same way as $Q_{T}$ in the proof of Theorem \ref{Thm:our-characterization} except that we fill $\mu^{(m)}/\mu^{(m-1)}$ with $m$ in \eqref{eq:labeling of Q_T} for $1\leq m\leq N$. Then the set $\{\,\widehat{Q}_T\,|\,T\in {\tt L}^\la_{\mu\nu}\,\}$ is equal to the one given in 
\cite[Theorem 4.13]{GJKKK14} to describe $f^\la_{\mu\nu}$. 
For example, 
\vskip 1mm
\noindent
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $3$};
\node at (\hhh*1.5,\vvv*0.5) {\small $3$};
\node at (\hhh*2.5,\vvv*0.5) {\small $4$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $T_1=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $4$};
\node at (\hhh*1.5,\vvv*0.5) {\small $2$};
\node at (\hhh*2.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $3$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $T_2=$};
\end{tikzpicture} 
\hskip 3mm
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*3.5,\vvv*0.5) {\small $1$};
%
\node at (\hhh*2.5,-\vvv*0.5) {\small $2$};
\node at (\hhh*3.5,-\vvv*0.5)  {\small $3$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $4$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,-\vvv*0) () {\small $\widehat{Q}_{T_1}=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*3.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*2.5,-\vvv*0.5) {\small $1$};
\node at (\hhh*3.5,-\vvv*0.5)  {\small $4$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,-\vvv*0) () {\small $\widehat{Q}_{T_2}=$};
\end{tikzpicture} 
\hskip 3mm
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*3.5,\vvv*0.5) {\small $1'$};
%
\node at (\hhh*2.5,-\vvv*0.5) {\small $1$};
\node at (\hhh*3.5,-\vvv*0.5)  {\small $1$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,-\vvv*0) {\small $Q_{T_1}=$};
%\node at (\hhh*4.5,-\vvv*2) () {.};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=0.8]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*3.5,\vvv*0.5) {\small $1$};
%
\node at (\hhh*2.5,-\vvv*0.5) {\small $1'$};
\node at (\hhh*3.5,-\vvv*0.5)  {\small $2$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $1$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,-\vvv*0) {\small $Q_{T_2}=$};
%\node at (\hhh*4.5,-\vvv*2) () {.};
\end{tikzpicture} 
\end{rem}

\subsection{Stembridge's description of $f^\la_{\mu\nu}$}
\begin{df}\label{df:St lattice property}
Let $w=w_1 \cdots w_N$ be a word in $\W_{\mc N}$
and $w_\rev$ be the reverse word of $w$.
Let $\widehat{w}$ be the word obtained from $w$ by replacing $k$ by $(k+1)'$ and $k'$ by $k$ for each $k \geq 1$.
Suppose that $w \widehat{w}_\rev = a_1\cdots a_{2N}$, and let ${m}_k(i)=c_k(a_{1}  \cdots  a_i)$ for $k \geq 1$ and $0 \leq i \leq 2N$. 
Then we say that $w$ satisfies the lattice property 
if 
\begin{equation}\label{Eqn:the lattice property}
\text{$m_{k+1}(i)=m_{k}(i)$ implies $|a_{i+1}|\neq k+1$ for $k\geq 1$ and $i\geq 0$}.
\end{equation}
Here we assume that $m_k(0)=0$.
\end{df}

\begin{df}
\label{Def:Stembridge's description}
For $\la, \mu, \nu \in \cP^+$, let ${\tt LRS}^\la_{\mu\nu}$ be the set of tableaux $Q$ such that 

{\rm (a)} $Q\in SST^+_{\mc N}(\la/\mu)$ with $c_{k}(Q)+c_{k'}(Q)=\nu_k$ for $k\geq 1$,

{\rm (b)} for $k\geq 1$, if $x$ is the rightmost letter in $w(Q)$ with $|x|=k$, then $x=k$,

{\rm (c)} $w(Q)$ satisfies the lattice property in Definition \ref{df:St lattice property}.
\end{df}
\noindent
We call ${\tt LRS}^\la_{\mu\nu}$ the set of {\em Littlewood--Richardson--Stembridge tableaux}.
Due to Stembridge it provided that  
for $\la, \mu, \nu \in \cP^+$ the shifted LR coefficient $f^\la_{\mu\nu}$ is equal to the number of tableaux in ${\tt LRS}^\la_{\mu\nu}$
(cf.~(\cite[Theorem 8.3]{Ste}).
So we have 
\begin{thm}[{\cite[Theorem 3.11]{CK17}}]\label{thm:F=LRS}
For $\lambda, \mu, \nu \in \cP^+$, we have \
$
{\tt F}^\lambda_{\mu \nu}={\tt LRS}^\lambda_{\mu \nu}.
$
\end{thm}

\begin{cor}\label{cor:equivalence of lattice conditions}
Let $w\in \W_{\mc N}$ be such that 
$(c_{k}(Q)+c_{k'}(Q))_{k\geq 1}\in \cP^+$, and 
for $k\geq 1$, if $x$ is the rightmost letter in $w$ with $|x|=k$, then $x=k$.
Then $w$ satisfies the ``lattice property'' in Definition \ref{Def:new lattice property} if and only if $w$ satisfies the lattice property in Definition \ref{df:St lattice property}.
\end{cor}

\begin{rem}
A bijection from ${\tt LRS}^{\la}_{\mu \nu}$ to ${\tt L}^\la_{\mu \nu}$ is also given in \cite[Theorem 4.7]{CNO14}, which coincides with the inverse of the map $T \mapsto Q_T$ in \eqref{eq:LRS recording} (see also the remarks in \cite[p.82]{CNO14}).
The proof of \cite[Theorem 4.7]{CNO14} use insertion schemes for two versions of  semistandard decomposition tableaux and another combinatorial model for $f^{\la}_{\mu\nu}$ by Cho \cite{Cho} as an intermediate object between ${\tt LRS}^{\la}_{\mu \nu}$ and ${\tt L}^\la_{\mu \nu}$.

On the other hand, we prove more directly that the map $T \mapsto Q_T$ in \eqref{eq:LRS recording} is a bijection from ${\tt L}^\la_{\mu \nu}$ to ${\tt LRS}^\la_{\mu\nu}$ by using a new characterization of the lattice property in Theorem \ref{thm:F=LRS}.
\end{rem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       Schur $P$-expansion of skew Schur functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Schur $P$-expansions of skew Schur functions}\label{section4}

For $r\geq 0$, let us denote by $\delta_r$ the partition $(r,r-1,\ldots,1)$ if $r \geq 1$,
and $(0)$ if $r=0$. We fix a nonnegative integer $r$.
Let $\la\in \cP$ be such that $D_{\delta_{r}}\subseteq D_\la \subseteq D_{((r+1)^{r+1})}$.
Here $((r+1)^{r+1})$ means the rectangular partition $(r+1,\ldots,r+1)$ with length $r+1$.
It is shown in~\cite{AS,Dew}
that the skew Schur function $s_{\lambda/\delta_k}$ has a nonnegative integral expansion in terms of Schur $P$-functions
\begin{equation*}
s_{\lambda/\delta_r} = \sum_{\nu\in\cP^+} a_{\lambda/\delta_r\,\nu} \hskip 0.5mm P_\nu, 
\end{equation*}
together with a combinatorial description of $a_{\lambda/\delta_r\,\nu}$.
Moreover it is shown that these skew Schur functions are the only ones (up to rotation of shape by $180^\circ$), which have Schur $P$-positivity. In this section, we give a new simple description of $a_{\lambda/\delta_r\,\nu}$ using $\q(n)$-crystals.

\begin{prop}[{\cite[Proposition 4.1]{CK17}}]\label{Prop:B-q(n)-crystal}
Let $\lambda \in \cP_n$ be such that $D_{\delta_{r}} \subseteq D_\lambda \subseteq D_{(r+1)^{r+1}}$.
Then the $\gl(n)$-crystal $B_n(\la/\delta_r)$, regarded as a subset of $\W_{[n]}$ together with ${\bf 0}$ is invariant under $\te_{\ov{1}}$ and $\tf_{\ov{1}}$. Hence $B_n(\la/\delta_r)$ is a $\q(n)$-crystal.
\end{prop}

Since $B_n(\la/\delta_r)$ is a $\mf q(n)$-crystal, the skew Schur polynomial $s_{\la/\delta_r}(x_1,\ldots,x_n)$ is a nonnegative integral linear combination of $P_\nu(x_1,\ldots,x_n)$. By applying  Remark \ref{rem:stability}, we have 

\begin{cor}
Under the above hypothesis, the skew Schur function $s_{\la/\delta_r}$ is Schur $P$-positive.
\end{cor}

\begin{df}\label{Def:Set-A}
Let $\la\in \cP$ be such that $D_{\delta_{r}} \subseteq D_\la \subseteq D_{((r+1)^{r+1})}$ and $\nu\in \cP^+$.
Let 
${\tt A}_{\la/\delta_r\,\nu}$ be the set of tableaux $Q$ such that 

{\rm (a)} $Q\in SST_{[r+1]}^+(\nu)$ with $c_{k}(Q)=\la_{r-k+2}-k+1$ for $1\leq k\leq r+1$,
 
{\rm (b)} $m_{k}(i)\leq m_{k+1}(i) +1$ for $1\leq k\leq r$ and $1\leq i\leq N$,  
where $w_\rev(Q)=w_1\cdots w_N$ and 
$m_k(i)=c_k(w_1\cdots w_i)$.
\end{df}

Then we have the following combinatorial description of $a_{\la/\delta_r\,\nu}$. 
\begin{thm}[{\cite[Theorem 4.4]{CK17}}]\label{Thm:Main-sect5}
For $\la\in \cP$ with $D_{\delta_{r}} \subseteq D_\la \subseteq D_{((r+1)^{r+1})}$ 
and $\nu\in \cP^+$, we have \
$a_{\la/\delta_r\,\nu}=\left|{\tt A}_{\la/\delta_r\,\nu}\right|.$
\end{thm}

\begin{exm}
Let $\lambda = (5,5,4,3,1)$ with $D_\la \subseteq D_{(5^5)}$ and $n=7$. 
For $\nu = (4,3,1)$,  
we have ${\tt L}_{\la/\delta_4\, \nu}=\{\,T_1, T_2\,\}$
and ${\tt A}_{\la/\delta_4\,\nu}=\{\,Q_{T_1}, Q_{T_2}\,\}$ as follows.
\begin{displaymath}
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4mm}
\def \vvv{4.8mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
\draw[-] (\hhh*4,\vvv*0) rectangle (\hhh*5,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*1) rectangle (\hhh*1,\vvv*0);
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-,black!20,fill=black!10] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
\draw[-] (\hhh*4,-\vvv*1) rectangle (\hhh*5,\vvv*0);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*2) rectangle (\hhh*1,-\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*2) rectangle (\hhh*2,-\vvv*1);
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
\draw[-] (\hhh*3,-\vvv*2) rectangle (\hhh*4,-\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*3) rectangle (\hhh*1,-\vvv*2);
\draw[-] (\hhh*1,-\vvv*3) rectangle (\hhh*2,-\vvv*2);
\draw[-] (\hhh*2,-\vvv*3) rectangle (\hhh*3,-\vvv*2);
%
\draw[-] (\hhh*0,-\vvv*4) rectangle (\hhh*1,-\vvv*3);
%\draw[-] (\hhh*1,-\vvv*4) rectangle (\hhh*2,-\vvv*3);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*4.5,\vvv*0.5) {\small $6$};
%
\node at (\hhh*3.5,-\vvv*0.5) {\small $5$};
\node at (\hhh*4.5,-\vvv*0.5) {\small $7$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $6$};
\node at (\hhh*3.5,-\vvv*1.5) {\small $6$};
%
\node at (\hhh*1.5,-\vvv*2.5) {\small $7$};
\node at (\hhh*2.5,-\vvv*2.5) {\small $7$};
%
\node at (\hhh*0.5,-\vvv*3.5) {\small $7$};
%
%\node at (\hhh*5.5,-\vvv*4) {,};
%
\node[left] at (-\hhh*0.2,-\vvv*1) {$T_1=$};
\end{tikzpicture} 
\hskip 3mm
\begin{tikzpicture}[baseline=0mm,scale=0.9]
\def \hhh{4mm}
\def \vvv{4.8mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-,black!20,fill=black!10] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
\draw[-] (\hhh*4,\vvv*0) rectangle (\hhh*5,\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*1) rectangle (\hhh*1,\vvv*0);
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-,black!20,fill=black!10] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
\draw[-] (\hhh*4,-\vvv*1) rectangle (\hhh*5,\vvv*0);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*2) rectangle (\hhh*1,-\vvv*1);
\draw[-,black!20,fill=black!10] (\hhh*1,-\vvv*2) rectangle (\hhh*2,-\vvv*1);
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
\draw[-] (\hhh*3,-\vvv*2) rectangle (\hhh*4,-\vvv*1);
%
\draw[-,black!20,fill=black!10] (\hhh*0,-\vvv*3) rectangle (\hhh*1,-\vvv*2);
\draw[-] (\hhh*1,-\vvv*3) rectangle (\hhh*2,-\vvv*2);
\draw[-] (\hhh*2,-\vvv*3) rectangle (\hhh*3,-\vvv*2);
%
\draw[-] (\hhh*0,-\vvv*4) rectangle (\hhh*1,-\vvv*3);
%\draw[-] (\hhh*1,-\vvv*4) rectangle (\hhh*2,-\vvv*3);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*4.5,\vvv*0.5) {\small $5$};
%
\node at (\hhh*3.5,-\vvv*0.5) {\small $6$};
\node at (\hhh*4.5,-\vvv*0.5) {\small $6$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $6$};
\node at (\hhh*3.5,-\vvv*1.5) {\small $7$};
%
\node at (\hhh*1.5,-\vvv*2.5) {\small $7$};
\node at (\hhh*2.5,-\vvv*2.5) {\small $7$};
%
\node at (\hhh*0.5,-\vvv*3.5) {\small $7$};
%
%\node at (\hhh*5.5,-\vvv*4) {,};
%
\node[left] at (-\hhh*0.2,-\vvv*1) {$T_2=$};
\end{tikzpicture} 
\hskip 8mm
\begin{tikzpicture}[baseline=4mm,scale=0.9]
\def \hhh{4mm}
\def \vvv{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $1$};
\node at (\hhh*1.5,\vvv*0.5) {\small $2$};
\node at (\hhh*2.5,\vvv*0.5) {\small $2$};
\node at (\hhh*3.5,\vvv*0.5) {\small $4$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $3$};
\node at (\hhh*2.5,-\vvv*0.5) {\small $3$};
\node at (\hhh*3.5,-\vvv*0.5) {\small $5$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $4$};
%
\node[left] at (-\hhh*0,-\vvv*0.2) {$Q_{T_1}=$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture} 
\hskip 3mm
\begin{tikzpicture}[baseline=4mm,scale=0.9]
\def \hhh{4mm}
\def \vvv{5mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
\draw[-] (\hhh*3,\vvv*0) rectangle (\hhh*4,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
\draw[-] (\hhh*2,-\vvv*1) rectangle (\hhh*3,\vvv*0);
\draw[-] (\hhh*3,-\vvv*1) rectangle (\hhh*4,\vvv*0);
%
\draw[-] (\hhh*2,-\vvv*2) rectangle (\hhh*3,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $1$};
\node at (\hhh*1.5,\vvv*0.5) {\small $2$};
\node at (\hhh*2.5,\vvv*0.5) {\small $2$};
\node at (\hhh*3.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $3$};
\node at (\hhh*2.5,-\vvv*0.5) {\small $4$};
\node at (\hhh*3.5,-\vvv*0.5) {\small $4$};
%
\node at (\hhh*2.5,-\vvv*1.5) {\small $5$};
%
\node[left] at (-\hhh*0,-\vvv*0.2) {$Q_{T_2}=$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{tikzpicture} 
\end{displaymath}
\noindent
Moreover, we have $s_{(5,5,4,3,1)/\delta_4} = 2 P_{(4,3,1)} + P_{(5,2,1)} + P_{(5,3)}.$
\end{exm}

\begin{rem}
Ardila--Serrano~\cite{AS} already gave a result on the Schur $P$-expansion of the staircase skew Schur function $s_{\delta_{r+1}/\mu}$
(see~\cite[Theorem 4.10]{AS}).
It is well known that $s_{\lambda/\delta_r} = s_{(\lambda/\delta_r)^\rot}$, where $\rot$ means the $180^\circ$ rotation operation on skew diagrams. It implies that the coefficient of $P_\nu$ of $s_{\lambda/\delta_r}$ is equal to that of $s_{\delta_{r+1}/\mu}$.
Via the map in~\cite[Corollary 4.8]{CK17}
we provided a bijection between them.
\end{rem}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   section{Schur expansion of Schur $P$-function}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Schur expansion of Schur $P$-function}\label{section5}

For $\la\in\cP^+$ and $\mu\in \cP$, let $g_{\lambda \mu}$ be the coefficient of $s_\mu$ in the Schur expansion of $P_\la$, that is,
\begin{equation*}
P_\lambda = \sum_{\mu} g_{\lambda \mu} s_\mu.
\end{equation*}
The purpose of this section is to give an alternate proof of the following combinatorial description of $g_{\la\mu}$ due to Stembridge.  

\begin{df} For $\lambda \in \cP^+$ and $\mu \in \cP$, let ${\tt G}_{\la\mu}$ be the set of tableaux $Q$ such that

{\rm (a)} $Q\in SST_{\mc N}(\mu)$ with $c_{k}(Q)+c_{k'}(Q)=\la_k$ for $k\geq 1$, 

{\rm (b)} for $k\geq 1$, if $x$ is the rightmost letter in $w(Q)$ with $|x|=k$, then $x=k$,

{\rm (c)} $w(Q)$ satisfies the lattice property.
\end{df}


\begin{thm}[{\cite[Theorem 9.3]{Ste}}]\label{thm-coeff-g}
For $\la \in \cP^+$ and $\mu \in \cP$, we have \
$g_{\la\mu} = \left| {\tt G}_{\la\mu} \right|$.
\end{thm}

Choose $n$ such that $\la\in \cP_n^+$ and $\mu \in \cP_n$.
Let ${\tt \bf L}_{\la\mu}=
\{\,T\,|\,T\in \B_n(\la),\,\tf_i T={\bf 0}\ (1\leq i\leq n-1),\ {\rm wt} (T) =w_0\mu \,\}.$
Then we have as a $\gl(n)$-crystal $\B_n(\la)\cong \bigsqcup_{\mu}B_n(\mu)^{\oplus |{\tt \bf L}_{\la\mu}|},$
and hence $g_{\la\mu}=|{\tt \bf L}_{\lambda \mu}|$ by linear independence of Schur polynomials.
The proof is similar to that of Theorem \ref{Thm:our-characterization}.
That is, the key is to construct a bijection between ${\tt \bf L}_{\lambda \mu}$ and ${\tt G}_{\la \mu}$.
For $T \in {\tt \bf L}_{\la\mu}$, let $w_\rev(T)=u_1\cdots u_N$ ($N=|\lambda|$). 
Since $T$ is a $\gl(n)$-lowest weight vector, we have by the tensor product of crystals that $u_{N-m+1}\otimes  \cdots \otimes u_N \in \B^{\otimes m}_n$ is a $\gl(n)$-lowest weight element for $1\leq m\leq N$. 
This implies that there exists $\mu^{(m)}\in \cP_n$ for $1\leq m\leq N$ such that 
$u_{N-m+1} \cdots u_N$ is equivalent as an element of $\gl(n)$-crystal to a $\gl(n)$-lowest weight element in $B_n(\mu^{(m)})$,
where $\mu^{(N)} = \mu$ and
$\mu^{(m)}$ is obtained by adding a box in the $(n-u_m+1)$-st row of $\mu^{(m-1)}$ with $\mu^{(0)}=\emptyset$. 
%Here all $\mu^{(m)}$'s are unshifted shape.
Define $Q_T$ to be a tableau of shape $\mu$ whose entry at $\mu^{(m)}/\mu^{(m-1)}$ is 
\begin{equation*}\label{eq:labeling of Q_T-2}
\begin{cases}
k', & \text{if $u_m$ belongs to $T^{(k)}\!\!\uparrow$},\\
k, & \text{if $u_m$ belongs to $T^{(k)}\!\!\downarrow$},\\
\end{cases}
\end{equation*}
for some $1\leq k\leq \ell(\la)$. 
Then we obtain the desired mapping $T \mapsto Q_T$.

\begin{exm}
Let $\la = (3,1)$ and $n=3$. The mapping $T \mapsto Q_T$ gives
\vskip 1mm
\noindent
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $3$};
\node at (\hhh*1.5,\vvv*0.5) {\small $3$};
\node at (\hhh*2.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $T_1=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $3$};
\node at (\hhh*1.5,\vvv*0.5) {\small $2$};
\node at (\hhh*2.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $T_2=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $3$};
\node at (\hhh*1.5,\vvv*0.5) {\small $2$};
\node at (\hhh*2.5,\vvv*0.5) {\small $3$};
%
\node at (\hhh*1.5,-\vvv*0.5) {\small $1$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $T_3=$};
\end{tikzpicture} 
\hskip 8mm
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
\draw[-] (\hhh*2,\vvv*0) rectangle (\hhh*3,\vvv*1);
%
\draw[-] (\hhh*0,-\vvv*1) rectangle (\hhh*1,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $1$};
\node at (\hhh*1.5,\vvv*0.5) {\small $1$};
\node at (\hhh*2.5,\vvv*0.5) {\small $1$};
%
\node at (\hhh*0.5,-\vvv*0.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $Q_{T_1}=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
%
\draw[-] (\hhh*0,-\vvv*1) rectangle (\hhh*1,\vvv*0);
\draw[-] (\hhh*1,-\vvv*1) rectangle (\hhh*2,\vvv*0);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $1'$};
\node at (\hhh*1.5,\vvv*0.5) {\small $1$};
%
\node at (\hhh*0.5,-\vvv*0.5) {\small $1$};
\node at (\hhh*1.5,-\vvv*0.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $Q_{T_2}=$};
\end{tikzpicture} 
\begin{tikzpicture}[baseline=0mm,scale=1]
\def \hhh{4.4mm}
\def \vvv{5.4mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\draw[-] (\hhh*0,\vvv*0) rectangle (\hhh*1,\vvv*1);
\draw[-] (\hhh*1,\vvv*0) rectangle (\hhh*2,\vvv*1);
%
\draw[-] (\hhh*0,-\vvv*1) rectangle (\hhh*1,\vvv*0);
%
\draw[-] (\hhh*0,-\vvv*2) rectangle (\hhh*1,-\vvv*1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node at (\hhh*0.5,\vvv*0.5) {\small $1'$};
\node at (\hhh*1.5,\vvv*0.5) {\small $1$};
%
\node at (\hhh*0.5,-\vvv*0.5) {\small $1$};
%
\node at (\hhh*0.5,-\vvv*1.5) {\small $2$};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\node[left] at (-\hhh*0,\vvv*0) () {\small $Q_{T_3}=$};
\end{tikzpicture} \ .
\\ \noindent
Thus $P_{(3,1)} = s_{(3,1)} + s_{(2,2)} + s_{(2,1,1)}.$
\end{exm}

\begin{rem}
Let $\la \in \cP^+$ be such that $D^+_\la \subseteq D^+_{\delta_{r+1}}$ for some $r \geq 0$. 
Let $\la^{\compp}$ be the complement of $\lambda$ in $D^+_{\delta_{r+1}}$.
It is shown in \cite{Dew} that $s_{\delta_{r+1}/\la} = \sum g_{\nu \la} P_{\nu^{\compp}},$
where the sum runs over all $\nu \in \cP^+$ with $|\nu| = |\lambda|$.
So we have 
$g_{\nu  \,  \la}=a_{\la^\comp/\delta_r\, (\nu^\compp)'},$
where $\lambda^\comp$ is the complement of $\lambda$ in $D_{((r+1)^{r+1})}$.
One may expect that there is a natural bijection between
${\tt G}_{\nu\, \la}$ and ${\tt A}_{\la^\comp/\delta_r\, (\nu^\compp)'}$ that we have not yet make explicit.
\end{rem}

\acknowledgements{
This work was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1501-01.
 The authors are grateful to the referees for careful comments.}
\printbibliography

\end{document}
