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%% define your title in the usual way
\title{Flagged Littlewood--Richardson tableaux and branching rule for orthogonal groups}

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\author{Il-Seung Jang\thanks{\href{mailto:is\_jang@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{1}}

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\address{\addressmark{1}Department of Mathematical Sciences, Seoul National University, Seoul, Korea}

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\abstract{
We give a new combinatorial formula for the branching rule from ${\rm GL}_n$ to ${\rm O}_n$ generalizing the Littlewood's restriction formula.
The formula is given in terms of Littlewood--Richardson tableaux with certain flag conditions which vanish in a stable range. 
}

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%\resume{\lipsum[2]}

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\keywords{quantum groups, crystal graphs, classical groups, branching rules}

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\begin{document}

\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%% Introduction
\section{Introduction}
Let $V^{\la}_{{\rm GL}_n}$ denote a complex finite-dimensional irreducible representation of the complex general linear group ${\rm GL}_n$ parametrized by a partition $\la$ of length $\ell(\la)\leq n$. 
Suppose that ${\rm G}_n$ is a closed subgroup ${\rm Sp}_n$ or ${\rm O}_n$, where $n$ is even for ${\rm G}_n={\rm Sp}_n$. 
%
Let $V^{\mu}_{\rm G_n}$ be a finite-dimensional irreducible ${\rm G}_n$-module parametrized by a partition $\mu$ with $\ell(\mu)\leq n/2$ for ${\rm G}_n={\rm Sp}_n$, and by a partition $\mu$ with $\ell(\mu)\leq n$ and $\mu'_1+\mu'_2\leq n$ for ${\rm G}_n={\rm O}_n$. Here $\mu'=(\mu'_i)_{i\geq 1}$ is the conjugate partition of $\mu$.  

Let 
\begin{equation}\label{eq:branching mult}
\left[V^{\la}_{{\rm GL}_n} : V^{\mu}_{\rm G_n} \right] = \dim {\rm Hom}_{{\rm G}_n}\left(V^{\mu}_{\rm G_n},V^{\la}_{{\rm GL}_n}\right)
\end{equation}
denote the multiplicity of $V^{\mu}_{\rm G_n}$ in $V^{\la}_{{\rm GL}_n}$.
In \cite{Lw-1,Lw-2}, Littlewood showed that if $\ell(\la)\leq n/2$, then 
\begin{equation}\label{eq:Littlewood restriction}
\begin{split}
\left[V^{\la}_{{\rm GL}_n} : V^{\mu}_{{\rm Sp}_n} \right]&=\sum_{\delta\in \cP^{(2)}}c^\la_{\delta'\mu}\quad, \quad \
\left[V^{\la}_{{\rm GL}_n} : V^{\mu}_{{\rm O}_n} \right]=\sum_{\delta\in \cP^{(2)}}c^\la_{\delta\,\mu}\,\,,
\end{split}
\end{equation}
where $c^{\alpha}_{\beta\gamma}$ is the Littlewood--Richardson coefficient corresponding to partitions  $\alpha,\beta,\gamma$, and $\cP^{(2)}$ denotes the set of partition with even parts. There have been numerous works on extending the Littlewood's restriction rules \eqref{eq:Littlewood restriction} for arbitrary $\la$ with $\ell(\la)\leq n$ (see \cite{EW,HTW} and also the references therein), but most of which are obtained in an algebraic way and hence given not in a subtraction-free way.

In \cite{Su}, Sundaram gave a beautiful combinatorial formula for \eqref{eq:branching mult} when ${\rm G}_n={\rm Sp}_n$, as the sum of the numbers of Littlewood--Richardson (LR) tableaux of shape $\la/\mu$ with content $\delta'$ satisfying certain constraints on their entries, which vanish in a stable range $\ell(\la)\leq n/2$. Recently, based on the results in \cite{K18-3, K15}, Lecouvey and Lenart obtained another formula for \eqref{eq:branching mult} when ${\rm G}_n={\rm Sp}_n$ in terms of LR tableaux with some flag conditions on their companion tableaux \cite{LL}. 
On the other hand, no orthogonal analogue of these formula has been known so far.

The main result in this abstract is to give a combinatorial formula for 
\eqref{eq:branching mult} when ${\rm G}_n={\rm O}_n$ for arbitrary $\la$ and $\mu$ in terms of LR tableaux with certain flag conditions on their companion tableaux which vanish in a stable range $\ell(\la)\leq n/2$.

For simplicity, let us state our main result when $n-2\mu'_1\geq 0$.
Note that the restriction on $n-2\mu_1'$ is not significant, since the result for $n-2\mu'_1< 0$ is almost identical.
Let ${\texttt {LR}}^{\la}_{\delta\mu^\pi}$ be the set of LR tableaux of shape $\la/\delta$ with content $\mu^\pi$, where $\mu^\pi$ is the skew Young diagram obtained by $180^\circ$-rotation of $\mu$. Then we have the following (\cref{thm:non-levi branching}).

\begin{thm}\label{thm:main intro}
For $U\in {\texttt {\em LR}}^{\la}_{\delta\mu^\pi}$, let $\sigma_i$ be the row index of the leftmost $\mu'_1-i+1$ in $U$ for $1 \le i \le \mu_1'$, and $\tau_j$ the row index of the second leftmost $\mu'_2-j+1$ in $U$ for $1 \le j \le \mu_2'$.
Let $m_1<\dots<m_{\mu'_1}$ be the sequence given by
$m_i=\min\{n-\sigma_i+1,2i-1\}$, 
and let $n_1\le\dots\le n_{\mu'_2}$ be the sequence such that 
$n_j$ is the $j$-th smallest number in 
$\{\,j+1,\dots,n\,\}\setminus\{\,m_{j+1},\dots,m_{\mu'_1}\,\}$.
Let $\underline{c}^{\la}_{\delta\mu}$ denote the number of $U\in {\texttt {\em LR}}^{\la}_{\delta\mu^\pi}$ such that
\begin{equation*}\label{eq:flagged conditions}
\tau_j + n_j \le n+1,
\end{equation*}
for $1\leq j\leq \mu'_2$.
Then we have
\begin{equation*}\label{eq:main result}
\left[V^{\la}_{{\rm GL}_n} : V^{\mu}_{{\rm O}_n} \right]=\sum_{\delta\in \cP^{(2)}}\underline{c}^\la_{\delta\mu}\,.
\end{equation*}
\end{thm}

The branching multiplicity \eqref{eq:branching mult} is equal to the one from $D_\infty$ to $A_{+\infty}$ from a viewpoint of see-saw dual pairs in Howe duality on a Fock space \cite{Wa}. 
We use the Kashiwara's crystal base theory of quantum groups and the {\em spinor model} for crystal graphs of type $D_\infty$ \cite{K16} to describe the latter multiplicity.
Unlike the case of ${\rm Sp}_n$ \cite{LL}, we have to develop in addition a non-trivial combinatorial algorithm on spinor model called {\em separation} in order to have a description of branching multiplicity in terms of LR tableaux satisfying the condition for $\underline{c}^{\la}_{\delta\mu}$. This is a key ingredient in the proof of \cref{thm:main intro}. We can also recover the formula \eqref{eq:Littlewood restriction} in a stable range directly from the above formula. A full version of this paper including detailed proofs has appeared in \cite{JK19}.
%%%%% End of Introduction

%%%%% Notations
\section{Notations} \label{sec:notations}
Let $\Z_+$ denote the set of non-negative integers.
Let $\cP$ be the set of partitions or Young diagrams. 
We let $\cP_{\ell}=\{\,\la\in\cP\,|\,\ell(\la)\leq \ell\,\}$ for $\ell\geq 1$, 
where $\ell(\la)$ is the length of $\la$,
let $\cP^{(2)}=\{\,\la\in\cP\,|\,\la=(\la_i)_{i \geq 1}, \la_i\in 2\Z_+ \ (i\geq 1)\,\}$.
%
For a skew Young diagram $\lambda/\mu$, we define ${SST}(\lambda/\mu)$ to be the set of semistandard tableaux of shape $\lambda/\mu$ with entries in $\mathbb{N}$. For $T\in SST(\lambda/\mu)$, let 
$w(T)$ be the word given by reading the entries of $T$ column by column from right to left and from top to bottom in each column, and let ${\rm sh}(T)$ denote the shape of $T$.  

Let $\la\in \cP$ be given. For $T\in SST(\lambda)$ and $a\in \N$, we denote by $a \rightarrow T$ the tableau obtained by the column insertion of $a$ into $T$ (cf. \cite{Ful}). For a word $w=w_1\dots w_r$, we define $(w\rightarrow T)=(w_r\rightarrow (\dots \rightarrow(w_1\rightarrow T)))$. For a semistandard tableau $S$, we define $(S\rightarrow T)=(w(S)\rightarrow T)$. 

Let $\la^\pi$ denote the skew Young diagram obtained from $\la$ by $180^\circ$ rotation. Let $H_\la$ and  $H_{\la^\pi}$ be the tableaux in $SST(\la)$ and $SST(\la^\pi)$, respectively, where the $i$-th entry from the top in each column is filled with $i$ for $i\geq 1$.

For $\la,\mu,\nu\in\cP$, let $\texttt{LR}^\la_{\mu \nu}$ be the set of Littlewood--Richardson tableaux $S$ of shape $\la/\mu$ with content $\nu$. There is a natural bijection from $\texttt{LR}^\la_{\mu \nu}$ to the set of $T\in SST(\nu)$ such that $(T\rightarrow H_\mu)=H_\la$, where each $i$ in the $j$th row of $S\in \texttt{LR}^\la_{\mu \nu}$ corresponds to $j$ in the $i$th row of $T$. We call such $T$ a companion tableau of $S\in \texttt{LR}^\la_{\mu \nu}$.

We also define $\texttt{LR}^\la_{\mu\nu^\pi}$ to be the set of $S\in SST(\la/\mu)$ with content $\nu^\pi$ such that $w(T)=w_1\dots w_r$ is an {anti-lattice word}, that is,
the number of $i$ in $w_k\dots w_r$ is greater than or equal to that of $i-1$ for each $k\geq 1$ and $1<i\leq \ell(\nu)$. 
Let us call $S$ a Littlewood--Richardson tableaux of shape $\la/\mu$ with content $\nu^\pi$.
As in case of $\texttt{LR}^\la_{\mu\nu}$, the map from $S\in \texttt{LR}^\la_{\mu\nu^\pi}$ to its companion tableau gives a natural bijection from $\texttt{LR}^\la_{\mu\nu^\pi}$ to the set of $T\in SST(\nu^\pi)$ such that $(T\rightarrow H_\mu)=H_\la$.
%
From now on, all the LR tableaux are assumed to be the corresponding companion tableaux unless otherwise specified. 

Let $S\in {\texttt {LR}}^{\la'}_{\mu'\nu'}$ be given, that is, $(S\rightarrow H_{\mu'})=H_{\la'}$.  
Let $S^1,\dots,S^p$ denote the columns of $S$ enumerated from the right. 
For $1\le i\le p$, let $H^i=(S^i \rightarrow H^{i-1})$ with $H^0=H_{\mu'}$  
so that $H^{p}=H_{\la'}$. 
Define $Q(S\rightarrow H_{\mu'})\in SST(\la/\mu)$ to be the tableau such that the horizontal strip ${\rm sh}(H^i)'/{\rm sh}(H^{i-1})'$ is filled with $1\leq i\leq p$. 
On the other hand, let $U\in {\texttt {LR}}^{\la}_{\mu\nu^\pi}$ be given, that is,  
${\rm sh}(U\rightarrow H_{\mu})=H_{\la}$. Let $U_i$ denote the $i$-th row of $U$ from the top, and let $H_i = (U_i\rightarrow H_{i-1})$ with $H_0=H_{\mu}$ for $1\le i\le p$. 
Define $Q(U\rightarrow H_{\mu})$ to be tableau such that the horizontal strip ${\rm sh}(H_i)/{\rm sh}(H_{i-1})$ is filled with $1\le i\le p$. 

Then we have a bijection
\begin{equation}\label{conjugation of LR}
\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
\psi : {\texttt {LR}}^{\la'}_{\mu'\nu'}  \ar@{->}[r] & {\texttt {LR}}^{\la}_{\mu\nu^\pi}},
\end{equation}
where for $S\in {\texttt {LR}}^{\la'}_{\mu'\nu'}$, $\psi(S)=U$ is given by 
a unique $U\in SST(\nu^\pi)$ such that $(U\rightarrow H_{\mu})=H_{\la}$ and 
$Q(U\rightarrow H_{\mu})=Q(S\rightarrow H_{\mu'})$.
%


%%%%% End of Notations

%%%%% Spinor model
\section{Spinor model} \label{sec:spinor}

%%%%% Definitions (of spinor model)
\subsection{Definitions}
Let us recall the spinor model of type $D_\infty$, which is a combinatorial model for 
the crystal of an integrable irreducible highest weight module over the quantum group of type $D_{\infty}$ 
(see \cite{K16} and \cite[Section 2.1]{JK19} for more details).

Let ${\mf g}$ be the Kac-Moody Lie algebra of type $D_{\infty}$.
We assume that the index set for simple roots is $I=\Z_+$, and the weight lattice is $P=\Z\Lambda_0\oplus \left( \bigoplus_{i\geq 1}\Z\epsilon_i \right)$.
The associated Dynkin diagram, set of simple roots $\Pi=\{\,\alpha_i\,|\,i \ge 0\,\}$, and fundamental weight $\Lambda_i$ $(i\ge 0)$ are given by
\begin{center}
\setlength{\unitlength}{0.16in} \hskip -3cm
\hskip 2cm \scalebox{1}{ \begin{picture}(24,5.8)
%\put(2,2){\makebox(0,0)[c]{${\mf d}_{\infty}\ :$}}
\put(6,0){\makebox(0,0)[c]{$\circ$}}
\put(6,4){\makebox(0,0)[c]{$\circ$}}
\put(8,2){\makebox(0,0)[c]{$\circ$}}
\put(10.4,2){\makebox(0,0)[c]{$\circ$}}
\put(14.85,2){\makebox(0,0)[c]{$\circ$}}
\put(17.25,2){\makebox(0,0)[c]{$\circ$}}
\put(19.4,2){\makebox(0,0)[c]{$\circ$}}
%\put(21.5,2){\makebox(0,0)[c]{$\bigcirc$}}
%
\put(6.35,0.3){\line(1,1){1.35}} \put(6.35,3.7){\line(1,-1){1.35}}
\put(8.4,2){\line(1,0){1.55}} \put(10.82,2){\line(1,0){0.8}}
%
\put(13.2,2){\line(1,0){1.2}} \put(15.28,2){\line(1,0){1.45}}
\put(17.7,2){\line(1,0){1.25}} \put(19.8,2){\line(1,0){1.25}}
%\put(21.95,2){\line(1,0){1.4}} %\put(24.3,2){\line(1,0){1.4}}
%
\put(12.5,1.95){\makebox(0,0)[c]{$\cdots$}}
%\put(21.5,1.95){\makebox(0,0)[c]{$\cdots$}}
\put(22,1.95){\makebox(0,0)[c]{$\cdots$}}
%
\put(6,5){\makebox(0,0)[c]{\tiny $\alpha_{0}$}}
\put(6,-1.2){\makebox(0,0)[c]{\tiny $\alpha_{1}$}}
\put(8.2,1){\makebox(0,0)[c]{\tiny $\alpha_{2}$}}
\put(10.4,1){\makebox(0,0)[c]{\tiny $\alpha_{3}$}}
\put(14.8,1){\makebox(0,0)[c]{\tiny $\alpha_{k-1}$}}
\put(17.15,1){\makebox(0,0)[c]{\tiny $\alpha_k$}}
\put(19.5,1){\makebox(0,0)[c]{\tiny $\alpha_{k+1}$}}
\end{picture} }

\begin{equation*}
\begin{split}
&\Pi=\{\, \alpha_0=-\epsilon_1-\epsilon_2, \  \alpha_i=\epsilon_i-\epsilon_{i+1} \ (i\geq 1)\, \}, \quad 
\La_i =
\begin{cases}
\La_0+\epsilon_1, & \text{if $i=1$}, \\
2\La_0+\epsilon_1+\cdots+\epsilon_i, & \text{if $i>1$}.
\end{cases}
\end{split}
\end{equation*}

\end{center}
% 
Let $\mathfrak{l}$ be the subalgebra of ${\mf g}$ associated to $\Pi\setminus\{\alpha_0\}$, which is of type $A_{+\infty}$.

For $n\geq 1$, let 
$$
{\mc P}({\rm O}_n)=\{\,\mu=(\mu_1,\cdots,\mu_n)\,|\,\mu_i\in\Z_+, \ \mu_1\geq\ldots \geq \mu_{n},\
\mu'_1+\mu'_2\leq n\,\}\,.
$$
For $\mu\in \mc{P}({\rm O}_n)$, put 
$
	\Lambda(\mu) = n\La_0+\mu'_1\epsilon_1+\mu'_2\epsilon_2 + \cdots\,.
$
Then we have $P_+=\{\,\La(\mu)\,|\, \mu\in \bigsqcup_n \mc{P}({\rm O}_n)\,\}$ the set of dominant integral weights for $\mf g$. 
We denote by ${\bf B}(\Lambda)$ the crystal of an integrable irreducible highest weight module over the quantum group $U_q({\mf g})$ with highest weight $\Lambda \in P_+$.



Let $T$ be a tableau of two-column skew shape $(2^{b+c}, 1^a) / (1^b)$ for $a, b, c \in \mathbb{Z}_+$.
We denote the left and right columns of T by $T^{\texttt{L}}$ and $T^{\texttt{R}}$ respectively.
%
Suppose that $T$ is semistandard and we can slide down $T^{\texttt{R}}$ by $k$ positions to have a semistandard tableau $T'$ of shape $(2^{b+c}, 1^{a-k})$$/(1^{b-k})$. We define ${\mf r}_T$ to to be the maximal such $k$.

Let
\begin{equation*} %\labe{eq:def_T}
\begin{split}
	\mathbf{T}(a)  \,\, & = \left\{\, T \,|\, T \in {SST}\big((2^{b+c}, 1^a) / (1^b)\big),\ b,c \in 2\Z_+,\ \mathfrak{r}_T \le 1 \,\right\}\quad (a\in\Z_+), \\
	\overline{\mathbf{T}}(0) \,\, &   = \bigsqcup_{b, c\, \in 2\mathbb{Z}_+} {SST}\big((2^{b+c+1}) / (1^b)\big),  \quad \ \ \mathbf{T}^{\textrm{sp}}  = \bigsqcup_{a \in \mathbb{Z}_+} {SST}((1^a)), \\
	\mathbf{T}^{\textrm{sp}+} \, &  = \{\, T \, |\,  T \in \mathbf{T}^{\textrm{sp}}, \, \mathfrak{r}_T = 0\, \}, \quad  \mathbf{T}^{\textrm{sp}-} = \{ \,T \, |\, T \in \mathbf{T}^{\textrm{sp}}, \, \mathfrak{r}_T = 1\, \},
\end{split}
\end{equation*}
where the integer ${\mf r}_T$ of $T \in {\bf T}^{\rm sp}$ is defined by the residue of ${\rm ht}(T)$ modulo $2$.
It is shown that ${\bf T}(a)$, $\ov{\bf T}(0)$ and ${\bf T}^{\rm sp}$ have ${\mf g}$-crystal structure \cite[Proposition 4.1]{K16} (cf. \cite[Section 2.3]{JK19}) such that 
\begin{equation*}
\begin{split}
	& \mathbf{T}(a) \cong \B(\La_{a}) \ \  (a \ge 2), \quad \mathbf{T}(0) \cong \B(2\La_{0}), \quad \overline{\mathbf{T}}(0) \cong \B(2\La_{1}), \quad \mathbf{T}(1) \cong \B(\La_{0}+\La_1), \\
	& \mathbf{T}^{\textrm{sp}+} \cong \B(\La_0)\,, \quad \mathbf{T}^{\textrm{sp}-} \cong \B(\La_1).
\end{split}
\end{equation*}

Let $\mu \in \mathcal{P}({\rm O}_n)$ be given. Let $q_\pm$ and $r_\pm$ be non-negative integers such that 
\begin{equation*} %\label{eq:def_q_r}
\begin{cases}	
n-2\mu'_1 = 2q_+ + r_+, & \textrm{if} \ n-2\mu'_1 \ge 0, \\ 
2\mu'_1 - n = 2q_- + r_-, & \textrm{if} \ n-2\mu'_1 < 0, 
\end{cases}	
\end{equation*} 
where $r_\pm = 0, 1$. 
Let $\ov{\mu} = (\overline{\mu}_i) \in \cP$ be such that 
$\ov{\mu}'_1 = n-\mu'_1$ and $\ov{\mu}'_i = \mu'_i$ for $i \ge 2$ and let $M_+ = \mu'_1$ and $M_- = \ov{\mu}'_1$. %, and $L = M_\pm + q_\pm$. 
Put
\begin{equation*}\label{eq:hat{T}}
\widehat{\mathbf{T}}(\mu, n) = 
\begin{cases}
\mathbf{T}(\mu_1) \times \cdots \times \mathbf{T}(\mu_{M_+}) \times \mathbf{T}(0)^{\times q_+} \times (\mathbf{T}^{\textrm{sp+}})^{\times r_+}, & \textrm{if} \ n - 2\mu'_1 \ge 0, \\
\mathbf{T}(\ov{\mu}_1) \times \cdots \times \mathbf{T}(\ov{\mu}_{M_-}) \times \overline{\mathbf{T}}(0)^{\times q_-} \times (\mathbf{T}^{\textrm{sp}-})^{\times r_-}, & \textrm{if} \ n - 2\mu'_1 < 0.
\end{cases}
\end{equation*}
We give the ${\mf g}$-crystal structure on $\widehat{\mathbf{T}}(\mu, n)$ by the tensor product rule of crystals by identifying $(\dots, T_2, T_1) \in \widehat{\bf T}(\mu, n)$ with $T_1 \otimes T_2 \otimes \dots$\,.

Let
$$
	\mathbf{T}(\mu, n) = 
	\{\, \mathbf{T} = (\dots,T_2,T_1) \in \widehat{\mathbf{T}}(\mu, n) \, | \, T_{i+1} \prec T_{i}   \ (i \ge 1) \,\},
$$
where $T_{i+1} \prec T_i$ means that the pair $(T_{i+1}, T_i)$ satisfies the {\em admissible conditions} given in \cite[Definition 3.4]{K16}.
%
It is shown in \cite{K16} that ${\bf T}(\mu, n)$ is a connected component in $\widehat{\bf T}(\mu, n)$ including the unique highest weight element of weight $\Lambda(\mu)$.
Hence we have the following.

\begin{thm} \label{thm:fundamental_theorem} \cite[Theorem 4.3--4.4]{K16} 
For $\mu \in \mathcal{P}({\rm O}_n)$,
we have $$\mathbf{T}(\mu, n) \cong \mathbf{B}(\Lambda(\mu)).$$  
\end{thm}
%
We call $\mathbf{T}(\mu, n)$ the {\em spinor model for $\mathbf{B}(\Lambda(\mu))$}.

\begin{ex} \label{ex:spinor}
{\em
Let $n=8$ and $\mu = (4, 3, 3, 2) \in \mathcal{P}({\rm O}_8)$. Then
$\Lambda(\mu) = \Lambda_4 + 2\Lambda_3 + \Lambda_2$.
%
Let ${\bf T} = (T_4,T_3, T_2, T_1)$ given by
\begin{equation*}
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
& \begin{ytableau}
\none &\none & \none & \none & \none & \none & \none & \none & \none & \none & \none & \tl{1} & \none  \\
\none &\none & \none & \none & \none & \none & \none & \none & \none & \none & \none & \tl{2} & \none  \\
\none &\none & \tl{1} & \none & \none & \tl{1} & \none & \none & \tl{1} & \none & \tl{1} & \tl{3} & \none \\
\none[\!\!\!\!\mathrel{\raisebox{-0.7ex}{$\scalebox{0.45}{\dots\dots\dots\dots}$}}] &\none & \tl{2} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.45}{\dots\dots}$}}] & \none & \tl{2} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.45}{\dots\dots}$}}] & \none & \tl{4} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.45}{\dots\dots}$}}]& \tl{2} & \tl{4} & \none[\ \mathrel{\raisebox{-0.7ex}{\quad $\scalebox{0.45}{\dots\dots\dots\dots}$\ ${}_{\scalebox{0.75}{$L$}}$}}] \\
\none &\tl{1} & \none & \none & \tl{1} & \none & \none & \tl{1} & \none & \none & \tl{3} & \none & \none  \\
\none &\tl{3} & \none & \none & \tl{3} & \none & \none & \tl{5} & \none & \none & \tl{5} & \none & \none  \\
\none &\tl{4} & \none & \none & \tl{4} & \none & \none & \tl{6} & \none & \none & \none & \none & \none  \\
\none &\tl{5} & \none & \none & \none & \none & \none & \none & \none & \none & \none & \none & \none  \\
\end{ytableau} \quad\quad \\
&\ \hskip 5mm T_4 \hskip 9mm T_3 \hskip 9mm T_2 \hskip 8mm T_1 \hskip 13mm 
\end{split}
\end{equation*} 
where the dotted line denotes the common horizontal line $L$.
In this case, $T_4 \prec T_3 \prec T_2 \prec T_1$ (cf. \cite[Definition 3.4]{K16})
and thus ${\bf T} \in {\bf T}(\mu, 8)$. 
}
\end{ex}


%%%%% End of Definition

%%%%% Separation lemma
\subsection{Separation lemma} \label{subsec:separation}
For simplicity, we assume that $n$ is even and $\mu \in {\mc P}({\rm O}_n)$ satisfies $n-2\mu_1' \ge 0$. The same result (\cref{lem:separation}) also holds for the other cases (see \cite[Section 3.3--3.4]{JK19} for more details).

\begin{df}
{\em 
Let
\begin{equation*}%\label{eq:l-highest weight}
{\bf H}(\mu,n)=\{\,{\bf T}\,|\,{\bf T}\in \mathbf{T}(\mu, n),\ \te_i{\bf T}={\bf 0}\ (i\neq 0)\,\}\,,
\end{equation*}
and call ${\bf T}\in {\bf H}(\mu,n)$ an {\em $\mf l$-highest weight element} in $\mathbf{T}(\mu, n)$.	
In other words, we have ${\bf T} \in {\bf H}(\mu, n)$ if and only if ${\bf T}\equiv_{\mf l} H_\la$ for some $\la\in \cP$.
Here
%$H_{\lambda} \in SST(\lambda)$, whose entries in the $i$-th row are all $i$,
%and 
$\equiv_{\mf l}$ means the ${\mf l}$-crystal equivalence or Knuth equivalence.
}
\end{df}

Let ${\bf T} = (T_l, \dots T_1) \in {\bf H}(\mu, n)$ with ${\rm sh}(T_i) = (2^{b_i+c_i}, 1^{a_i}) / (1^{b_i})$ for $1 \le i \le l$.
We denote by $T_i^{\texttt{R}}(k)$ (resp. $T_i^{\texttt{L}}(k)$) the $k$-th entry of $T_i^{\texttt{R}}$ (resp. $T_i^{\texttt{L}}$) from the bottom.
%
Let us introduce an algorithm on $(T_{i+1}, T_i)$, 
which is roughly speaking sliding the {\em tail} of $T_i$ to the left by one position.
\begin{itemize}
	\item[(S1)] If $T_{i+1}^{\texttt{R}}(1) < T_i^{\texttt{L}}(a_i)$, then we move the subtableau $\{ T_i^{\texttt{L}}(k) : 1 \le k \le a_i \}$ of $T_i^{\texttt{L}}$ to be located below $T_{i+1}^{\texttt{R}}$.
		For example, 
		\begin{equation*}
\hskip 0.2cm
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
	\none & \none & \none & \none & \none  &  \tl{1} & \none\\
	\none & \none & \none & \none & \none &   \tl{2} & \none\\
	\none & \none & \tl{\blue 1} & \none &  \tl{\red 1} & \tl{3} & \none\\
	\none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] & \none & \tl{\blue {\bf 2}} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] & \tl{\red 2} & \tl{4} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] \\
	\none & \tl{1} & \none & \none &  \tl{\red {\bf 3}} & \none & \none\\
	\none & \tl{4} & \none & \none &  \tl{\red 5} & \none & \none\\
	\none & \none & \none & \none & \none & \none & \none & \none\\
	\none & \none[] & \none[\, \tl{$T_{i+1}$}] & \none & \none & \none[\,\, \tl{$T_i$} \quad ] & \none[] & \none
\end{ytableau}
\end{split} \quad {\xmapsto{\hspace*{1.5cm}}} \quad
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
	\none & \none & \none & \none & \none & \tl{1} & \none \\
	\none &\none & \none & \none & \none & \tl{2} & \none \\
	\none &\none & \tl{\blue 1} & \none & \tl{1} & \tl{3} & \none \\
	\none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] &\none & \tl{\blue 2} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] & \tl{2} & \tl{4} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] \\
	\none &\tl{1}  & \tl{\red 3}  & \none &  \none & \none & \none \\
	\none &\tl{4}  & \tl{\red 5}  & \none &  \none & \none & \none \\
	\none &\none & \none & \none & \none & \none & \none \\
	\none &\none & \none & \none & \none & \none & \none \\
\end{ytableau}
\end{split}
\end{equation*}
Here $T_{i+1}^{\texttt{R}}(1) = 2 < T_i^{\texttt{L}}(a_i) = 3$ with $a_i = 2$.

	
	\item[(S2)] If $T_{i+1}^{\texttt{R}}(1) > T_i^{\texttt{L}}(a_i)$, then 
		we slide up the subtableau $\{ T_i^{\texttt{L}}(k) : k \ge a_i \}$ of $T_i^{\texttt{L}}$ by two positions and put $T_{i+1}^{\texttt{R}}(1)$ below it. 
		%
		Also we slide down the subtableau $T_{i+1}^{\texttt{R}} \setminus \{ T_{i+1}^{\texttt{R}}(1) \}$ of $T_{i+1}^{\texttt{R}}$ by two positions and put the subtableau $\{ T_i^{\texttt{L}}(k) : 1 \le k \le a_i-1 \}$ below it. 
		For example,
		\begin{equation*}
\hskip 3mm
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none & \none & \none & \none & \tl{1} \\
\none & \none & \none & \none & \tl{2} \\
\none & \tl{\blue 1} & \none & \tl{\red 1} & \tl{3} \\
\none & \tl{\blue {\bf 4}} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] & \tl{\red 2} & \tl{4} \\
\tl{1} & \none & \none &  \tl{\red {\bf 3}} & \none \\
\tl{5} & \none & \none &  \tl{\red 6} & \none \\
\tl{6} & \none & \none &  \tl{\red 8} & \none \\
\tl{7} & \none & \none &  \none & \none \\
\none & \none & \none & \none & \none & \none \\
\none[ ] & \none[\, \tl{$T_{i+1}$}] & \none & \none & \none[\tl{$T_i$}\,\,\,] & \none[ ] 
\end{ytableau}
\end{split}\ \  {\xmapsto{\hspace*{0.25cm}}}\ \
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none & \none & \none & \tl{\red 1} & \tl{1} \\
\none & \none & \none & \tl{\red 2} & \tl{2} \\
\none & \tl{\blue 1} & \none & \tl{\red 3} & \tl{3} \\
\none & \tl{\blue 4} & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.4}{\dots\dots}$}}] & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.4}{\dots\dots}$}}] & \tl{4} \\
\tl{1} & \none & \none & \none & \none \\
\tl{5} & \none & \none & \tl{\red 6} & \none \\
\tl{6} & \none & \none & \tl{\red 8} & \none \\
\tl{7} & \none & \none & \none & \none \\
\none & \none & \none & \none & \none \\
\none & \none & \none & \none & \none \\
\end{ytableau}
\end{split}\ \ \xmapsto{\hspace*{0.25cm}}\ \
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none & \none & \none &  \tl{\red 1} & \tl{1} \\
\none & \none & \none &  \tl{\red 2} & \tl{2}  \\
\none & \tl{\blue 1} & \none & \tl{\red 3} & \tl{3} \\
\none & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.4}{\dots\dots}$}}] & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.4}{\dots\dots}$}}] & \tl{\blue 4} & \tl{4} \\
\tl{1} & \none & \none & \none & \none \\
\tl{5} & \tl{\red 6} & \none & \none & \none \\
\tl{6} & \tl{\red 8} & \none & \none & \none \\
\tl{7} & \none & \none & \none & \none \\
\none & \none & \none &  \none & \none \\
\none & \none & \none & \none & \none \\
\end{ytableau}
\end{split}\ \ \xmapsto{\hspace*{0.25cm}}\ \
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none & \none & \none &  \tl{\red 1} & \tl{1} \\
\none & \none & \none &  \tl{\red 2} & \tl{2} \\
\none & \none & \none & \tl{\red 3} & \tl{3} \\
\none & \none & \none[\mathrel{\raisebox{-0.6ex}{$\scalebox{0.45}{\dots\dots}$}}] & \tl{\blue 4} & \tl{4} \\
\tl{1} & \tl{\blue 1} & \none & \none & \none \\
\tl{5} & \tl{\red 6} & \none  & \none & \none \\
\tl{6} & \tl{\red 8} & \none  & \none & \none \\
\tl{7} & \none & \none & \none  & \none \\
\none & \none & \none & \none & \none & \none \\
\none & \none & \none & \none & \none & \none \\
\end{ytableau}
\end{split}
\end{equation*} 
Here $T_{i+1}^{\texttt{R}}(1) = 4 > T_i^{\texttt{L}}(a_i) = 3$ with $a_i = 3$.
%
\end{itemize}
Note that the single-column tableaux $T_{i+1}^{\texttt{L}}$ and $T_i^{\texttt{R}}$ are invariant under the above algorithm.

We identify ${\bf T}$ with $(T_l^{\texttt{L}}, T_l^{\texttt{R}}, \dots T_1^{\texttt{L}}, T_1^{\texttt{R}})$.
Let $\widetilde{\bf T}$ be the sequence of single-column tableaux obtained from ${\bf T}$ by applying the above algorithm to each pair $(T_{i+1}, T_i)$ from $l-1$ to $1$, and then removing $T_l^{\texttt{L}}$.
%
By \cite[Lemma 3.10, Corollary 3.11]{JK19}, we have $\widetilde{\bf T} \in {\bf H}(\td{\mu},n-1)$, where $\widetilde{\mu} = (\mu_2, \mu_3, \dots)$\,.
%
Hence we can apply the above algorithm to $\widetilde{\bf T}$ again, and repeat this process as far as possible to get a tableau $\ov{\bf T}$.
%}	
%\end{rem}

\begin{lem}[Separation lemma] \label{lem:separation}
Under the above hypothesis, $\ov{\bf T}$ satisfies the following conditions:
\begin{itemize}
\item[\em (1)] $\ov{\bf T} \in SST(\eta)$, where $\eta$ is the skew Young diagram given in \eqref{eq:shape after separation},

\item[\em (2)] $\ov{\bf T}$ is Knuth equivalent to ${\bf T}$, that is, $\ov{\bf T}\equiv_{\mf l}{\bf T}$,

\item[\em (3)] Let $\ov{\bf T}^{\texttt{\em body}}$ and $\ov{\bf T}^{\texttt{\em tail}}$ be the subtableaux of $\ov{\bf T}$ located above and below the horizontal line L, respectively. Then 
	$\ov{\bf T}^{\texttt{\em body}}=H_{(\delta')^\pi}$ for some $\delta\in \cP^{(2)}$, and 
$\ov{\bf T}^{\texttt{\em tail}}\in \texttt{\em LR}^{\la'}_{\delta' \mu'}$ if ${\bf T}\equiv_{\mf l} H_{\la'}$ for some $\la\in \cP$.
\end{itemize}
%
\begin{equation}\label{eq:shape after separation}
\raisebox{.9ex}{$\eta=$} 
\resizebox{.38\hsize}{!}{
\def\lr#1{\multicolumn{1}{|@{\hspace{.75ex}}c@{\hspace{.75ex}}|}{\raisebox{-.04ex}{$#1$}}}
\def\l#1{\multicolumn{1}{|@{\hspace{.75ex}}c@{\hspace{.75ex}}}{\raisebox{-.04ex}{$#1$}}}
\def\r#1{\multicolumn{1}{@{\hspace{.75ex}}c@{\hspace{.75ex}}|}{\raisebox{-.04ex}{$#1$}}}
\raisebox{-.6ex}
{$\begin{array}{cccccccccc}
\cline{8-9}
& & & & & & &\l{\ \ } & \r{ } & \\
\cline{6-7}
 & & & & & \l{\ \ } & & & \r{ }\\
\cline{5-5} 
& & & & \l{\ \ } &  &  & & \r{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{}_{(\delta')^\pi} } \\
\cline{3-4}
& & \l{\ \ } & & & & & & \r{ } &  \\
\cline{2-2}\cdashline{1-10}[0.5pt/1pt]\cline{6-9}
& \l{\ \ } &  & & \r{\ \ }  &  & &  \\ 
\cline{4-5}
& \l{\ \ } & \r{\!\!\!\!\!\!\!\!\! {}_{\mu'}} & & &  & &  \\
& \l{\ \ } & \r{\ \ } & & & & &  \\  
\cline{3-3} 
& \lr{\ \ } & & & &  & &  \\  
\cline{2-2} 
\end{array}$}}\ \ \raisebox{-.7ex}{\scalebox{0.75}{$L$}}
\end{equation}
\end{lem}
Note that ${\bf T} \equiv_{\mf l} \ov{\bf T} \equiv_{\mf l} \ov{\bf T}^{\texttt{body}} \otimes \ov{\bf T}^{\texttt{tail}}$ by (2), and it is not difficult to check that (3) implies that the map 
\begin{equation} \label{eq:injection}
\xymatrixcolsep{2pc}\xymatrixrowsep{0pc}\xymatrix{
\mathbf{T}   \ar@{|->}[r] & \ov{\bf T}^{\texttt{tail}} 
}
\end{equation}
is injective (see \cite[Lemma 6.5]{JK19}).
We will describe the image of the injection \eqref{eq:injection} in \cref{sec:branching}. 

\begin{ex}{\rm
Let $n=8$ and $\mu = (4, 3, 3, 2) \in \mathcal{P}({\rm O}_8)$. 
Let $\mathbf{T} = (T_4, T_3, T_2, T_1) \in {\bf H}(\mu,8)$ given as in \cref{ex:spinor}.
Then since $T_4^{\texttt{R}}(1) = 2 > 1 = T_3^{\texttt{L}}(3)$, $T_3^{\texttt{R}}(1) = 2 > 1 = T_2^{\texttt{L}}(3)$ and $T_2^{\texttt{R}}(1) = 4 > 3 = T_1^{\texttt{L}}(2)$, we apply the algorithm (S2) to each pair $(T_{i+1}, T_i)$ for $i=1, 2, 3$.
%
Consequently we have $\widetilde{\bf T}$ given by 
\begin{equation*}
\begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none &\none &\none &\none & \none & \none & \none & \none & \none & \none & \none & \none & \tl{1} &\none & \tl{1} & \none  \\
\none &\none &\none &\none & \none & \none & \none & \none & \none & \none & \none & \none & \tl{2} &\none & \tl{2} & \none  \\
\none & \none & \none &\none & \tl{1} & \none & \none &\none & \tl{1} & \none & \none &\none & \tl{3} &\none & \tl{3} & \none  \\
\none & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \none & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \tl{2} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \none & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \tl{2} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \none & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \tl{4} & \none[\mathrel{\raisebox{-0.7ex}{$\scalebox{0.3}{\dots\dots}$}}] & \tl{4} & \none \\
\tl{\color{gray}{1}} &\none & \tl{1} & \none & \none &\none & \tl{1} & \none & \none &\none & \tl{1} & \none & \none & \none & \none  \\
\tl{\color{gray}{3}} &\none & \tl{3} & \none & \none &\none & \tl{5} & \none & \none &\none & \tl{5} & \none & \none & \none & \none   \\
\tl{\color{gray}{4}} &\none & \tl{4} & \none & \none &\none & \tl{6} & \none & \none & \none & \none & \none & \none & \none   \\
\tl{\color{gray}{5}} & \none & \none & \none & \none & \none & \none & \none & \none & \none & \none & \none  \\
\end{ytableau}
\end{split}
\end{equation*}  
with the left-most column (in gray) removed.
By \cite[Lemma 3.10, Corollary 3.11]{JK19}, we have $\widetilde{\bf T} \in {\bf H}(\td{\mu},7)$ with $\td{\mu}=(3,3,2)$.

Repeating this process, we have
\begin{equation*}
\hspace{5cm} \ov{\bf T} = \hspace{-6.3cm} \begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
\none & \none & \none & \none & \none & \none & \tl{1} & \tl{1}   \\
\none & \none & \none & \none & \none & \none & \tl{2} & \tl{2}   \\
\none & \none &\none & \none & \tl{1} & \tl{1} & \tl{3} & \tl{3}  \\
\none &\none & \none &\none & \tl{2} & \tl{2} & \tl{4} & \tl{4}   \\
\tl{1} & \tl{1} & \tl{1} & \tl{1} &\none & \none & \none & \none   \\
\tl{3} & \tl{3} & \tl{5} & \tl{5} &\none & \none & \none & \none   \\
\tl{4} & \tl{4} & \tl{6} & \none & \none & \none & \none & \none  \\ 
\tl{5} & \none & \none & \none & \none & \none & \none & \none     \\
\end{ytableau}
\end{split}\,,
\end{equation*} 
where 
\begin{equation*}
\hspace{3.5cm} \ov{\bf T}^{\texttt{body}} = \hspace{-5cm} \begin{split}
\ytableausetup {mathmode, boxsize=0.9em} 
\begin{ytableau}
	\none & \none & \none & \none & \none & \none & \tl{1} & \tl{1}   \\
\none & \none & \none & \none & \none & \none & \tl{2} & \tl{2}   \\
\none & \none &\none & \none & \tl{1} & \tl{1} & \tl{3} & \tl{3}  \\
\none &\none & \none &\none & \tl{2} & \tl{2} & \tl{4} & \tl{4}   \\
\end{ytableau}
\end{split}\,\,, \quad \quad \quad 
 \ov{\bf T}^{\texttt{tail}} = \begin{split}
\begin{ytableau}
	\tl{1} & \tl{1} & \tl{1} & \tl{1} &\none & \none & \none & \none   \\
\tl{3} & \tl{3} & \tl{5} & \tl{5} &\none & \none & \none & \none   \\
\tl{4} & \tl{4} & \tl{6} & \none & \none & \none & \none & \none  \\ 
\tl{5} & \none & \none & \none & \none & \none & \none & \none     \\
\end{ytableau}
\end{split}
\end{equation*}
}
\end{ex}

%%%%% End of Separation lemma

%%%%% End of Spinor model

%%%%% Branching multiplicities
\section{Combinatorial formula of branching multiplicities} \label{sec:branching}

%%%%% Description
\subsection{Branching from $D_{\infty}$ to $A_{+\infty}$}
In this section, we assume $n \in \mathbb{Z}_+$.
Let $\mu \in {\mc P}({\rm O}_n)$, $\la\in \cP_{n}$ and $\delta\in \cP^{(2)}_{n}$ be given. 
We denote by ${\delta}^{\texttt{rev}} = (\delta_1^{\texttt{rev}},\dots,\delta_n^{\texttt{rev}})$ the reverse sequence of $\delta=(\delta_1, \dots,\delta_n)$, that is, $\delta^{\texttt{rev}}_i=\delta_{n-i+1}$, for $1\leq i\leq n$. We put $p=\mu'_1$, $q=\mu'_2$, and $r=(\ov{\mu})'_1$ if $n-2\mu'_1<0$. 

Let 
\begin{equation*}
\texttt{LR}^{\mu}_{\lambda}(\mathfrak{d})
=\{\,{\bf T}\,|\,{\bf T}\in {\bf H}(\mu,n),\ {\bf T}\equiv_{\mf l} H_{\la'}\,\}\,, \quad 
c^{\mu}_{\lambda}(\mathfrak{d}) = |\texttt{LR}^{\mu}_{\lambda}(\mathfrak{d})|\,.
\end{equation*} 
Note that $c^{\mu}_{\lambda}(\mathfrak{d})$ is equal to the multiplicity of irreducible highest weight $\mf l$-module with highest weight $\sum_{i \ge 1}\lambda_i^{'}\epsilon_i$ in the irreducible highest weight $\mf g$-module with highest weight $\La(\mu)$.

\begin{df}\label{bounded orthogonal LR}
{\rm 
For $S \in \texttt{LR}^{\lambda'}_{\delta'\mu'}$, 
let $s_1\leq \dots \leq s_p$ denote the entries in the first row, and $t_1\leq \dots \leq t_q$ the entries in the second row of $S$.
%
Let $1 \le m_1 < \cdots < m_p < n$ be the sequence defined inductively from $p$ to $1$ as follows:
\begin{equation*}
m_i = \max\{\,k\,|\,\delta^{\texttt{rev}}_k\in X_i,\ \delta^{\texttt{rev}}_k<s_i\,\},
\end{equation*}
where 
\begin{equation*}
X_i =
\begin{cases}
\{\,\delta^{\texttt{rev}}_{i},\dots,\delta^{\texttt{rev}}_{2i-1}\,\}\setminus \{\delta^{\texttt{rev}}_{m_{i+1}},\dots,\delta^{\texttt{rev}}_{m_p}\}, & \text{if $1\leq i\leq r$}, \\
\{\,\delta^{\texttt{rev}}_{i},\dots,\delta^{\texttt{rev}}_{n-p+i}\,\}\setminus \{\delta^{\texttt{rev}}_{m_{i+1}},\dots,\delta^{\texttt{rev}}_{m_p}\}, & \text{if $r<i\leq p$},
\end{cases}
\end{equation*}
Here we assume that $r=p$ when $n-2\mu'_1\geq 0$.

Let $n_1<\dots<n_q$ be the sequence such that $n_j$ is the $j$-th smallest integer in $\{j+1, \cdots, n\}\setminus \{ m_{j+1}, \cdots, m_p \}$ for $1 \le j \le q$.

Then we define $\ov{{\texttt {LR}}}^{\lambda'}_{\delta'\mu'}$ to be a subset of ${\texttt {LR}}^{\lambda'}_{\delta'\mu'}$ consisting of $S$ satisfying  
\begin{equation*} \label{eq:condition_on_second_row}
t_j > \delta_{n_j}^{\texttt{rev}},
\end{equation*}
for $1 \le j \le q$.
We put $\overline{c}^{\lambda}_{\delta\mu} = |\ov{\texttt{LR}}^{\lambda'}_{\delta'\mu'}|$.
}
\end{df}

The following is the main result in this abstract, which characterizes the image of injection \eqref{eq:injection}.

\begin{thm} \label{thm:main1} 
For $\mu \in {\mc P}({\rm O}_n)$ and $\la\in \cP_n$,
we have a bijection
\begin{equation*}
\xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
\texttt{{\em LR}}^{\mu}_{\lambda}(\mathfrak{d}) \ar@{->}[r] & \ \underset{\scalebox{0.6}{$\delta \in \cP_{n}^{(2)}$}}{\bigsqcup} \ov{\texttt{{\em LR}}}^{\lambda'}_{\delta'\mu'}  \\ 
\mathbf{T}   \ar@{|->}[r] & \ov{\bf T}^{\texttt{\em tail}} 
}.
\end{equation*}		
\end{thm}

\begin{cor}\label{cor:main result}
Under the above hypothesis, we have
\begin{equation} \label{eq:formula for branching multiplicities}
c^{\mu}_{\lambda}(\mathfrak{d}) 
= \sum_{\delta \in \cP^{(2)}_n}\overline{c}^{\lambda}_{\delta\mu}\,.
\end{equation}
\end{cor}
%

Let us give the alternative description of $c^{\mu}_{\la}(\mf{d})$ which is simpler than 
$\ov{\texttt{LR}}^{\la'}_{\delta'\mu'}$.

\begin{df}\label{flagged orthogonal LR+}{\rm 
For $U\in {\texttt {LR}}^{\la}_{\delta\mu^\pi}$, let 
$\sigma_1>\dots>\sigma_p$ denote the entries in the rightmost column and $\tau_1>\dots>\tau_q$ the  second rightmost column of $U$, respectively.
Let $m_1<\dots<m_p$ be the sequence defined by
\begin{equation*}
m_i=
\begin{cases}
\min\{n-\sigma_i+1,2i-1\}, & \text{if $1\leq i\leq r$},\\
\min\{n-\sigma_i+1,n-p+i\}, & \text{if $r< i\leq p$}.
\end{cases}
\end{equation*} 
and let $n_1<\dots<n_q$ be the sequence such that $n_j$ is the $j$-th smallest number in $\{\,j+1,\dots,n\,\}\setminus\{\,m_{j+1},\dots,m_{p}\,\}$.
%
Then we define $\underline{\texttt {LR}}^{\la}_{\delta\mu}$ to be the subset of ${\texttt {LR}}^{\la}_{\delta\mu^\pi}$ consisting of $U$ such that
\begin{equation}\label{eq:flagged conditions}
\tau_j + n_j\le n+1,
\end{equation}
for $1\leq j\leq q$.
We put 
$\underline{c}^{\lambda}_{\delta \mu}=|\underline{\texttt {LR}}^{\la}_{\delta \mu}|$.
}
\end{df}
%
\begin{rem}{\rm
Recall that $\texttt{LR}^\la_{\delta\mu^\pi}$ is the set of $S\in SST(\la/\delta)$ with content $\mu^\pi$ such that $w(T)=w_1\dots w_r$ is an {anti-lattice word}, while in \cref{flagged orthogonal LR+}, \,${\texttt {LR}}^{\la}_{\delta\mu^\pi}$ is given by the set of the companion tableaux $U$ of $S$.
}
\end{rem}


\begin{thm}\label{thm:main result-flagged}
For $\mu\in {\mc P}({\rm O}_n)$, $\la\in \cP_{n}$ and $\delta\in \cP^{(2)}_{n}$, the bijection $\psi : {\texttt {\em LR}}^{\la'}_{\mu'\nu'}  \longrightarrow {\texttt {\em LR}}^{\la}_{\mu\nu^\pi}$ in \eqref{conjugation of LR} induces a bijection from $\overline{\texttt {\em LR}}^{\la'}_{\delta'\mu'}$ to $\underline{\texttt{\em LR}}^{\lambda}_{\delta\mu}$. 
\end{thm}

\begin{cor}\label{cor:main result-2}
Under the above hypothesis, we have
\begin{equation} \label{eq:main result-2}
c^{\mu}_{\lambda}(\mathfrak{d}) 
= \sum_{\delta \in \cP^{(2)}_n}\underline{c}^{\lambda}_{\delta\mu}\,.
\end{equation}
%
In particular, if $\ell(\lambda) \le \frac{n}{2}$, then we have the Littlewood's restriction formula \eqref{eq:Littlewood restriction} for ${\rm G_n} = {\rm O}_n$ from \eqref{eq:main result-2}.
\end{cor}

\begin{ex} \label{ex:example for LR undervar}
	{\rm
	Let $n=8$, $\mu = (2, 2, 2, 1, 1) \in \mathcal{P}({\rm O}_8)$, $\lambda = (5, 4, 4, 3, 2, 2) \in \cP_8$, and $\delta = (4,2,2,2,2) \in \cP_8^{(2)}$. 
	
		Let us consider the Littlewood--Richardson tableau $U \in \texttt{LR}^{\lambda}_{\delta \mu^{\pi}}$ given by
		\begin{equation} \label{eq:ex of LR undervar}
			\hskip 5cm U = \hskip -7.4cm
			\begin{split}
				\ytableausetup {mathmode, boxsize=0.9em} 
				\begin{ytableau}
					\none & \none \\
					 \none & \tl{1}  \\
					 \none & \tl{2}  \\
					 \tl{2} & \tl{3}  \\
					 \tl{3} & \tl{4} \\
					\tl{6} & \tl{6} \\
					\none & \none \\
				\end{ytableau} 
			\end{split}\,\,,
		\end{equation}
where
$
(\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5) = (6, 4, 3, 2, 1)
$ and
$(\tau_1, \tau_2, \tau_3) = (6, 3, 2).
$
Note that the shape of $U$ is $\mu^{\pi} = (1,1,2,2,2)$ and the content is $\lambda / \delta = (1,2,2,1,0,2)$.
Then the sequences $(m_i)_{1 \le i \le 5}$ and $(n_j)_{1 \le j \le 3}$ (\cref{flagged orthogonal LR+}) are given by $(1, 3, 5, 7, 8)$ 
	and $(2, 4, 6)$ respectively.
It is easy to check that $U$ satisfies the condition \eqref{eq:flagged conditions}. Hence $U \in \underline{\texttt {LR}}^{\la}_{\mu \delta}$.
	
On the other hand, let $S$ be the Littlewood--Richardson tableau in $\ov{\texttt{LR}}^{\lambda'}_{\delta' \mu'}$ (recall \cref{bounded orthogonal LR}) with the enumeration of the columns as follows: 
\begin{equation*}
	\hskip 5cm S  = \hskip -5.5cm \begin{split}
	\ytableausetup {mathmode, boxsize=0.9em} 
		\begin{ytableau}
			\tl{\red{1}} & \none &\tl{\red{3}} &\none & \tl{\red{3}} &\none & \tl{\red{3}} &\none & \tl{\red{5}}   \\
			\tl{\blue{2}} & \none &\tl{\blue{4}} &\none & \tl{\blue{4}} & \none & \none &\none & \none    \\
			\none & \none &\none &\none & \none & \none &\none  &\none & \none \\
			\none[\scalebox{0.6}{$S^5$}] & \none &\none[\scalebox{0.6}{$S^4$}] & \none & \none[\scalebox{0.6}{$S^3$}] & \none &\none[\scalebox{0.6}{$S^2$}] &\none & \none[\scalebox{0.6}{$S^1$}] \\
		\end{ytableau}
	\end{split}\,,
\end{equation*}
where $(s_1, s_2, s_3, s_4, s_5) = (1, 3, 3, 3, 5)$ and $(t_1, t_2, t_3) = (2, 4, 4)$.
%
Then $\psi(S)$  \eqref{conjugation of LR} is obtained by
\begin{equation*}
	\hskip 2.5cm \left(
	\begin{split}
		\ytableausetup {mathmode, boxsize=0.9em} 
		\begin{ytableau}
			\none[\scalebox{0.6}{\color{gray}{$\tl{8}\cdots$}}] & \none[\color{black}{\tl{1}}] & \none[\color{black}{\tl{1}}] & \none[\color{black}{\tl{1}}] & \none[\color{black}{\tl{1}}] & \tl{\red{1}}  \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{7}\cdots$}}] & \none[\color{black}{\tl{2}}] & \none[\color{black}{\tl{2}}]  & \tl{\red{2}}  & \tl{\blue{2}} & \none  \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{6}\cdots$}}] & \none[\color{black}{\tl{3}}] & \none[\color{black}{\tl{3}}]  & \tl{\red{3}} & \tl{\blue{3}} & \none    \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{5}\cdots$}}] & \none[\color{black}{\tl{4}}] & \none[\color{black}{\tl{4}}]  & \tl{\red{4}} & \none & \none \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{4}\cdots$}}] & \none[\color{black}{\tl{5}}] & \none[\color{black}{\tl{5}}] & \none  & \none & \none \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{3}\cdots$}}] & \tl{\red{6}} & \tl{\blue{6}} & \none & \none  & \none \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{2}\cdots$}}] & \none & \none & \none & \none  & \none  \\
			\none[\scalebox{0.6}{\color{gray}{$\tl{1}\cdots$}}] & \none & \none & \none & \none  & \none  \\
			\none & \none & \none[\quad \scalebox{0.7}{$\psi(S) \rightarrow H_{\delta}$}] & \none & \none  & \none  \\
		\end{ytableau}
	\end{split}   \quad , 
	\begin{split}
		\ytableausetup {mathmode, boxsize=1.0em} 
		\begin{ytableau}
			\none & \none & \none & \none & \none &  \tl{\red{1}}  \\
			\none & \none & \none &  \tl{\red{2}}  & \tl{\blue{3}} & \none  \\
			\none & \none & \none &  \tl{\red{3}} & \tl{\blue{4}} & \none    \\
			\none & \none & \none &  \tl{\red{4}} & \none & \none \\
			\none & \none & \none &  \none & \none & \none \\
			\none & \tl{\red{5}} & \tl{\blue{5}} & \none  & \none & \none \\
			\none & \none & \none & \none & \none  & \none  \\
			\none & \none & \none & \none & \none  & \none  \\
			\none & \none & \none[\quad \scalebox{0.7}{$Q(\psi(S) \rightarrow H_{\delta})$}] & \none & \none  & \none  \\
		\end{ytableau} 
	\end{split} 
	\right)
	\ \xymatrixcolsep{3pc}\xymatrixrowsep{0pc}\xymatrix{
		\ar@{->}[r] & } \ 
	\begin{split}
		\ytableausetup {mathmode, boxsize=1.0em} 
			\begin{ytableau}
				 \none & \tl{\red{1}}  \\
				 \none & \tl{\red{2}}  \\
				 \tl{\blue{2}} & \tl{\red{3}}  \\
				 \tl{\blue{3}} & \tl{\red{4}} \\
				\tl{\blue{6}} & \tl{\red{6}} \\
				\none & \none \\
			\end{ytableau} 
	\end{split} = \psi(S). \hskip 13.5cm
\end{equation*}
Note that $\psi(S)$ is same with $U$ \eqref{eq:ex of LR undervar}.
%
Under the above correspondence, 
we observe that $\sigma_i \ (1 \le i \le 5)$ and $\tau_j \ (1 \le j \le 3)$ record
the positions of $s_i$ and $t_j$ in $\delta'$, respectively, and vice versa.
This implies that $S \in \ov{\texttt{LR}}^{\lambda'}_{\delta' \mu'}$ if and only if $\psi(S) \in \underline{\texttt{LR}}^{\lambda}_{\delta\mu}$.
} 
\end{ex}

\begin{rem}
{\rm
(1) We may have an analogue of \cref{thm:main1} for type $B$ and $C$, that is, a multiplicity formula with respect to the branching from $B_\infty$ and $C_{\infty}$ to $A_{+\infty}$, respectively (see Remark 4.14 in \cite{JK19} for more details).

(2) When $n$ is odd, there is a bijection between $\texttt{{LR}}^{\mu}_{\lambda}(\mathfrak{d})$ and a set of LR tableaux with certain conditions, where $\la$ appears as an inner shape of LR tableaux \cite{J}. This alternative description of $\texttt{{LR}}^{\mu}_{\lambda}(\mathfrak{d})$ is used to construct a bijection between the set of pairs of standard tableau of shape $\la$ and ${\bf T}\in \texttt{{LR}}^{\mu}_{\lambda}(\mathfrak{d})$ and the set of vacillating tableaux of shape $\mu$. 
}
\end{rem}



%%%%%%% Branching multiplicities of non-Levi type
\subsection{Branching from ${\rm GL}_n$ to ${\rm O}_n$}
We assume that the base field is $\C$.
Let $V^\la_{{\rm GL}_n}$ denote the finite-dimensional irreducible ${\rm GL}_n$-module corresponding to $\la\in \cP_n$, and $V_{{\rm O}_n}^\mu$ the finite-dimensional irreducible module ${\rm O}_n$-module corresponding to $\mu\in {\mc P}({\rm O}_n)$. 

Then we have the following new combinatorial description of $\left[ V_{\rm{GL}_n}^{\la} : V_{{\rm O}_n}^{\mu} \right]$.

\begin{thm}\label{thm:non-levi branching}
For $\la\in \cP_{n}$ and $\mu\in {\mc P}({\rm O}_n)$, we have
\begin{equation*}
\left[ V_{\rm GL_n}^{\lambda} : V_{\rm O_n}^{\mu} \right] 
= \sum_{\delta \in \cP^{(2)}_n}\ov{c}^{\lambda}_{\delta\mu}
= \sum_{\delta \in \cP^{(2)}_n}\underline{c}^{\lambda}_{\delta\mu}.
\end{equation*}
\end{thm}
\begin{proof} It follows from the branching rule of see-saw pairs $(D_{\infty}, A_{+\infty})$ and $({\rm GL}_n, {\rm O}_n)$ \cite[Theorem 5.3]{K18-3}
\[
\left[ V_{\rm GL_n}^{\lambda} : V_{\rm O_n}^{\mu} \right] = c_{\lambda}^{\mu}(\mathfrak{d})\,,
\]
and \cref{cor:main result,cor:main result-2}.
\end{proof}
%
\begin{rem}{\rm
As an application of the branching multiplicity, we obtain a new combinatorial realization for the Lusztig $t$-weight multiplicity $K_{\mu 0}(t)$ of type $B_n$ and $D_n$ with highest weight $\mu$ and weight $0$ or generalized exponents (see \cite[Section 5]{JK19}). This gives an orthogonal analogue of the result for type $C_n$ in \cite{LL}.
}
\end{rem}

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


%%%%%%% End of Branching multiplicities of non-Levi type

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