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\title[Crystal analysis of type $C$ Stanley symmetric functions]{Crystal Analysis of type $C$ Stanley Symmetric Functions}

\author{Graham Hawkes\thanks{\href{mailto:hawkes@math.ucdavis.edu}{hawkes@math.ucdavis.edu}}, Kirill Paramonov\thanks{\href{mailto:kbparamonov@ucdavis.edu}{kbparamonov@ucdavis.edu}}
  \and Anne Schilling\thanks{\href{mailto:anne@math.ucdavis.edu}{anne@math.ucdavis.edu}. This research was partially supported by NSF grant  DMS--1500050.}}

\address{Department of Mathematics, University of California, One Shields Ave, Davis, CA 95616, U.S.A.}
\keywords{Stanley symmetric functions, crystal bases, Kra\'skiewicz insertion, mixed Haiman insertion, unimodal tableaux,
primed tableaux}

\received{\today}

\abstract{Combining results of T. K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, 
indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur 
functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the 
coefficients in the Schur expansion in terms of highest weight crystal elements.}

\begin{document}


\maketitle

\section{Introduction}
Schubert polynomials of types $B$ and  $C$ were independently introduced by Billey and Haiman~\cite{Billey.Haiman.1995} 
and Fomin and Kirillov~\cite{Fomin.Kirillov.1996}. Stanley symmetric functions~\cite{Stanley.1984} are stable limits 
of Schubert polynomials, designed to study properties of reduced words of Coxeter group elements. In his Ph.D. thesis,
T. K. Lam~\cite{Lam.1995} studied properties of Stanley symmetric functions of types $B$ (and similarly $C$) and $D$.
In particular he showed, using Kra\'skiewicz insertion~\cite{Kraskiewicz.1989,Kraskiewicz.1995}, that the type $B$ Stanley 
symmetric functions have a positive integer expansion in terms of $P$-Schur functions. On the other hand, 
Stembridge~\cite{Stembridge.1989} proved that the $P$-Schur functions expand positively in terms of Schur functions. 
Combining these two results, it follows that Stanley symmetric functions of type $B$ (and similarly type $C$)
have a positive integer expansion in terms of Schur functions.

Schur functions $s_\lambda(\mathbf{x})$, indexed by partitions $\lambda$, are ubiquitous in combinatorics and 
representation theory. They are the characters of the symmetric group and can also be interpreted as characters of type 
$A$ crystals. In~\cite{Morse.Schilling.2016}, this was exploited to provide a combinatorial interpretation in terms of 
highest weight crystal elements of the coefficients in the Schur expansion of Stanley symmetric functions in type $A$.
In this paper, we carry out a crystal analysis of the Stanley symmetric functions $F_w^C(\mathbf{x})$ of type $C$, indexed
by a Coxeter group element $w$. In particular, we use Kra\'skiewicz insertion~\cite{Kraskiewicz.1989,Kraskiewicz.1995}
and Haiman's mixed insertion~\cite{Haiman.1989} to find a crystal structure on primed tableaux, which in turn implies
a crystal structure $\mathcal{B}_w$ on signed unimodal factorizations of $w$ for which $F^C_w(\mathbf{x})$ is a character.
Moreover, we present a type $A$ crystal isomorphism $\Phi \colon \mathcal{B}_w \rightarrow \bigoplus_\lambda 
\mathcal{B}_{\lambda}^{\oplus g_{w\lambda}}$ for 
some combinatorially defined nonnegative integer coefficients $g_{w\lambda}$; here $\mathcal{B}_\lambda$ is the
type $A$ highest weight crystal of highest weight $\lambda$ . This implies the desired decomposition 
$F^C_w(\mathbf{x}) = \sum_\lambda g_{w\lambda} s_\lambda (\mathbf{x})$ (see \cref{corollary.main2})
and similarly for type $B$.

In \cref{section.background}, we review type $C$ Stanley symmetric functions and 
type $A$ crystals. In \cref{section.isomorphism} we describe our crystal isomorphism by combining a slight
generalization of the Kra\'skiewicz insertion~\cite{Kraskiewicz.1989,Kraskiewicz.1995} and Haiman's mixed 
insertion~\cite{Haiman.1989}. The main result regarding the crystal structure under Haiman's mixed insertion 
is stated in \cref{theorem.main2}. The combinatorial interpretation of the coefficients $g_{w\lambda}$ 
is given in \cref{corollary.main2}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}
\label{section.background}

\subsection{Type $C$ Stanley symmetric functions}

The \defn{Coxeter group} $W_C$ of type $C_n$ (or $B_n$), also known as the hyperoctahedral group or the group 
of signed permutations, is a finite group generated by $\{s_0, s_1, \ldots, s_{n-1}\}$ subject to the quadratic relations
$s_i^2 = 1$ for all $i \in I = \{0,1,\ldots,n-1\}$, the commutation relations $s_i s_j = s_j s_i$ provided $|i-j|>1$, and the
braid relations $s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}$ for all $i>0$ and $s_0 s_1 s_0 s_1 = s_1 s_0 s_1 s_0$.

It is often convenient to write down an element of a Coxeter group as a sequence of indices of $s_i$ in the product 
representation of the element. For example, the element $w = s_2 s_1 s_2 s_1 s_0 s_1 s_0 s_1$ is represented by the 
word ${\bf w} = 2120101$. A word of shortest length $\ell$ is referred to as a \defn{reduced word} and $\ell(w):=\ell$ 
is referred as the length of $w$. The set of all reduced words of the element $w$ is denoted by $R(w)$.

We say that a reduced word $a_1 a_2 \ldots a_\ell$ is \defn{unimodal} if there exists an index $v$, such that 
$a_1 > a_2 > \cdots > a_v < a_{v+1} < \cdots < a_\ell.$
Consider a reduced word $\textbf{a} = a_1 a_2 \ldots a_{\ell(w)}$ of a Coxeter group element $w$. 
A \defn{unimodal factorization} of $\textbf{a}$ is a factorization 
$\mathbf{A} = (a_1 \ldots a_{\ell_1}) (a_{\ell_1+1} \ldots a_{\ell_2}) \cdots (a_{\ell_r + 1} \ldots a_L)$ such that each factor 
$(a_{\ell_i+1} \ldots a_{\ell_{i+1}})$ is unimodal. Factors can be empty.

For a fixed Coxeter group element $w$, consider all reduced words $R(w)$, and denote the set of all unimodal 
factorizations for reduced words in $R(w)$ as $U(w)$. 
Given a factorization $\mathbf{A} \in U(w)$, define the \defn{weight} of a factorization $\wt(\mathbf{A})$ to be the vector 
consisting of the number of elements in each factor. Denote by $\nz(\mathbf{A})$ the number of non-empty factors of 
$\mathbf{A}$. 

\begin{example}
For the factorization $\mathbf{A} = (2102)()(10) \in U(s_2 s_1 s_2 s_0 s_1 s_0)$, we have $\wt(\mathbf{A}) = (4,0,2)$ 
and $\nz(\mathbf{A}) = 2$.
\end{example}

Following~\cite{Billey.Haiman.1995, Fomin.Kirillov.1996, Lam.1995}, the \defn{type $C$ Stanley symmetric function} 
associated to $w\in W_C$ is defined as
\begin{equation}
\label{equation.StanleyC}
	F^C_w(\mathbf{x}) = \sum_{\mathbf{A} \in U(w)} 2^{\nz(\mathbf{A})} 
        \mathbf{x}^{\wt(\mathbf{A})}.
\end{equation}
Here $\mathbf{x} = (x_1, x_2, x_3, \ldots)$ and $\mathbf{x}^{\mathbf{v}} = x_1^{v_1} x_2^{v_2} x_3^{v_3} \cdots$.
It is not obvious from the definition why the above functions are symmetric. We refer reader to~\cite{Billey.2014},
where this fact follows easily from an alternative definition. \defn{Type $B$ Stanley symmetric functions} are also labeled by 
$w\in W_C$ given by $F_w^B(\mathbf{x}) = 2^{-o(w)} F_w^C(\mathbf{x})$, where $o(w)$ is the number of zeroes in a 
reduced word for $w$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Type $A$ crystal of words}

Crystal bases~\cite{kashiwara.1994} play an important role in many areas of mathematics. For example, they make it 
possible to analyze representation theoretic questions using combinatorial tools. Here we only review the crystal of words 
in type $A_n$ and refer the reader for more background on crystals to~\cite{Bump.Schilling.2017}.

Consider the set of words $\mathcal{B}_n^h$ of length $h$ in the alphabet $\{1,2,\ldots,n+1\}$.
We impose a crystal structure on $\mathcal{B}_n^h$ by defining lowering operators $f_i$ and raising operators $e_i$
for $1\leqslant i \leqslant n$ and a weight function. The weight of $\mathbf{b} \in \mathcal{B}_n^h$ is the tuple
$\wt(\mathbf{b}) = (a_1,\ldots, a_{n+1})$, where $a_i$ is the number of letters $i$ in $\mathbf{b}$.
The crystal operators $f_i$ and $e_i$ only depend on the letters $i$ and $i+1$ in $\mathbf{b}$.
Consider the subword $\mathbf{b}^{\{i,i+1\}}$ of $\mathbf{b}$ consisting only of the letters $i$ and $i+1$.
Successively bracket any adjacent pairs $(i+1) i$ and remove these pairs from the word.
The resulting word is
of the form $i^a (i+1)^b$ with $a,b\geqslant 0$. Then $f_i$ changes this subword within $\mathbf{b}$ to $i^{a-1} (i+1)^{b+1}$
if $a>0$ leaving all other letters unchanged and otherwise annihilates $\mathbf{b}$. The operator $e_i$ changes this 
subword within $\mathbf{b}$ to $i^{a+1} (i+1)^{b-1}$ if $b>0$ leaving all other letters unchanged and otherwise annihilates 
$\mathbf{b}$. We call an element $\mathbf{b}\in \mathcal{B}_n^h$ \defn{highest weight} if $e_i(\mathbf{b})=\mathbf{0}$ for all
$1\leqslant i\leqslant n$.

 \begin{theorem} \cite{Kashiwara.Nakashima.1994}
 A word $\mathbf{b} = b_1 \ldots b_h \in \mathcal{B}_n^h$ is highest weight if and only if it is a Yamanouchi word.
 That is, for any index $k$ with $1 \leqslant k \leqslant h$ the weight of a subword $b_k b_{k+1} \ldots b_h$ is a partition.
 \end{theorem}

Two crystals $\mathcal{B}$ and $\mathcal{C}$ are said to be \defn{isomorphic} if there exists a bijective map 
$\Phi \colon \mathcal{B} \rightarrow \mathcal{C}$ that preserves the weight function and commutes with the crystal 
operators $e_i$ and $f_i$. A \defn{connected component} $X$ of a crystal is a set of elements where for any two
$\mathbf{b},\mathbf{c} \in X$ one can reach $\mathbf{c}$ from $\mathbf{b}$ by applying a sequence of $f_i$ and $e_i$.

\begin{theorem} \cite{Kashiwara.Nakashima.1994}
Each connected component of $\mathcal{B}_n^h$ has a unique highest weight element.
Furthermore, if $\mathbf{b}, \mathbf{c} \in \mathcal{B}_n^h$ are highest weight elements such that
$\wt(\mathbf{b}) = \wt(\mathbf{c})$, then the connected components generated by $\mathbf{b}$ and 
$\mathbf{c}$ are isomorphic.
\end{theorem}

We denote a connected component with a highest weight element of highest weight $\lambda$ by
$\mathcal{B}_\lambda$. The \defn{character} of the crystal $\mathcal{B}$ is defined to be the polynomial
$\chi_{\mathcal{B}} (\mathbf{x}) = \sum_{\mathbf{b} \in \mathcal{B}} \mathbf{x}^{\wt(\mathbf{b})}$
in the variables $\mathbf{x}=(x_1,x_2,\ldots,x_{n+1})$.

\begin{theorem}[\cite{Kashiwara.Nakashima.1994}]
The character of $\mathcal{B}_{\lambda}$ is equal to the Schur polynomial $s_\lambda (\mathbf{x})$ (or Schur 
function in the limit $n\to \infty$).
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Crystal isomorphism} 
\label{section.isomorphism}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Kra\'skiewicz insertion}
\label{section.kraskiewicz}

In this section, we describe Kra\'skiewicz insertion. To do so, we first need to define 
the \defn{Edelman--Greene insertion}~\cite{Edelmann.Greene.1987}. It is defined for a word 
$\mathbf{w} = w_1 \ldots w_\ell$ and a letter $k$ such that the concatenation $w_1 \ldots w_\ell k$ is an $A$-type 
reduced word. The Edelman--Green insertion of a letter $k$ into an {\it increasing} word 
$\mathbf{w} = w_1 \ldots w_\ell$, denoted by $\mathbf{w} \leftsquigarrow k$, is constructed as follows:
\begin{enumerate}
	\item If $w_\ell < k$, then $\mathbf{w} \leftsquigarrow k = \mathbf{w'},$ where 
	$\mathbf{w'} = w_1 w_2 \ldots w_\ell\ k$.
	\item If $k>0$ and $k\, k+1 = w_i \, w_{i+1}$ for some $1\leqslant i < \ell$, then 
	$\mathbf{w} \leftsquigarrow k = k+1  \leftsquigarrow\mathbf{w}$. 
	\item Else let $w_i$ be the leftmost letter in $\mathbf{w}$ such that $w_i>k$. Then
	$\mathbf{w} \leftsquigarrow k = w_i \leftsquigarrow \mathbf{w'}$, where 
	$\mathbf{w'} = w_1 \ldots w_{i-1}\ k\ w_{i+1} \ldots w_\ell$.
\end{enumerate}
In the cases above, when $\mathbf{w} \leftsquigarrow k = k'  \leftsquigarrow\mathbf{w'}$,
the symbol $k'  \leftsquigarrow\mathbf{w'}$ indicates a word $\mathbf{w'}$ together with a ``bumped''
letter $k'$.

Next we consider a reduced unimodal word $\mathbf{a} = a_1 a_2 \ldots a_\ell$ with 
$a_1 > a_2 >\cdots > a_v < a_{v+1} < \cdots < a_\ell$. 
The \defn{Kra\'skiewicz row insertion} \cite{Kraskiewicz.1989,Kraskiewicz.1995} is defined for a unimodal word 
$\mathbf{a}$ and a letter $k$ such that the concatenation $a_1 a_2 \ldots a_\ell k$ is a $C$-type reduced word. 
The Kra\'skiewicz row insertion of $k$ into $\mathbf{a}$ (denoted similarly as $\mathbf{a} \leftsquigarrow k$), is
performed as follows:
\begin{enumerate}
	\item If $k=0$ and there is a subword $101$ in $\mathbf{a}$, then $\mathbf{a} \leftsquigarrow 0 = 
	0 \leftsquigarrow \mathbf{a}$.
	\item If $k \neq 0$ or there is no subword $101$ in $\mathbf{a}$, denote the decreasing part $a_1 \ldots a_v$ 
	as $\mathbf{d}$ and the increasing part $a_{v+1} \ldots a_\ell$ as $\mathbf{g}$. Perform the Edelman-Greene 
	insertion of $k$ into $\mathbf{g}$.
	\begin{enumerate}
		\item If $a_\ell < k$, then $\mathbf{g} \leftsquigarrow k = a_{v+1} \ldots a_\ell k =: \mathbf{g'}$ and
		$\mathbf{a} \leftsquigarrow k = \mathbf{d} \mathbf{g} \leftsquigarrow k = \mathbf{d\ g'} =: \mathbf{a'}$.
		\item If there is a bumped letter and $\mathbf{g} \leftsquigarrow k = k' \leftsquigarrow \mathbf{g'}$, negate all 
		letters in $\mathbf{d}$ (call the resulting word $-\mathbf{d}$) and perform the Edelman-Greene insertion 
		$-\mathbf{d} \leftsquigarrow -k'$. Note that there will always be a bumped letter, and so 
		$-\mathbf{d} \leftsquigarrow -k' = -k'' \leftsquigarrow -\mathbf{d'}$ for some decreasing word $\mathbf{d'}$. 
		The result of the Kra\'skiewicz insertion is: $\mathbf{a} \leftsquigarrow k = \mathbf{d}[\mathbf{g} \leftsquigarrow k] 
		= \mathbf{d}[k' \leftsquigarrow \mathbf{g'}] =  - [\mathbf{-d} \leftsquigarrow -k']\ \mathbf{g'} =
		[k'' \leftsquigarrow \mathbf{d'}]\mathbf{g'} = k'' \leftsquigarrow \mathbf{a'}$, where $\mathbf{a'} := \mathbf{d'g'}$.
	\end{enumerate}
\end{enumerate}

\begin{example}
$31012 \leftsquigarrow 0 =0 \leftsquigarrow 31012,
31012 \leftsquigarrow 1 = 1 \leftsquigarrow 32012.$
\end{example}

The insertion is constructed to ``commute'' a unimodal word with a letter: If 
$\mathbf{a} \leftsquigarrow k = k' \leftsquigarrow \mathbf{a'}$, the two elements of the type $C$ Coxeter group 
corresponding to the words $\mathbf{a}\ k$ and $k' \mathbf{a'}$ are the same.

The type $C$ Stanley symmetric functions~\eqref{equation.StanleyC} are defined in terms of unimodal factorizations.
To put the formula on a completely combinatorial footing, we need to treat the powers of $2$ by introducing signed
unimodal factorizations. A \defn{signed unimodal factorization} of $w\in W_C$ is a unimodal factorization 
$\mathbf{A}$ of $w$, in which every non-empty factor is assigned either a $+$ or $-$ sign. Denote the set of all 
signed unimodal factorizations of $w$ by $U^{\pm} (w)$.

For a signed unimodal factorization $\mathbf{A} \in U^{\pm} (w)$, define $\wt(\mathbf{A})$ to be the vector with $i$-th 
coordinate equal to the number of letters in the $i$-th factor of $\mathbf{A}$. Notice from~\eqref{equation.StanleyC}
\begin{equation}
\label{equation.Upm}
	F^C_{w}(\mathbf{x}) = \sum_{\mathbf{A} \in U^{\pm}(w)} \mathbf{x}^{\wt(\mathbf{A})}.
\end{equation}

We will use the Kra\'skiewicz insertion to construct a map between signed unimodal factorizations of a Coxeter group 
element $w$ and pairs of certain types of tableaux $(\mathbf{P},\mathbf{T})$. We define these types of tableaux next.

A \defn{shifted diagram} $\mathcal{S}(\lambda)$ associated to a partition $\lambda$ with distinct parts is the set of 
boxes in positions $\{(i,j) \mid \ 1\leqslant i\leqslant \ell(\lambda), \ i\leqslant j\leqslant \lambda_i+i-1\}$. Here, we use 
English notation, where the box $(1,1)$ is always top-left.

Let $X^\circ_n$ be an ordered alphabet of $n$ letters $X^\circ_n = \{0< 1 < 2< \cdots < n-1\}$, and let $X'_n$ be an 
ordered alphabet of $n$ letters together with their primed counterparts as $X'_n = \{1' < 1 < 2'< 2< \cdots <n' < n\}$.

Let $\lambda$ be a partition with distinct parts.
A \defn{unimodal tableau} $\mathbf{P}$ of shape $\lambda$ on $n$ letters is a filling of $\mathcal{S}(\lambda)$ with 
letters from the alphabet $X^\circ_n$ such that the word $P_i$ obtained by reading the $i$th row from the top
of $\mathbf{P}$ from left to right, is a unimodal word, and $P_i$ is the longest unimodal subword in the concatenated 
word $P_{i+1} P_i$ \cite{Billey.2014} (cf. also with decomposition tableaux~\cite{Serrano.2010,Cho.2013}). 
The \defn{reading word} of a unimodal tableau $\mathbf{P}$ is given by $\pi_{\mathbf{P}} = P_\ell P_{\ell-1} \ldots P_1$. 
A unimodal tableau is called \textit{reduced} if $\pi_{\mathbf{P}}$ is a type $C$ reduced word corresponding to the Coxeter 
group element $w_{\mathbf{P}}$. Given a fixed Coxeter group element $w$, denote the set of reduced unimodal tableaux 
$\mathbf{P}$ of shape $\lambda$ with $w_{\mathbf{P}} = w$ as $\mathcal{UT}_w (\lambda)$.

A \defn{signed primed tableau} $\mathbf{T}$ of shape $\lambda$ on $n$ letters 
(cf. semistandard $Q$-tableau~\cite{Lam.1995}) is a filling of $\mathcal{S}(\lambda)$ 
with letters from the alphabet $X'_n$ such that:
\begin{enumerate}
	\item The entries are weakly increasing along each column and each row of $\mathbf{T}$.
	\item Each row contains at most one $i'$ for every $i = 1,\ldots,n$.
	\item Each column contains at most one $i$ for every $i = 1,\ldots,n$.
\end{enumerate}
The reason for using the word ``signed'' in the name is to distinguish the set of primed tableaux above from the 
``unsigned'' version described later in the chapter.

Denote the set of signed primed tableaux of shape $\lambda$ by $\mathcal{PT^{\pm}} (\lambda)$. Given an element 
$\mathbf{T} \in \mathcal{PT^{\pm}} (\lambda)$, define the weight of the tableau $\wt(\mathbf{T})$ as the vector with
$i$-th coordinate equal to the total number of letters in $\mathbf{T}$ that are either $i$ or $i'$.

\begin{example}
\scalebox{0.7}{$\Bigg(\young(43201,:212,::0),\ \young(11\twop\threep3,:\twop2\threep,::4)\Bigg)$} is a pair consisting 
of a unimodal tableau and a signed primed tableau both of shape $(5,3,1)$.
\end{example}

For a reduced unimodal tableau $\mathbf{P}$ with rows $P_\ell, P_{\ell-1}, \ldots, P_1$, the Kra\'skiewicz insertion of 
a letter $k$ into tableau $\mathbf{P}$ (denoted again by $\mathbf{P} \leftsquigarrow k$) is performed as follows:
\begin{enumerate}
	\item Perform Kra\'skiewicz insertion of the letter $k$ into the unimodal word $P_1$. If there is no bumped letter and 
	$P_1 \leftsquigarrow k = P'_1$, the algorithm terminates and the new tableau $\mathbf{P'}$ consists of rows 
	$P_\ell, P_{\ell-1}, \ldots, P_2, P'_1$. If there is a bumped letter and $P_1 \leftsquigarrow k  = k' \leftsquigarrow P'_1$,
	continue the algorithm by inserting $k'$ into the unimodal word $P_2$.
	\item Repeat the previous step for the rows of $\mathbf{P}$ until either the algorithm terminates, in which case the new 
	tableau $\mathbf{P}'$ consists of rows $P_\ell, \ldots, P_{s+1}, P'_s, \ldots, P'_1$, or, 
        the insertion continues until we bump a letter $k_e$ from $P_\ell$, in which case we then put $k_e$ on a new row of 
	the shifted shape of $\mathbf{P'}$, so that the resulting tableau $\mathbf{P'}$ consists of rows 
	$k_e, P'_\ell, \ldots, P'_1$. 
\end{enumerate}

\begin{example}
\scalebox{0.7}{$\young(43201,:212,::0) \leftsquigarrow 0 = \young(43210,:210,::01)$.}
\end{example}

\begin{lemma} \cite{Kraskiewicz.1989}
Let $\mathbf{P}$ be a reduced unimodal tableau with reading word $\pi_\mathbf{P}$ for an element $w\in W_C$.
Let $k$ be a letter such that $\pi_\mathbf{P}k$ is a reduced word. Then the tableau 
$\mathbf{P'} = \mathbf{P} \leftsquigarrow k$ is a reduced unimodal tableau, for which the reading word $\pi_{\mathbf{P'}}$ 
is a reduced word for $w s_k$.
\end{lemma}

\begin{lemma} \cite[Lemma 3.17]{Lam.1995}
 \label{lemma.ins}
Let $\mathbf{P}$ be a unimodal tableau, and $\mathbf{a}$ a unimodal word such that $\pi_{\mathbf{P}}\mathbf{a}$ is reduced. 
Let $(x_1,y_1), \ldots, (x_r, y_r)$ be the (ordered) list of boxes added when  $\mathbf{P} \leftsquigarrow {\mathbf{a}}$ 
is computed. Then there exists an index $v$, such that $x_1 < \cdots < x_v \geqslant \cdots \geqslant x_r $ and 
$y_1 \geqslant \cdots \geqslant y_v < \cdots < y_r$.
\end{lemma}

Let $\mathbf{A} \in U^{\pm} (w)$ be a signed unimodal factorization with unimodal factors 
$\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n$. We recursively construct a sequence 
$(\emptyset, \emptyset) = (\mathbf{P}_0, \mathbf{T}_0),\ (\mathbf{P}_1, \mathbf{T}_1), \ldots, (\mathbf{P}_n, \mathbf{T}_n) 
= (\mathbf{P}, \mathbf{T})$ of tableaux, where 
$\mathbf{P}_s \in \mathcal{UT}_{(\mathbf{a}_1 \mathbf{a}_2 \ldots \mathbf{a}_s)} (\lambda^{(s)})$ and $\mathbf{T}_s 
\in \mathcal{PT}^{\pm} (\lambda^{(s)})$ are tableaux of the same shifted shape $\lambda^{(s)}$.

To obtain the \defn{insertion tableau} $\mathbf{P}_s$, insert the letters of $\mathbf{a}_s$ one by one from left to right, into 
$\mathbf{P}_{s-1}$. Denote the shifted shape of $\mathbf{P}_{s}$ by $\lambda^{(s)}$. Enumerate the boxes in 
the skew shape $\lambda^{(s)} / \lambda^{(s-1)}$ in the order they appear in $\mathbf{P}_s$. Let these boxes be 
$(x_1,y_1), \ldots, (x_{\ell_s}, y_{\ell_s})$.

\begingroup\emergencystretch=1em
Let $v$ be the index that is guaranteed to exist by \cref{lemma.ins} when we compute 
$\mathbf{P}_{s-1} \leftsquigarrow {\mathbf{a_s}}$. The \defn{recording tableau} $\mathbf{T}_{s}$ is a primed 
tableau obtained from $\mathbf{T}_{s-1}$ by adding the boxes $(x_1, y_1), \ldots,$ $(x_{v-1}, y_{v-1})$, each filled with 
the letter $s'$, and the boxes $(x_{v+1},y_{v+1}), \ldots,$ $(x_{\ell_s},  y_{\ell_s})$, each filled with the letter $s$. 
The special case is the box $(x_v,y_v)$, which could contain either $s'$ or $s$. 
The letter is determined by the sign of the factor $\mathbf{a}_s$: If the sign is $-$, the box is filled with the letter $s'$, and if 
the sign is $+$, the box is filled with the letter $s$.
We call the resulting map the \defn{primed Kra\'skiewicz map} $\mathrm{KR}'$.
\endgroup

\begin{example}
Given a signed unimodal factorization $\mathbf{A} = (-0) (+212) (-43201)$, the sequence of tableaux is
\vspace{-1ex}
\[{\footnotesize(\emptyset,\emptyset), \quad (\ \young(0),\young(\onep)\ ), \quad \Big( \ \young(212,:0), 
\young(\onep\twop2,:2)\ \Big), \quad \Bigg(\ \young(43201,:212,::0),\young(\onep\twop2\threep3,:2\threep3,::\threep)\ \Bigg)}\]
\end{example}

If the recording tableau is constructed, instead, by simply labeling its boxes with $1,2,3,\ldots$ in the order these boxes 
appear in the insertion tableau, we recover the original Kra\'skiewicz map  \cite{Kraskiewicz.1989,Kraskiewicz.1995}, 
which is a bijection $\mathrm{KR}\colon R(w) \rightarrow \bigcup_{\lambda} \big[\mathcal{UT}_w (\lambda) \times 
\mathcal{ST} (\lambda)\big]$, where $\mathcal{ST}(\lambda)$ is the set of \defn{standard shifted tableau} of shape 
$\lambda$, i.e., the set of fillings of $\mathcal{S} (\lambda)$ with letters
$1,2, \ldots,|\lambda|$ such that each letter appears exactly once, each row filling is increasing, and each 
column filling is increasing. 

\begin{theorem}
\label{theorem.KR}
The primed Kra\'skiewicz map is a bijection
$\mathrm{KR}'\colon U^{\pm}(w) \rightarrow 
\bigcup_{\lambda} \big[\mathcal{UT}_w (\lambda) \times \mathcal{PT}^{\pm} (\lambda)\big]$.
\end{theorem}

\cref{theorem.KR} and~\eqref{equation.Upm} imply the following relation:
\begin{equation}
\label{equation.PTpm}
	F^C_{w}(\mathbf{x}) = \sum_{\lambda} \big|\mathcal{UT}_w (\lambda) \big| \sum_{\mathbf{T} \in 
	\mathcal{PT}^{\pm}(\lambda)} \mathbf{x}^{\wt(\mathbf{T})}.
\end{equation}

\begin{remark}
The sum $\sum_{\mathbf{T} \in \mathcal{PT}^{\pm}(\lambda)} \mathbf{x}^{\wt(\mathbf{T})}$ is also known as the
$Q$-Schur function. The expansion~\eqref{equation.PTpm} was shown in~\cite{Billey.Haiman.1995}.
\end{remark}

At this point, we are halfway there to expand $F^C_{w}(\mathbf{x})$ in terms of Schur functions.
In the next section we introduce a crystal structure on the set $\mathcal{PT} (\lambda)$ of unsigned primed tableaux.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mixed insertion}
\label{section.implicit}

Set $\mathcal{B}^h = \mathcal{B}^h_{\infty}$. Similar to the well-known RSK-algorithm, mixed insertion~\cite{Haiman.1989} 
gives a bijection between $\mathcal{B}^h$ and the set of pairs of tableaux $(\mathbf{T}, \mathbf{Q})$, but in this case 
$\mathbf{T}$ is an (unsigned) primed tableau of shape $\lambda$ and $\mathbf{Q}$ is a standard shifted tableau of 
the same shape. 

An \defn{(unsigned) primed tableau} of shape $\lambda$ (cf. semistandard $P$-tableau~\cite{Lam.1995} or
semistandard marked shifted tableau~\cite{Cho.2013}) is a signed primed tableau $\mathbf{T}$ of shape $\lambda$ with 
only unprimed elements on the main diagonal. Denote the set of primed tableaux of shape $\lambda$ by 
$\mathcal{PT}(\lambda)$. The weight function  $\wt(\mathbf{T})$ of $\mathbf{T} \in \mathcal{PT}(\lambda)$ is inherited 
from the weight function of signed primed tableaux, that is, it is the vector with $i$-th coordinate equal to the number of 
letters $i'$ and $i$ in $\mathbf{T}$. We can simplify~\eqref{equation.PTpm} as
\begin{equation}
\label{equation.PT}
	F^C_{w}(\mathbf{x}) = \sum_{\lambda} 2^{\ell(\lambda)} \big|\mathcal{UT}_w (\lambda) \big|  
	\sum_{\mathbf{T} \in \mathcal{PT}(\lambda)} \mathbf{x}^{\wt(\mathbf{T})}.
\end{equation}

\begin{remark} The sum $\sum_{\mathbf{T} \in \mathcal{PT}(\lambda)} \mathbf{x}^{\wt(\mathbf{T})}$ is also known as a 
$P$-Schur function.
\end{remark}

Given any word $b_1 b_2 \ldots b_h$ in the alphabet $X = \{1<2<3<\cdots\}$, we recursively construct a sequence of tableaux 
$(\emptyset, \emptyset) = (\mathbf{T}_0, \mathbf{Q}_0),$ $(\mathbf{T}_1, \mathbf{Q}_1), \ldots, (\mathbf{T}_h, \mathbf{Q}_h) 
= (\mathbf{T}, \mathbf{Q})$, where $\mathbf{T}_s \in \mathcal{PT}(\lambda^{(s)})$ and $\mathbf{Q}_s \in 
\mathcal{ST}(\lambda^{(s)})$. 
To obtain the tableau $\mathbf{T}_{s}$, insert the letter $b_s$ into $\mathbf{T}_{s-1}$ as follows. 
First, insert $b_s$ into the first row of $\mathbf{T}_{s-1}$, bumping out the leftmost element $y$ that is strictly greater 
than $b_i$ in the alphabet $X' = \{1' < 1 < 2' < 2< \cdots \}$.
\begin{enumerate}
	\item If $y$ is not on the main diagonal and $y$ is not primed, then insert it into the next row, bumping out the leftmost 
    	element that is strictly greater than $y$ from that row.
	\item If $y$ is not on the main diagonal and $y$ is primed, then insert it into the next column to the right, bumping out 
	the topmost element that is strictly greater than $y$ from that column.
	\item If $y$ is on the main diagonal, then it must be unprimed. Prime $y$ and insert it into the column on the right, 
	bumping out the topmost element that is strictly greater than $y$ from that column.
\end{enumerate}
If a bumped element exists, treat it as a new $y$ and repeat the steps above -- if the new $y$ is unprimed, row-insert 
it into the row below its original cell, and if the new $y$ is primed, column-insert it into the column to the right of its original cell.

The insertion process terminates either by placing a letter at the end of a row, bumping no new element, or forming a 
new row with the last bumped element.
The shapes of $\mathbf{T}_{s-1}$ and $\mathbf{T}_s$ differ by one box. Add that box to $\mathbf{Q}_{s-1}$ with a letter 
$s$ in it, to obtain the standard shifted tableau $\mathbf{Q}_s$.

\begin{example}
For a word $332332123$, some of the tableaux in the sequence $(\mathbf{T}_i, \mathbf{Q}_i)$ are\newline
\scalebox{0.7}{$\qquad \qquad \Big(\ \young(2\threep,:3),\young(12,:3) \ \Big), \quad \Big(\ \young(22\threep3,:33),
\young(1245,:36) \ \Big), \quad \Bigg(\ \young(1\twop2\threep3,:2\threep3,::3),\young(12459,:368,::7) \ \Bigg)$}.
\end{example}

\begin{theorem} \cite{Haiman.1989}
The construction above gives a bijection
$\mathrm{HM} \colon \mathcal{B}^h \rightarrow \bigcup_{\lambda\vdash h} \big[ \mathcal{PT}(\lambda) \times 
\mathcal{ST}(\lambda) \big].$
\end{theorem}

The bijection $\mathrm{HM}$ is called a \defn{mixed insertion}. If $\mathrm{HM}(\mathbf{b}) = (\mathbf{T},\mathbf{Q})$, 
denote $P_{\mathrm{HM}} (\mathbf{b}) = \mathbf{T}$ and $R_{\mathrm{HM}}(\mathbf{b}) = \mathbf{Q}$.
Just as for the RSK-algorithm, the mixed insertion has the property of preserving the recording tableau within each 
connected component of the crystal $\mathcal{B}^h$.

\begin{theorem}
\label{theorem.main0}
The recording tableau $R_{\mathrm{HM}} (\cdot)$ is constant on each connected component of the crystal 
$\mathcal{B}^h$.
\end{theorem}

Let us fix a recording tableau $\mathbf{Q}_{\lambda} \in \mathcal{ST} (\lambda)$. Define a map 
$\Psi_\lambda \colon \mathcal{PT}(\lambda) \rightarrow \mathcal{B}^{h}$ as $\Psi_\lambda (\mathbf{T}) = 
\mathrm{HM}^{-1} (\mathbf{T}, \mathbf{Q}_\lambda)$. By \cref{theorem.main0}, the set  
$\mathrm{Im}(\Psi_{\lambda})$ consists of several connected components of $\mathcal{B}^h$. 
The map $\Psi_{\lambda}$ can thus be taken as a crystal isomorphism, and we can define the crystal operators 
and weight function on $\mathcal{PT}(\lambda)$ as
\begin{equation}
\label{equation.ef}
	e_i(\mathbf{T}) := (\Psi_\lambda^{-1} \circ e_i \circ \Psi_\lambda) (\mathbf{T}), \quad f_i(\mathbf{T}) 
	:= (\Psi_\lambda^{-1} \circ f_i \circ \Psi_\lambda) (\mathbf{T}), \quad \wt(\mathbf{T}) := (\wt \circ \Psi_\lambda) (\mathbf{T}).
\end{equation}

Although it is not clear that the crystal operators constructed above are independent of the choice of $\mathbf{Q}_\lambda$, 
in the next section we will construct explicit crystal operators on the set $\mathcal{PT}(\lambda)$ that satisfy the relations 
above and do not depend on the choice of $\mathbf{Q}_\lambda$.

\begin{example}
For $\mathbf{T} =$ \scalebox{0.6}{$\young(1\twop2\threep3,:2\threep3,::3)$}, choose $\mathbf{Q}_{\lambda} = $
\scalebox{0.6}{$\young(12345,:678,::9)$}.
Then $\Psi_\lambda (\mathbf{T}) = 333332221$ and $e_1 \circ \Psi_\lambda (\mathbf{T}) = 333331221$. Thus,
$e_1 (\mathbf{T}) = (\Psi_\lambda^{-1} \circ e_1 \circ \Psi_\lambda) (\mathbf{T})  =$
\scalebox{0.6}{$\young(112\threep3,:2\threep3,::3)$}, $f_1(\mathbf{T}) = f_2(\mathbf{T}) = \mathbf{0}.$
\end{example}

To summarize, we obtain a crystal isomorphism between the crystal $(\mathcal{PT}(\lambda), e_i, f_i, \wt)$, denoted again 
by $\mathcal{PT}(\lambda)$, and a direct sum $\bigoplus_\mu \mathcal{B}_{\mu}^{\oplus h_{\lambda\mu}}$. We will provide a
combinatorial description of the coefficients $h_{\lambda\mu}$ in the next section. This implies the relation on characters 
of the corresponding crystals $\chi_{\mathcal{PT}(\lambda)} = \sum_\mu h_{\lambda\mu} s_\mu$. Thus we can
rewrite~\eqref{equation.PT} one last time
\begin{equation*}
	F^C_{w}(\mathbf{x}) = \sum_{\lambda} 2^{\ell(\lambda)} \big|\mathcal{UT}_w (\lambda) \big|  \sum_{\mu} 
	h_{\lambda\mu} s_\mu = \sum_\mu \Big( \sum_\lambda 2^{\ell(\lambda)} \big|\mathcal{UT}_w (\lambda) 
	\big|\ h_{\lambda\mu} \Big) s_\mu.
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Explicit crystal operators on shifted primed tableaux}
\label{section.explicit}

We consider the alphabet $X'=\{1' < 1 < 2' < 2 < 3' < \cdots\}$ of primed and unprimed letters. It is useful to think about the 
letter $(i+1)'$ as a number $i + 0.5$. Thus, we say that letters $i$ and $(i+1)'$ differ by half a unit and letters $i$ and 
$(i+1)$ differ by a whole unit.

Given an (unsigned) primed tableau $\mathbf{T}$, the \defn{reading word}  $\mathrm{rw}(\mathbf{T})$ is constructed as:
\begin{enumerate}
\item List all primed letters in the tableau, column by column, in decreasing order within each column, moving from the 
rightmost column to the left, and with all the primes removed (i.e. all letters are increased by half a unit).
(Call this part of the word the \defn{primed reading word}.)
\item Then list all unprimed elements, row by row, in increasing order within each row, moving from the bottommost 
row to the top. (Call this part of the word the \defn{unprimed reading word}.)
\end{enumerate}

To find the letter on which the crystal operator $f_i$ acts, apply the bracketing rule for letters $i$ and $i+1$ within the 
reading word $\mathrm{rw}(\mathbf{T})$. 
If all letters $i$ are bracketed in $\mathrm{rw}(\mathbf{T})$, then $f_i(\mathbf{T}) = \mathbf{0}$. 
Otherwise, the rightmost unbracketed letter $i$ in $\mathrm{rw}(\mathbf{T})$ corresponds to an $i$ or an $i'$ in 
$\mathbf{T}$, which we call \defn{bold unprimed} $i$ or \defn{bold primed} $i$ respectively.
If the bold letter $i$ is unprimed, denote the cell it is located in as $x$.
If the bold letter $i$ is primed, we \textit{conjugate} the tableau $\mathbf{T}$ first.

The \defn{conjugate} of a primed tableau $\mathbf{T}$ is obtained by reflecting the tableau over the main diagonal, 
changing all primed entries $k'$ to $k$ and changing all unprimed elements $k$ to $(k+1)'$ (i.e. increase the content of all 
boxes by half a unit). The main diagonal is now the North-East boundary of the tableau. Denote the resulting tableau as 
$\mathbf{T}^*$.

Under the transformation $\mathbf{T} \to \mathbf{T}^*$, the bold primed $i$ is transformed into bold unprimed $i$.
Denote the cell it is located in as $x$.

Given any cell $z$ in a shifted primed tableau $\mathbf{T}$ (or conjugated tableau $\mathbf{T}^*$), denote by 
$c(z)$ the content of $z$.
Denote by $z_E$ the cell to the right of $z$, $z_W$ the cell to its left, $z_S$ the cell below, and $z_N$ the cell above.
Denote by $z^*$ the corresponding conjugated cell in $\mathbf{T}^*$ (or in $\mathbf{T}$).
Now, consider the box $x_E$ (in $\mathbf{T}$ or in $\mathbf{T}^*$) and notice that $c(x_E) \geqslant (i+1)'$.\\

\noindent
\textbf{Crystal operator $f_i$ on primed tableaux:}
\begin{enumerate}
	\item If $c(x_E) =  (i+1)'$, the box $x$ must lie outside of the main diagonal and the box right below $x_E$ cannot
	have content equal to $(i+1)'$. Change $c(x)$ to $(i+1)'$ and
	change $c(x_E)$ to $(i+1)$ (i.e. increase the content of $x$ and $x_E$ by half a unit). 
	\item If $c(x_E) \neq (i+1)'$ or $x_E$ is empty, then there is a 
	maximal connected ribbon (expanding in South and West directions) with the following properties:
	\begin{enumerate}
		\item The North-Eastern most box of the ribbon (the tail of the ribbon) is $x$.
		\item The contents of all boxes within a ribbon besides the tail are either $(i+1)'$ or $(i+1)$.
	\end{enumerate}
	Denote the South-Western most box of the ribbon (the head) as $x_H$. 
	\begin{enumerate}
		\item If $x_H = x$, change $c(x)$ to $(i+1)$ (i.e. increase the 
		content of $x$ by a whole unit).
		\item If $x_H \neq x$ and $x_H$ is on the main diagonal (in case of a tableau $\mathbf{T}$), change $c(x)$ 
		to $(i+1)'$ (i.e. 
		increase the content of $x$ by half a unit).
		\item Otherwise, the content $c(x_H)$ must be $(i+1)'$ due to the bracketing rule. We change $c(x)$ to $(i+1)'$ 
		and change $c(x_H)$ to $(i+1)$ (i.e. increase the content of $x$ and $x_H$ by half a unit).\\
	\end{enumerate}
\end{enumerate}

\vspace{-0.7cm}

In the case when the bold $i$ in $\mathbf{T}$ is unprimed, we apply the above crystal operator rules to $\mathbf{T}$
to find $f_i(\mathbf{T})$

\begin{example}
In the following examples, we mark the bold $i$ (if it exists):\newline
\scalebox{0.7}{$
		\qquad\qquad  f_2 \Big(\ \young(1\twop2\threep,:2\threep3)\ \Big) =  \mathbf{0},
		\quad 
		\ f_2 \Big(\ \young(1\twop\boldtwo\threep,:2\threep4)\ \Big) = \ \young(1\twop\threep3,:2\threep4)\ ,
		\quad
		\ f_2 \Big(\ \young(112\boldtwo,:3\fourp4)\ \Big) = \ \young(1123,:3\fourp4)\ ,$}
		\newline \mbox{} \newline
\scalebox{0.7}{$
		\qquad \qquad f_2 \Bigg(\ \young(11\twop\boldtwo3,:22\threep,::33)\ \Bigg) = \ 
		\young(11\twop\threep3,:22\threep,::33)\ ,	
		\quad
		\ f_2 \Bigg(\ \young(111\boldtwo3,:22\threep,::3\fourp)\ \Bigg) = \ \young(111\threep3,:223,::3\fourp)\ .$}
\end{example}

In the case when the bold $i$ is primed in $\mathbf{T}$, we first conjugate $\mathbf{T}$ and then apply the above crystal 
operator rules on $\mathbf{T}^*$, before reversing the conjugation. 
Note that Case 2b is impossible for $\mathbf{T}^*$, since the main diagonal is now on the North-East.

\begin{example}
Let $	\mathbf{T} =$ \scalebox{0.7}{$\young(1\boldtwop23,:3\fourp,::4)$},
then $\mathbf{T}^* =$ \scalebox{0.7}{$\young(\twop,\boldtwo\fourp,\threep4\fivep,\fourp)$}
and $f_2 (\mathbf{T}) =$ \scalebox{0.7}{$\young(12\threep3,:3\fourp,::4)$}\ .
\end{example}

\begin{theorem}
\label{theorem.main2}
	 For any $\mathbf{b} \in \mathcal{B}^h$ with $P_{\mathrm{HM}}(\mathbf{b}) = \mathbf{T}$ and 
	 $f_i(\mathbf{b})\neq \mathbf{0}$, the operator $f_i$ defined on  above satisfies
	\begin{equation*}
	P_{\mathrm{HM}}(f_i(\mathbf{b})) = f_i(\mathbf{T}).
	\end{equation*}
	Also, $f_i(\mathbf{b}) = \mathbf{0}$ if and only if $f_i(\mathbf{T})=\mathbf{0}$.
\end{theorem}

The crystal operators $e_i(\mathbf{T})$ are defined similarly. 
Consider the reading word $\mathrm{rw}(\mathbf{T})$ and apply the bracketing rule on the letters $i$ and $i+1$.
If all letters $i+1$ are bracketed in $\mathrm{rw}(\mathbf{T})$, then $e_i(\mathbf{T}) = \mathbf{0}$. Otherwise, the action
of $e_i$ on $\mathbf{T}$ can be obtained from the action of $f_i$ on $-\mathbf{T}$. For more details we refer to the long 
version of the paper.

\begin{theorem}
\label{theorem.main3}
	Given a primed tableau $\mathbf{T}$ with $f_i(\mathbf{T}) \neq \mathbf{0}$, the operators $e_i$ satisfy
	$e_i(f_i(\mathbf{T})) = \mathbf{T}.$
\end{theorem}

\begin{corollary}
\label{theorem.main4}
	 For any $\mathbf{b} \in \mathcal{B}^h$ with $\mathrm{HM}(\mathbf{b}) = (\mathbf{T},\mathbf{Q})$, the operator 
	 $e_i$ defined above satisfies $\mathrm{HM}(e_i(\mathbf{b})) = (e_i(\mathbf{T}), \mathbf{Q})$,
	given the left-hand side is well-defined.
\end{corollary}

The consequence of \cref{theorem.main2}, as discussed in \cref{section.implicit}, is a crystal 
isomorphism $\Psi_\lambda \colon \mathcal{PT}(\lambda) \rightarrow \bigoplus \mathcal{B}_{\mu}^{\oplus h_{\lambda\mu}}$. 
Now, to determine the nonnegative integer coefficients $h_{\lambda\mu}$, it is enough to count the highest weight elements
in $\mathcal{PT}(\lambda)$ of given weight $\mu$. 

\begin{proposition}
	A primed tableau $\mathbf{T} \in \mathcal{PT}(\lambda)$ is a highest weight element if and only if its reading word 	
	$\mathrm{rw}(\mathbf{T})$ is a Yamanouchi word. That is, for any suffix of $\mathrm{rw}(\mathbf{T})$, its weight 
	is a partition.
\end{proposition}

Thus we define $h_{\lambda\mu}$ to be the number of primed tableaux $\mathbf{T}$ of shifted shape $\mathcal{S}(\lambda)$ 
and weight $\mu$ such that $\mathrm{rw}(\mathbf{T})$ is Yamanouchi.

\begin{example}
	Let $\lambda = (5,3,2)$ and $\mu = (4,3,2,1)$. There are three primed tableaux of shifted shape 
	$\mathcal{S}((5,3,2))$ and weight $(4,3,2,1)$ with a Yamanouchi reading word, namely
	\scalebox{0.7}{$\young(1111\twop,:22\threep,::3\fourp)$}, \scalebox{0.7}{$\young(1111\threep,:222,::3\fourp)$} and
	\scalebox{0.7}{$\young(1111\fourp,:222,::33)$}.
	Therefore $h_{(5,3,2)(4,3,2,1)} = 3$.
\end{example}

We summarize our results for the type $C$ Stanley symmetric functions as follows.
\begin{corollary}
\label{corollary.main2}
	The expansion of $F^C_w(\mathbf{x})$ in terms of Schur symmetric functions is
	\begin{equation}
	\label{equation.FC}
	F^C_w(\mathbf{x}) = \sum_\lambda g_{w\lambda} s_\lambda (\mathbf{x}), \quad
	\text{where} \quad
	g_{w\lambda} = \sum_\mu 2^{\ell(\mu)} \big|\mathcal{UT}_w (\mu) \big| \ h_{\mu\lambda}\ .
	\end{equation}
\end{corollary}

\begin{example}\label{exa}
Consider the word $w=0101=1010$. There is only one unimodal tableau corresponding to $w$, namely 
$\mathbf{P} =$ \scalebox{0.7}{$\young(101,:0)$}, which belongs to $\mathcal{UT}_{0101} (3,1)$. Thus, 
$g_{w\lambda} = 4h_{(3,1)\lambda}$. There are only three possible highest weight primed tableaux of shape $(3,1)$, 
namely \scalebox{0.7}{$\young(111,:2),\ \young(11\twop,:2)$}
and \scalebox{0.7}{$\young(11\threep,:2)$}, which implies that $h_{(3,1)(3,1)}= h_{(3,1)(2,2)} = h_{(3,1)(2,1,1)} = 1$ and 
$h_{(3,1)\lambda} = 0$ for other weights $\lambda$. The expansion of $F^C_{0101}(\mathbf{x})$ is thus
\begin{equation*}
	F^C_{0101} = 4s_{(3,1)} + 4s_{(2,2)} + 4s_{(2,1,1)}.
\end{equation*}
\end{example}

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\end{document}
