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\title{Symplectic keys and Demazure atoms in type $C$}

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\author{Jo\~{a}o Miguel Santos
	\thanks{\href{mailto:jmsantos@mat.uc.pt}{jmsantos@mat.uc.pt}.}\addressmark{1}}


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\address{\addressmark{1}CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal}

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\abstract{We compute, mimicking the Lascoux--Sch\"utzenberger type $A$ combinatorial  procedure, left and right keys for a Kashiwara--Nakashima tableau in type $C$. These symplectic keys have a role similar to the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type $C$. The right and the left symplectic keys are related through the Lusztig involution. A type $C$ Sch\"utzenberger evacuation is defined to realize that involution.}

\resume{Nous calculons, en imitant la proc\'edure combinatoire du type $A$ de Lascoux--Sch\"utzenberger, les cl\'es gauche et droite pour un tableau de Kashiwara--Nakashima du type $C$. Ces cl\'es symplectiques ont un r\^ole similaire aux cl\'es des tableaux de Young semistandard. Plus pr\'ecis\'ement, nos cl\'es symplectiques fournissent un crit\`ere de tableau pour l'ordre de Bruhat sur le groupe hyperocta\'edrique et ses classes, et d\'ecrivent les atomes et les caract\`eres de Demazure du type $C$. Les cl\'es symplectiques droite et gauche sont li\'ees par l'involution de Lusztig. Une \'evacuation de Sch\"utzenberger du type $C$ est d\'efinie pour r\'ealiser cette involution.}

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\keywords{Keys, Demazure crystal graph, Demazure characters and atoms in type C}



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\section{Introduction}

To generate the characters of a given finite dimensional irreducible representation of the symplectic Lie algebra $sp(2n, \C)$, two different types of symplectic tableaux have been proposed: the King tableaux \cite{king1976weight} and the De Concini tableaux \cite{de1979symplectic}.
We work with symplectic Kashiwara and Nakashima tableaux, 
which are a variation of  De Concini tableaux, and with its crystal structure. That crystal structure allows a plactic monoid compatible with insertion and sliding algorithms, and Robinson--Schensted type correspondence, studied by Lecouvey in terms of crystal isomorphisms \cite{lecouvey2002schensted}.

Kashiwara \cite{kashiwara1992crystal} and Littelmann \cite{ littelmann1995crystal} have shown that Demazure characters \cite{demazure1974nouvelle}, for any Weyl group, can be lifted to certain subsets of Kashiwara--Nakashima tableaux, called Demazure crystals. Demazure characters (key polynomials) are then generated over Demazure crystals. In type $C_n$, they are non symmetric Laurent polynomials, with respect to the action of the Weyl group, which can be seen as "partial" symplectic characters. 
Given a partition $\lambda$, let $v$ be in the orbit of $\lambda$ under the action of the Weyl group, the Demazure crystal, $\mathfrak{B}_v$, is a union of disjoint sets, \emph{Demazure crystal atoms}, $\widehat{\mathfrak{B}}_u$,  over an interval in the Bruhat order, on the cosets modulo the stabilizer of $\lambda$. This order, induced on the orbit of $\lambda$, gives $\mathfrak{B}_v=\displaystyle \biguplus_{\lambda\le u\le v} \widehat{\mathfrak{B}}_u$. 




In type $A_{n-1}$, Lascoux and Sch\"{u}tzenberger  identified tableaux with nested columns as key tableaux \cite{lascoux1990keys}, and defined the right key map that sends tableaux to key tableaux. Their right key map can be used to describe the type $A$ Demazure atoms $\widehat{\mathfrak{B}}_u,\,u\in \N^n$ \cite[Theorem 3.8]{lascoux1990keys}.
Azenhas, in a presentation in {\em The 69th S\'eminaire Lotharingien de Combinatoire} \cite{azenhas2012key}, identified some type $C$ Kashiwara--Nakashima tableaux as key tableaux, but does not give a construction of the right key map. Motivated by Azenhas \cite{azenhas2012key} and inspired by Lascoux and Sch\"utzenberger \cite{lascoux1990keys}, we give a construction of left and right keys of a type $C$ Kashiwara--Nakashima tableau. Our construction, based on type $C$ frank words, introduced in \cref{SecKeys}, and Sheats \emph{jeu de taquin}, allows us to prove \cref{DemKey}, a type $C$ analogue of \cite[Theorem $3.8$]{lascoux1990keys}. We also show, in \cref{SecLusztig}, that both keys are related via the Sch\"utzenberger evacuation in type $C$, or Lusztig involution, explicitly realized here using Baker-Lecouvey insertion or Sheats \emph{jeu de taquin}.
During the preparation of the paper \cite{santos2019symplectic}, Jacon and Lecouvey informed us about their paper \cite{jacon2019keys}, where, with a different approach, they find the same key map in type $C$. 
In the model of alcove paths, Lenart defined an initial key and a final key \cite{lenart2007combinatorics}, for any Lie type, related via the Lusztig involution. 

The paper is organized as follows. In \cref{SecBruhat}, we discuss the Weyl group of type $C$, $B_n$, the Bruhat order on $B_n$ and on its cosets modulo the stabilizer of $\lambda$, the Kashiwara--Nakashima tableaux and the symplectic key tableaux. Those key tableaux are used in \cref{minimalBn} to explicitly construct the minimal length coset representatives and, 
recalling some results from Proctor \cite{proctor1980interactions}, \cref{keycoset} gives a tableau criterion for the Bruhat order on $B_n$ and on those cosets. 
\cref{SecRS} briefly recalls Baker-Lecouvey insertion, the Sheats \textit{jeu de taquin} and Robinson--Schensted type $C$ correspondence, to discuss the plactic and coplactic monoids of type $C$. These monoids describe connected components and crystal isomorphic connected components of type $C$ Kashiwara crystal, for a $U_q(sp_{2n})$-module.
In \cref{SecKeys}, we extend the concept of frank word, in type $A$, to type $C$, and 
our \cref{rightkeymap} gives right and left key maps. Using the right key map, \cref{DemKey}, our main result, describes the tableaux that contribute to a Demazure crystal atom and to a Demazure crystal in type $C$. In \cref{SecLusztig}, we develop a type $C$ evacuation within the plactic monoid, an analogue of the $J$-operation discussed by Sch\"utzenberger for semistandard Young tableaux in \cite{schutzenberger1977correspondance}. \cref{key&Lusz} shows that the evacuation of the right key of a Kashiwara--Nakashima tableau is the left key of the evacuation of the same tableau. 

\textbf{Caution:} Operators, maps and group actions act on the right.

\section{Weyl group of type C, Bruhat order and symplectic key tableau}\label{SecBruhat}

Fix $n\in \N_{>0}$. Define the sets $[n]=\{1< \dots< n\}$ and $[\pm n]=\{1<\dots< n< \overline{n}< \dots< \overline{1}\}$ where $\overline{i}$ is just another way of writing $-i$.
The hyperoctahedral group is the group, $B_n$, with generators $s_i,\, 1\leq i\leq n$, subject to the relations:
$s_i^2 = 1, \, 1 \le i \le n;\; (s_is_{i+1})^3 = 1, 1 \le i \le n - 2;\;(s_{n-1}s_n)^4=1;(s_is_j)^2 = 1, \,1 \le i < j \le n,\, |i - j| > 1$. This group is a Coxeter group and we consider the (strong) Bruhat order on its elements \cite{bjorner2006combinatorics}. \cref{keycoset} gives a symplectic tableau criterion for this order in $B_n$.
The elements of $B_n$ can be seen as odd bijective maps from $[\pm n]$ to itself. 
The subgroup with the generators $s_1, \dots, s_{n-1}$ is the symmetric group $\mathfrak{S}_n$. 
The groups $\mathfrak{S}_n$ and $B_n$ are the Weyl groups for the root systems of types $A_{n-1}$ and $C_n$, respectively.
Given $\sigma\in B_n$, $\sigma=[a_1\, a_2\,\dots \,a_n]$, where $a_i=(i)\sigma$ for $i \in [n]$, is the window notation of $\sigma$. Given a vector $v\in \Z^n$,  $s_i$, with $i\in [n]$, acts on $v$, $vs_i$, swapping the $i$-th and the $(i+1)$-th entries, if $i\in [n-1]$, or changing the sign of the last entry, if $i=n$. The length of $\sigma\in B_n$, $(\sigma)\ell$, is the least number of generators of $B_n$ needed to go from $[1\,2\,\dots\,n]$, the identity map, to $\sigma$. Any expression of $\sigma$ as a product of $(\sigma)\ell$ generators of $B_n$ is called reduced. 
\subsection{Kashiwara--Nakashima tableau in type C}
We recall the symplectic tableaux used by
Kashiwara and Nakashima to label the vertices of the type $C$ crystal graphs \cite{kashiwara1991crystal}, which are a variation of De Concini tableaux \cite{de1979symplectic}.
A vector $\lambda=(\lambda_1,\dots, \lambda_n)\in\Z^n$ is a partition of $|\lambda|=\sum\limits_{i=1}^n \lambda_i$ if $\lambda_1\geq \lambda_2\geq\dots\geq\lambda_n\geq 0$.
A partition $\lambda$ is identified with its Young diagram of shape $\lambda$, an array of boxes, left justified, in which the $i$-th row, from top to bottom, has $\lambda_i$ boxes.
For example, the Young diagram of shape $\lambda=(2,2,1)$ is $\YT{0.14in}{}{
	{{},{}},
	{{},{}},
	{{}}}$.
%\end{example}
Given $\mu$ and $\nu$ two partitions with $\nu\leq \mu$ entrywise, we write $\nu\subseteq \mu$. The Young diagram of shape $\mu/\nu$ is obtained after removing the boxes of the Young diagram of $\nu$ from the Young diagram of $\mu$.
A semistandard Young skew tableau of shape $\mu/\nu$ on the completely ordered alphabet $A$ is a filling of the diagram $\mu/\nu$ with letters from $A$, such that the entries are strictly increasing in each column and weakly increasing in each row. When $|\nu|=0$ we obtain a  
semistandard Young tableau (SSYT) of shape $\mu$.
Denote by $SSYT(\mu/\nu, A)$
the set of all semistandard skew Young tableaux $T$ of shape $\mu/\nu$, with entries in $A$. When $A=[n]$, we write $SSYT(\mu/\nu, n)$.


From now on we consider tableaux on the alphabet $[\pm n]$.
A \emph{column} is a strictly increasing sequence of numbers in $[\pm n]$ and it is usually displayed vertically.
A column is said to be \emph{admissible} if the following column condition (1CC) holds for that column:

\begin{defin}[1CC]\label{1CC}
	Let $C$ be a column. The $1CC$ holds for $C$ if for all pairs $i$ and $\overline{i}$ in $C$, where $i$ is in the $a$-th row counting from the top of the column, and $\overline{i}$ in the $b$-th row counting from the bottom, we have $a+b\leq i$.
\end{defin}
If a column $C$ is admissible then $C$ has at most $n$ letters.
If not, we say that $C$ is not admissible at $z$, where $z$ is the minimal positive integer such that $z$ and $\overline{z}$ exist in $C$ and there are more than $z$ numbers in $C$ with absolute value less or equal than $z$. 
For instance, the column $\YT{0.15in}{}{
	{{1}},
	{{2}},
	{{\overline{1}}}}$ is not admissible at $1$.
We now define splittable columns:
\begin{defin}
	Let $C$ be a column and let $I = \{z_1 > \dots > z_r\}$ be the
	set of unbarred letters $z$ such that the pair $(z, \overline{z})$ occurs in $C$. The column
	$C$ can be split when there exists a set of $r$ unbarred
	letters $J = \{t_1 > \dots > t_r\} \subseteq [n]$ such that $t_1$ is the greatest letter of $[n]$ satisfying $t_1 < z_1$,  $t_1 \not\in C$, and $\overline{t_1}\not\in C$; and for $i=2, \dots, r$, $t_i$ is the greatest letter of $[n]$ satisfying $t_i < \min(t_{i-1},   z_i)$,  $t_i \not\in C$, and $\overline{t_i} \not\in C$.\end{defin}
%
A column $C$ is admissible if and only if $C$ can be split  \cite[Lemma 3.1]{sheats1999symplectic}.
If $C$ can be split then we define right column of $C$, $Cr$, and the left column of $C$, $ C\ell$. The column $Cr$ is obtained by replacing, in $C$,  $\overline{z_i}$ with $\overline{t_i}$ for each letter $z_i \in I $ and reordering, if needed; $ C\ell$ is obtained after replacing $z_i$ with $t_i$ for each letter $z_i \in I $ and reordering, if needed.
%
If $C$ is admissible then $ C\ell\leq C \leq Cr$ by entrywise comparison. If $C$ does not have symmetric entries, then $C$ is admissible and  $ C\ell= C =Cr$.
Let $T$ be a skew tableau with all of its columns admissible. The split form of a skew tableau $T$, $(T)spl$, is the skew tableau obtained after replacing each column $C$ of $T$ by the two columns $ C\ell\,Cr$. The tableau $(T)spl$ has double the amount of columns of $T$.
%	\begin{definition}
A semistandard skew tableau $T$ is a Kashiwara--Nakashima (KN) skew tableau if its split form is a semistandard skew tableau. We define $\mathcal{KN}(\mu/\nu, n)$ to be the set of all KN tableaux of shape $\mu/\nu$ in the alphabet $[\pm n]$. When $|\nu|=0$ we obtain $\mathcal{KN}(\mu, n)$. 
%	\end{definition}
When $T\in SSYT(\mu/\nu, [\pm n])$ with no symmetric entries in any of its columns, $T$ is a KN skew tableau. In particular $SSYT(\mu/\nu, n)\subseteq \mathcal{KN}(\mu/\nu, n)$.

The weight of a word $w$, $(w)\text{wt}$, on the alphabet $[\pm n]$ is the vector in $\Z^n$ where the entry $i$ is the multiplicity of the letter $i$ minus the multiplicity of the letter $\overline{i}$, for $i\in [n]$. The length of $w$ is its number of letters.
The column reading word of a KN tableau $T$, $(T)cr$, is obtained reading down columns, right to left. %the word read in $T$ in the Japanese way.
The weight of $T$ is the vector $(T)\text{wt}:=((T)cr)\text{wt}$. Let
$T=\YT{0.15in}{}{
	{{2},{2}},
	{{3},{3}},
	{{\overline{3}}}}$ and $n=3$. The split form of
$T$ is the tableau $(T)spl=\YT{0.15in}{}{
	{{1}, {2},{2},{2}},
	{{2},{3},{3},{3}},
	{{\overline{3}},{{\overline{1}}}}}$. Hence $T\in \mathcal{KN}((2,2,1),3)$.
Also $(T)cr=23\,23\overline{3}$ and $(T)\text{wt}=((T)cr)\text{wt}=(0,2,1)$.%

Given a partition $\lambda\in\Z^n$, the $B_n$-orbit of $\lambda$ is the set $\lambda B_n:=\{\lambda\sigma\mid \sigma\in B_n\}$.
\begin{defin}
	A key tableau in type $C$, on the alphabet $[\pm n]$, is a KN tableau in $\mathcal{KN}(\lambda,n)$, for some partition $\lambda$, in which the set of elements of each column is contained in the set of elements of the previous column and the letters $i$ and $\overline{i}$ do not appear simultaneously as entries, for any $i\in [n]$.
	%\begin{example}
	See the example at the end of \cref{Ssec Bruhat}.
	%\end{example}
\end{defin}
%
%The set of key tableaux in type $A$ is the subset of the key tableaux in type $C$ consisting of the tableaux having only positive entries, hence they are SSYT for the alphabet $[n]$.
%
%Every vector $v$ of  $\Z^n$ is in the $B_n$-orbit of exactly one partition.  %, $\lambda_v$, which is the partition
%the one obtained  by sorting the absolute values of all entries of $v$. 
%\begin{example}
%\end{example}
%Consider a partition $\lambda\in\Z^n$. The $B_n$-orbit of $\lambda$ is the set $\lambda B_n:=\{\lambda\sigma\mid \sigma\in B_n\}$. Every vector of $\Z_n$ is in the orbit of exactly one partition,  $\lambda_v$, which is the partition obtained after ordering the absolute value of all entries of $v$.
%\begin{example}
%	Let $v=(1,\overline{3},0, 3, \overline{2})$. Then $\lambda_v=(3,3,2,1,0)$.
%\end{example}
Given $v=(v_1, \dots, v_n)\in \Z^n$, put in the first $|v_i|$ columns the letter $i$ if $v_i>0$ or $\overline{i}$ if $v_i<0$. This defines a key tableau of weight $v$, $(v)K$.
\begin{prop}\label{UniKw}
	Let $v\in \lambda B_n$. There is exactly one key tableau $(v)K$ whose weight is $v$. The shape of $(v)K$ is $\lambda$. $(\lambda)K$ is the only KN tableau of weight and shape $\lambda$. % also called Yamanouchi tableau of shape $\lambda$. 
	The map $v\mapsto (v)K$ is a bijection between $\lambda B_n$ and key tableaux in $\mathcal{KN}(\lambda, n)$.
\end{prop}



\subsection{The Bruhat order on $B_n$ and cosets of $B_n$}\label{Ssec Bruhat}
Given a partition $\lambda\in \Z^n$, let $W_\lambda=\{\rho\in B_n\mid \lambda\rho=\lambda\}$ be the stabilizer of $\lambda$, under the action of $B_n$,  a standard parabolic subgroup of $B_n$ generated by a subset of simple generators. 
Let $W_\lambda\setminus B_n=\{W_\lambda\sigma:\sigma\in B_n\}$ be the set of right cosets of $B_n$ determined by the subgroup $W_\lambda$. Given a right coset in $W_\lambda\setminus B_n$, all its elements return the same vector when acting on $\lambda$. Hence the vectors $v$ in the $B_n$-orbit of $\lambda$ define a labelling for the right cosets. 
Therefore, the symplectic key tableaux in $\mathcal{KN}(\lambda, n)$ and the cosets of $B_n$, modulo $W_\lambda$, are in bijection: $(v)K\leftrightarrow v \leftrightarrow W_\lambda\sigma_v$, where $\sigma_v$ is the minimal length coset representative.
Key tableaux, $(v)K$, $v\in\lambda B_n$, may be used
to explicitly construct the minimal length coset representatives of  $W_\lambda\setminus B_n$, a generalization of what Lascoux does for vectors in $\mathbb{N}^n$ (hence $\sigma_v \in \mathfrak{S}_n$).
\begin{prop}\label{minimalBn}
	Let $v\in \lambda B_n$ and $T$ the tableau obtained after adding the column $\YT{0.15in}{}{
		{{1}},
		{{2}},
		{{\vdots}},
		{{n}}}$ to the left of $(v)K$. The aforementioned minimal length coset representative $\sigma_v$ is given by the reading word $T$, where entries with the same absolute value are read just once. 
\end{prop}
Let	$v=(3,\overline{3},0, 0, \overline{2})$.
Then
$(v)K=\YT{0.15in}{}{
	{{1},{1},{1}},
	{{\overline{5}},{\overline{5}},{\overline{2}}},
	{{\overline{2}},{\overline{2}}}}$, $T=\YT{0.15in}{}{
	{{1}, {1},{1},{1}},
	{{2},{\overline{5}},{\overline{5}},{\overline{2}}},
	{{3},{\overline{2}},{\overline{2}}},
	{{4}},{{5}}}
$ and $\sigma_v=[1\overline{2}\overline{5}34]$.


Given  $v$ and $u$  in $\lambda B_n$, we write $v\le u$ to mean $\sigma_v\leq\sigma_u$ in the Bruhat order. Put $\Lambda_n=(n, n-1, \dots, 1)$.
Thanks to Theorem $3BC$ of Proctor's Ph.D. thesis \cite{proctor1980interactions}, we have a tableau criterion for the Bruhat order on vectors in the same $B_n$-orbit.
\begin{teo}\cite[Theorem $3BC$]{proctor1980interactions}\label{keycoset}
	Let $v,u\in\lambda B_n$. Then $\sigma_v\leq\sigma_u$ if and only if $(v)K\leq (u)K$, by entrywise comparison. In particular, for $\sigma, \rho \in B_n$, $\sigma\leq \rho \Leftrightarrow (\Lambda_n \sigma)K\leq (\Lambda_n \rho)K$.
\end{teo}
\noindent
For instance,	$v\!=\!(3,\overline{3},0, 0, \overline{2})\!\leq\! u\!=\!(\overline{3},2,0,\overline{3},0)$, because 
$(v)K\!\!=\!\!\YT{0.15in}{}{
	{{1},{1},{1}},
	{{\overline{5}},{\overline{5}},{\overline{2}}},
	{{\overline{2}},{\overline{2}}}}$ $
\le$ $ (u)K\!\!=\!\!\YT{0.15in}{}{
	{{2},{2},{\overline{4}}},
	{{\overline{4}},{\overline{4}},{\overline{1}}},
	{{\overline{1}},{\overline{1}}}}
$.

\section{Type C crystal graphs, plactic and coplactic monoids}\label{SecRS}
Let $[\pm n]^\ast$ be the free monoid on the alphabet $[\pm n]$. 
Recall the type $C_n$ simple roots $\{\alpha_i=\mathbf{e_1}-\mathbf{e_2}, i\in [n-1]\}\cup \{\alpha_n=2\mathbf{e_n}\}$. Here a Kashiwara crystal of type $C_n$ is a nonempty set $\mathfrak{B}$ together with the following maps and statistics \cite{bump2017crystal}:
$e_i, f_i:\mathfrak{B}\rightarrow \mathfrak{B}\sqcup\{0\}$, 
$ \varepsilon_i, \varphi_i:\mathfrak{B}\rightarrow \Z$,
$\text{wt}: \mathfrak{B}\rightarrow \Z^n$,
%\end{align*}
where $i\in [n]$ and $0\notin \mathfrak{B}$ is an auxiliary element, such that:
%\begin{enumerate}
if $a, b \in \mathfrak{B}$ then $(a)e_i=b\Leftrightarrow (b)f_i=a$ and in this case $(b)\text{wt}=(a)\text{wt}+\alpha_i$, $(b)\varepsilon_i=(a)\varepsilon_i-1$ and $(b)\varphi_i=(a)\varphi_i+1$;
for all $a \in \mathfrak{B}$, we have $(a)\varphi_i=\langle (a)\text{wt}, \frac{2\alpha_i}{\langle \alpha_i, \alpha_i\rangle}\rangle+(a)\varepsilon_i$, where $\langle ,\rangle$ is the usual inner product in $\R^n$.
For all $a \in \mathfrak{B}$, we have 
$(a)\varphi_i=\max\{k\in \Z_{\geq 0}\mid (a)f_i^k\neq 0\}$ and  $(a)\varepsilon_i=\max\{k\in \Z_{\geq 0}\mid (a)e_i^k\neq 0\}$.
An element $u\in \mathfrak{B}$ such that $(u)e_i=0$ (or $(u)f_i=0$) for all $i\in [n] $ is called  a \emph{highest weight element} (or \emph{lowest weight element}).
We associate with $\mathfrak{B}$ a coloured oriented graph with weighted vertices in $\mathfrak{B}$ and edges labelled by $i\in [n]$: $b\overset{i}{\rightarrow} b'$ if and only if $b'=(b)f_i$, $i\in [n]$, $b, b'\in\mathfrak{B}$. This is the \emph{crystal graph} of $\mathfrak{B}$. The $C_n$ standard crystal $\mathbb{B}$ is
$1\xrightarrow{1} 2\xrightarrow{2} \dots\xrightarrow{n-1} n\xrightarrow{n} \overline{n}\xrightarrow{n-1}\dots \xrightarrow{1}1$, with set
$\mathbb{B}=[\pm n]$, 
where $(i)\text{wt}=\bf e_i$, $(\overline{i})\text{wt}=-\bf e_i$. The highest weight word is the word $1$, and the lowest weight word is $\overline{1}$.

Let $\mathfrak{B}$ and $\mathfrak{C}$ be two crystals associated to the same root system. The \emph{tensor product} $\mathfrak{B}\otimes \mathfrak{C}$ is a crystal whose set is the Cartesian product $\mathfrak{B}\times\mathfrak{C}$, where its elements are $b\otimes c$, $b\in\mathfrak{B}$ and $c\in\mathfrak{C}$, with $(b\otimes c)\text{wt}=(b)\text{wt}+(c)\text{wt}$. The crystal operator $f_i$ is defined by $(b\otimes c)f_i=\begin{cases}
(x)f_i\otimes y\; \text{if}\; (c)\varphi_i\leq(b)\varepsilon_i\\
x\otimes (y)f_i\; \text{if}\; (c)\varphi_i>(b)\varepsilon_i
\end{cases}$ $\!\!\!\!\!\!\!\!$, and $e_i$ is its inverse. Using the tensor product we can define the crystal $\mathbb{B}^{\otimes k}$ of words of length $k$. Thus, we define how the crystal operators $f_i$ and $e_i$ act on any finite word. This operators can be described via the signature rule, see \cite{bump2017crystal}. Let $G_n=\bigoplus\limits_{k\geq 0}\mathbb{B}^{\otimes k}$ be the type $C_n$ crystal of all words in $[\pm n]^\ast$. The  crystal $G_n$ is the union of connected components where each component has a unique highest (lowest) weight word. Two connected components are isomorphic if and only if they have the same highest weight \cite{lecouvey2002schensted}.

The Robinson--Schensted (RS) type $C$ correspondence is a bijection between words $w\in G_n$ of length $k$, and tuples consisting of a KN tableau $(w)P$ % of partition shape
and an \emph{oscillating tableau} $Q$, of length $k$, with the same final shape as $(w)P$, see \cite{lecouvey2002schensted}.
We denote this map by $w\mapsto ((w)P, Q)$, where $(w)P$ can be computed via Sheats \emph{jeu de taquin} or \emph{Baker-Lecouvey insertion}. 
The symplectic \textit{jeu de taquin} \cite{lecouvey2002schensted, sheats1999symplectic} is a weight-preserving procedure that allows us to change the shape of a  KN skew tableau and eventually rectify it, i.e., make it to have partition shape. It is a variation of the ordinary \textit{jeu de taquin} for skew SSYTs.
The rectification is independent of the order in which the inner corners of $\nu$  are filled \cite[Corollary 6.3.9]{lecouvey2002schensted}. 

The Baker-Lecouvey insertion \cite{baker2000insertion,lecouvey2002schensted} is a bumping algorithm that, given a word $w$ in the alphabet $[\pm n]$, returns the KN tableau $(w)P$.
It depends on the symplectic jeu de taquin.  This insertion is similar to the usual column insertion for SSYTs. 
However, when inserting a letter it may happen that we remove a cell from the inserted tableau, instead of adding. The length of $((w)P)cr$ might be less than the length of $w$, but the weight is preserved, $(w)\text{wt}=((w)P)\text{wt}$. 
If the word $w$ does not have symmetric letters, then the insertion works just like the column insertion for SSYTs.
If $l$ is the length of $w$, $(w)P$ is the rectification of the skew tableau of shape $\Lambda_l/\Lambda_{l-1}$ and reading word $w$ \cite[Corollary 6.3.9]{lecouvey2002schensted}. %\end{proposition}
%\begin{proposition}\label{rr}
More generally, if $T\in \mathcal{KN}(\mu/\nu, n)$, the rectification of $T$ coincides with $((T)cr)P$.



Given $w_1, w_2 \in [\pm n]^\ast$, the relation
$w_1 \sim w_2\Leftrightarrow (w_1)P=(w_2)P$ defines an equivalence relation on $[\pm n]^\ast$ known as Knuth equivalence. The type $C$ plactic monoid is the quotient $[\pm n]^\ast/\sim$ where each Knuth (plactic) class is uniquely identified with a KN tableau \cite{lecouvey2002schensted}. Hence two Knuth-related words have the same weight. It is also described as the quotient of $[\pm n]^\ast$ by the elementary Knuth relations; see \cite{lecouvey2002schensted} for details. If $w_1 \sim w_2$ then they occur in the same place in two isomorphic connected components of $G_n$ 
\cite{lecouvey2002schensted}, \emph{i.e.}, $(w_1)e_i\sim (w_2)e_i$ and $(w_1)f_i\sim(w_2)f_i$, $i \in [n]$.

Two words $w_1,w_2\in [\pm n]^\ast$ are% crystal connected or
coplactic equivalent if and only if they belong to the same connected component of $G_n$. %, \emph{i.e.}, one is obtained from  another by some sequence of crystal operators $f_i$ and $e_j$, $i, j \in [n]$.  
The connected components of $G_n$ are the coplactic classes in the RS correspondence that identify words with the same oscillating tableau \cite[Proposition 5.2.1]{lecouvey2002schensted}. 

Choose a word $w\in [\pm n]^\ast$ where the shape of $(w)P$ is $\lambda$. If we replace every word of its coplactic class with its insertion tableau we obtain the crystal $\mathfrak{B}^\lambda$ of tableaux  $\mathcal{KN}(\lambda, n)$. The crystal $\mathfrak{B}^\lambda$ does not depend on the initial choice of word $w$ in the plactic class of $w$ \cite[Theorem 6.3.8]{lecouvey2002schensted}.
A word $w$ of $G_n$ is a highest weight word if and only if the weight of all its prefixes (including itself) is a partition. In this case, \section{Right and left keys and Demazure atoms in type C}\label{SecKeys}
We generalize Lascoux--Sch\"{u}tzenberger frank words, in type $A$ \cite{lascoux1990keys},   to type $C$ to create  right and left key maps in type $C$. %, that send KN tableaux to key tableaux in type C.
Our \cref{DemKey} detects  the type $C$ KN tableaux for Demazure atoms. It is the type $C$ version of  Lascoux and Sch\"{u}tzenberger \cite[Theorem 3.8]{lascoux1990keys}.
\begin{defin}
	The word $w\in [\pm n]^\ast$ is a type $C$ frank word if the lengths of its maximal column factors form a multiset equal to the multiset formed by the lengths of the columns of the tableau $(w)P$.
\end{defin}%\vspace{-.2 cm}
For instance, $(23\overline{2}\overline{3}1)P=(\overline{1}113\overline{3})P=\YT{0.14in}{}{
	{{1},{1},{\overline{1}}},
	{{3}},
	{{\overline{3}}}}$. Since $23\overline{2}\overline{3}1$ and $\overline{1}113\overline{3}$ have one column of length $3$ and two columns of length $1$, they are frank words. 

Given a frank word $w$, the number of letters of $w$ is the same as the number of cells of $(w)P$. %, hence the case $3$ of the Baker-Lecouvey insertion doesn't happen.
%
%\begin{proposition}
%	Let $w$ be frank word on the alphabet $[\pm n]$. 
This implies that all columns of $w$ are admissible.
The following proposition is an extension of \cite[Proposition 7]{fulton1997young} on SSYTs to KN tableaux.
\begin{prop}\label{Fultonices}
	Let $T\in\mathcal{KN}(\lambda, n)$. Let $\mu/\nu$ be a skew diagram with same number of columns of each length as $T$. Then there is a unique KN skew tableau $S$ with shape $\mu/\nu$ that rectifies to $T$ and $(S)cr$ is a frank word. 
\end{prop}


\begin{cor}\label{lastcolumn} Let $S$ be as in the previous proposition. 
	The last column of $S$ depends only on the length of that column.
\end{cor}

Fixed a KN tableau $T$, consider the set of all possible last columns taken from skew tableaux with same number of columns of each length as $T$. \cref{lastcolumn} implies that this set has one element for each distinct column length of $T$. For each column $C$ in this set, consider the column $Cr$, its right column. The next proposition implies that this set of right columns is nested, if we see each column as the set formed by its letters.

\begin{prop}\label{nestedcolumns}
	Consider $T$ a two-column KN skew tableau $C_1C_2$ with empty cells in the first column. Slide via symplectic \textit{jeu de taquin} the bottommost of those empty cell, obtaining a two-column KN skew tableau $C_1'C_2'$. Then $C_2'r\subseteq C_2r$.
\end{prop}
%\begin{proof}
%	If the sliding was vertical then $C_2'=C_2$, hence $rC_2'=rC_2$. If the sliding was horizontal, call $\beta$ to the number on the cell right of the empty cell on $spl(T)$. Call $\Phi$ to the function that takes an admissible column to the associated coadmissible column. 
%	
%	If $\beta=b$ is unbarred then $C_2'=\Phi^{-1}(\Phi(C_2)\setminus\{b\})$. In this case $\Phi(C_2')=\Phi(C_2)\setminus\{b\}$, hence $rC_2$ and $rC_2'$ have the same barred part. Consider $z_1<\dots<z_\ell$ the unbarred letters that appear on $C_2$ and not on $\Phi(C_2)$. When we take $b$ from $\Phi(C_2)$, if $\overline{b}\in \Phi(C_2)$ our set of letters $z_1<\dots<z_\ell$ will lose an element, giving the inclusion of the unbarred part of $C_2'$ in $C_2$; if $\overline{b}\not\in \Phi(C_2)$, then $b \in C_2$ and in $C_2'$ the least $z_i>b$ may reduce to $b$, and subsequent $z_j$ may reduce to $z_{j-1}$. Hence we have the inclusion of the unbarred part of $C_2'$ in $C_2$.
%	
%	If $\beta=\overline{b}$ is barred then $C_2'=C_2\setminus\{\overline{b}\}$. In this case $rC_2$ and $rC_2'$ have the same unbarred part.  Consider $\overline{t_1}>\dots>\overline{t_\ell}$ the barred letters that appear on $\Phi(C_2)$ and not on $C_2$. When we take $\overline{b}$ from $C_2$, if $b\in C_2$ our set of $\overline{t_1}>\dots>\overline{t_\ell}$ letters will lose an element, giving the inclusion of the barred part of $rC_2'$ in $rC_2$;  if $b\not\in C_2$, then $\overline{b} \in \Phi(C_2)$ and in $C_2'$ the least $\overline{z_i}>\overline{b}$ may reduce to $\overline{b}$, and subsequent bigger $\overline{z_j}$'s may reduce to $\overline{z_{j+1}}$. Hence we have the inclusion of the barred part of $\Phi(C_2')$ in $\Phi(C_2)$.
%\end{proof}
%This proposition defines a map that sends a KN tableau to key tableau in type $C$, identified as the (symplectic) right key of a KN tableau.
\noindent Next, one gives the type $C$ right key map. It extends the one defined for type $A$ in \cite{lascoux1990keys}.
\begin{teo}[Right key map]\label{rightkeymap}
	Given a KN tableau $T$, if we replace each column with a column of the same size taken from the right columns of the last columns of all skew tableaux associated to $T$, then we obtain a key tableau. This tableau is the right key tableau of $T$ and we denote it by $(T)K_+$. (See \cref{ExRightkey}.)
\end{teo}
%\begin{proof}
%	The previous proposition implies that the columns of $K_+(T)$ are nested and do not have symmetric entries. So, it is indeed a KN key tableau.
%\end{proof}
\begin{obs} Recall the set up of \cref{Fultonices}. If the shape of $S$, $\mu/\nu$, is such that every two consecutive columns have at least one cell in the same row, then each column of $S$ is a maximal column factor of the word $(S)cr$, hence $(S)cr$ is a frank word. Moreover, the columns of $S$ appear in reverse order in $(S)cr$. Therefore, given a KN tableau $T$, the columns of $(T)K_+$ consist of right columns of the first columns of the frank words associated to $T$.
\end{obs}
%In the same spirit of the right key, we define the left key of a KN tableau. %Given a tableau $T$, we can replace each column with a column taken from the left columns of the first columns of all skew-tableaux with the same number of columns of each length as $T$. 
In the set up of \cref{nestedcolumns}, we also can prove that $ C_1\ell\subseteq  C_1'\ell$, hence the set of left columns of the first columns of all skew tableaux with the same number of columns of each length as $T$ will be nested.
The left key $(T)K_-$ is obtained after replacing each column of $T$ with a column of the same size taken from this set.
\begin{ex}\label{ExRightkey}
	The tableau $T=\YT{0.15in}{}{
		{{1},{3},{\overline{1}}},
		{{3},{\overline{3}}},
		{{\overline{3}}}}$ has the following six KN skew tableaux with same number of columns of each length as $T$, each one corresponding to a permutation of its column lengths, and each one is associated to the frank word given by its column reading. 
	
	\begin{tikzpicture}
	\node at (0,0) {$\YT{0.15in}{}{
			{{1},{3},{\overline{1}}},
			{{3},{\overline{3}}},
			{{\overline{3}}}}$};	
	\node at (4,1) {$\begin{tikzpicture}[scale=.38, baseline={([yshift=-.8ex]current bounding box.center)}]
		\draw (0,0) rectangle +(1,1);
		\draw (0,1) rectangle +(1,1);
		\draw (0,2) rectangle +(1,1);
		\draw (1,2) rectangle +(1,1);
		\draw (2,2) rectangle +(1,1);
		\draw (2,3) rectangle +(1,1);
		\node at (.5,.5) {$\overline{3}$};
		\node at (.5,1.5) {$3$};
		\node at (.5,2.5) {$1$};
		\node at (1.5,2.5) {$\overline{3}$};
		\node at (2.5,2.5) {$\overline{1}$};
		\node at (2.5,3.5) {$3$};
		\end{tikzpicture}$};
	\node at (8,1) {$\begin{tikzpicture}[scale=.38, baseline={([yshift=-.8ex]current bounding box.center)}]
		\draw (0,0) rectangle +(1,1);
		\draw (1,0) rectangle +(1,1);
		\draw (1,1) rectangle +(1,1);
		\draw (1,2) rectangle +(1,1);
		\draw (2,1) rectangle +(1,1);
		\draw (2,2) rectangle +(1,1);
		\node at (.5,.5) {$2$};
		\node at (1.5,.5) {$\overline{2}$};
		\node at (1.5,1.5) {$\overline{3}$};
		\node at (1.5,2.5) {$1$};
		\node at (2.5,1.5) {$\overline{1}$};
		\node at (2.5,2.5) {$3$};
		\end{tikzpicture}$};
	\node at (4,-1) {$\begin{tikzpicture}[scale=.38, baseline={([yshift=-.8ex]current bounding box.center)}]
		\draw (0,0) rectangle +(1,1);
		\draw (0,1) rectangle +(1,1);
		\draw (1,0) rectangle +(1,1);
		\draw (1,1) rectangle +(1,1);
		\draw (1,2) rectangle +(1,1);
		\draw (2,2) rectangle +(1,1);
		\node at (.5,.5) {$2$};
		\node at (.5,1.5) {$1$};
		\node at (1.5,.5) {$\overline{2}$};
		\node at (1.5,1.5) {$\overline{3}$};
		\node at (1.5,2.5) {$3$};
		\node at (2.5,2.5) {$\overline{1}$};
		\end{tikzpicture}$};
	\node at (8,-1) {$$\begin{tikzpicture}[scale=.38, baseline={([yshift=-.8ex]current bounding box.center)}]
		\draw (0,0) rectangle +(1,1);
		\draw (0,1) rectangle +(1,1);
		\draw (1,1) rectangle +(1,1);
		\draw (2,1) rectangle +(1,1);
		\draw (2,2) rectangle +(1,1);
		\draw (2,3) rectangle +(1,1);
		\node at (.5,.5) {$2$};
		\node at (.5,1.5) {$1$};
		\node at (1.5,1.5) {$\overline{2}$};
		\node at (2.5,1.5) {$\overline{1}$};
		\node at (2.5,2.5) {$\overline{3}$};
		\node at (2.5,3.5) {$3$};
		\end{tikzpicture}$$};
	\node at (12,0) {$\SYT{0.15 in}{}{{{3}},{{\overline{3}},{1}},{{\overline{1}},{\overline{2}},{2}}}$};
	\draw [->] (1,.5) -- (3,1);
	\draw [->] (1,-.5) -- (3,-1);
	\draw [->] (5,1) -- (7,1);
	\draw [->] (5,-1) -- (7,-1);
	\draw [->] (9,1) -- (11,.5);
	\draw [->] (9,-1) -- (11,-.5);	
	\end{tikzpicture}
	%		\vspace{-.3cm}
	
	The right key of $T$ has as columns $\YT{0.15in}{}{
		{{3}},
		{{\overline{3}}},
		{{\overline{1}}}}\!r$, $\YT{0.15in}{}{
		{{3}},
		{{\overline{1}}}}\!r$ and $\YT{0.15in}{}{
		{{\overline{1}}}}\!r$, hence $(T)K_+=\YT{0.15in}{}{
		{{3},{3},{\overline{1}}},
		{{\overline{2}},{\overline{1}}},
		{{\overline{1}}}}$.
	%%	Consider the Kashiwara--Nakashima tableau $T=\YT{0.17in}{}{
	%%		{{1},{2}},
	%%		{{3},{\overline{2}}},
	%%		{{\overline{3}}}}$. 
	%%	We can obtain from $T$ the skew tableau $\SYT{0.17in}{}{
	%%		{{2}},
	%%		{{3},{1}},
	%%		{{\overline{2}}, {\overline{3}}}}$ via symplectic jeu de taquin. Hence, the right key tableau associated to $T$ has as columns $r\!\YT{0.17in}{}{
	%%		{{2}},
	%%		{{\overline{2}}}}$ and $r\!\YT{0.17in}{}{
	%%		{{2}},
	%%		{{3}},
	%%		{{\overline{2}}}}$. 
	%%	Hence $K_+(T)=\YT{0.17in}{}{
	%%		{{2},{2}},
	%%		{{3},{\overline{1}}},
	%%		{{\overline{1}}}}$.
	
	The left key of
	$T$ has as columns
	$\YT{0.15in}{}{
		{{1}},
		{{3}},
		{{\overline{3}}}}\!\ell$, $\YT{0.15in}{}{
		{{1}},
		{{2}}}\!\ell$ and $\YT{0.15in}{}{
		{{2}}}\!\ell$, hence $(T)K_-=\YT{0.15in}{}{
		{{1},{1},{2}},
		{{2},{2}},
		{{\overline{3}}}}$.
\end{ex}
\subsection{Demazure crystals and right key tableaux}
Let $\lambda\in\Z^n$ be a partition and $v\in\lambda B_n$.
We define $(v)\mathfrak{U}=\{T\in \mathcal{KN}(\lambda,n)\mid (T)K_+=(v)K\}$ the set of KN tableaux of $B^\lambda$ with right key $(v)K$.
Given a subset $X$ of $\mathfrak{B}^\lambda$, consider the operator $\mathfrak{D}_i$ on $X$, $i\in [n]$, defined by
$X\mathfrak{D}_i=\{x\in \mathfrak{B}^\lambda\mid (x)e_i^k\in X \,\,\text{for some $k\geq 0$}\}$ \cite{bump2017crystal}.
If $v=\lambda \sigma $ where $\sigma =s_{i_1}\dots s_{i_{(\sigma)\ell}}\in B_n$ is a reduced word, we define the Demazure crystal to be
$\mathfrak{B}_v=\{(\lambda)K\}\mathfrak{D}_{i_1}\dots \mathfrak{D}_{i_{(\sigma)\ell}}$.
This definition is independent of the reduced word for $\sigma$ \cite[Theorem 13.5]{bump2017crystal}, 
as well of the coset representative of $W_\lambda\sigma$, that is, $\mathfrak{B}_{\lambda \sigma }=\mathfrak{B}_{\lambda \sigma_v }$. 
	
From \cite[Proposition 2.5.1]{bjorner2006combinatorics}, if $\rho\leq\sigma$ in $B_n$ then $\rho_u=\sigma_u\leq\sigma_v$ where $u=\lambda\rho$. Since $(x)e_i^0=x$, if $\rho\leq\sigma$ then  $\mathfrak{B}_{\lambda \rho}=\mathfrak{B}_{\lambda \rho_u}\subseteq\mathfrak{B}_{\lambda \sigma_v}=\mathfrak{B}_{ v}$.
Thus we define the Demazure crystal  atom $\hat{\mathfrak{B}}_{ v}$ to be
$\hat{\mathfrak{B}}_{ v}=\hat{\mathfrak{B}}_{ \lambda \sigma}:=\mathfrak{B}_{\lambda \sigma_v }\setminus\bigcup\limits_{\rho_u<\sigma_v }\mathfrak{B}_{\lambda \rho_u}=\mathfrak{B}_{v }\setminus\bigcup\limits_{u<v }\mathfrak{B}_{u}\stackrel{\text{\cref{keycoset}}}{=}\mathfrak{B}_{v }\setminus\bigcup\limits_{(u)K<(v)K }\mathfrak{B}_{u}.$


 Thanks to the right key map, \cref{rightkeymap}, we now describe the Demazure  crystal atom in type $C$.
%using the right key map (\cref{rightkeymap}).
Lascoux and Sch\"{u}tzenberger,  \cite[Theorem 3.8]{lascoux1990keys}, have given the type $A$ version. Full details can be seen in the full version of this extended abstract \cite{santos2019symplectic}.
\begin{teo}[Main Theorem]\label{DemKey}
	Let $v\in \lambda B_n$.
	Then $(v)\mathfrak{U}=\hat{\mathfrak{B}}_{v}$.
\end{teo}

Given $v\in \lambda B_n$ define the Demazure character (or key polynomial), $\kappa_v$, and the Demazure atom in type $C$, $\widehat{\kappa}_v$, as the generating functions of the KN tableau weights in $\mathfrak{B}_v$ and $\widehat{\mathfrak{B}}_v$, respectively:  $\kappa_v=\sum\limits_{T\in \mathfrak{B}_{v}}x^{(T)\text{wt}},\,
\hat{\kappa}_v=\sum\limits_{T\in \hat{\mathfrak{B}}_{v}}x^{(T)\text{wt}}$. \cref{DemKey} detects the KN tableaux in $\mathfrak{B}^\lambda$ contributing to the Demazure atom $\hat{\kappa}_v=\sum\limits_{\substack{(T)K_+=(v)K\\
		T\in \mathfrak{B}^\lambda }}x^{(T)\text{wt}}$.
Moreover, one has
$\kappa_v=\sum\limits_{u\leq v}\hat{\kappa}_u=\sum\limits_{\substack{u\leq v\\T\in (u)\mathfrak{U}}}x^{(T)\text{wt}}=\sum\limits_{\substack{(u)K\leq (v)K\\T\in (u)\mathfrak{U}}}x^{(T)\text{wt}}=\sum\limits_{(T)K_+\leq (v)K}x^{(T)\text{wt}}$.
In \cref{cristal21}, $\mathfrak{B}_{(1,\overline{2})}=\{T\in\mathfrak{B}^\lambda\mid (T)K_+\leq ((1,\overline{2}))K\}=\left\{\YT{0.15 in}{}{{{1},{1}},{{2}}}, \YT{0.15 in}{}{{{1},{2}},{{2}}}, \YT{0.15 in}{}{{{1},{1}},{{\overline{2}}}}, \YT{0.15 in}{}{{{1},{\overline{2}}},{{2}}}, \YT{0.15 in}{}{{{1},{\overline{2}}},{{\overline{2}}}}\right\}$.

\begin{ex}\label{cristal21}
	Consider the crystal graph of $\mathfrak{B}^{(2,1)}$:
	\begin{multicols}{2}
		${%\scriptsize \begin{array}{cccc}
			\begin{tikzpicture}
			[scale=.55,auto=left]
			\node (n0) at (0,14.5) {$\YT{0.17 in}{}{{{1},{1}},{{2}}}$};
			\node (n1l) at (-3.5,13.5)  {$\YT{0.17 in}{}{{{1},{2}},{{2}}}$};
			\node (n1r) at (3.5,13.5)  {$\YT{0.17 in}{}{{{1},{1}},{{\overline{2}}}}$};
			\node (n2l) at (-3.5,11)  {$\YT{0.17 in}{}{{{1},{\overline{2}}},{{2}}}$};
			\node (n2r) at (3.5,11) {$\YT{0.17 in}{}{{{1},{2}},{{\overline{2}}}}$};
			\node (n3r) at (3.5,8.5)  {$\YT{0.17 in}{}{{{2},{2}},{{\overline{2}}}}$};
			\node (n4rr) at (5,6)  {$\YT{0.17 in}{}{{{2},{2}},{{\overline{1}}}}$};
			\node (n4r) at (2,6)  {$\YT{0.17 in}{}{{{2},{\overline{2}}},{{\overline{2}}}}$};
			\node (n5r) at (3.5,3.5)  {$\YT{0.17 in}{}{{{2},{\overline{2}}},{{\overline{1}}}}$};
			\node (n6r) at (3.5,1)  {$\YT{0.17 in}{}{{{\overline{2}},{\overline{2}}},{{\overline{1}}}}$};
			\node (n3ll) at (-5,8.5)  {$\YT{0.17 in}{}{{{1},{\overline{2}}},{{\overline{2}}}}$};
			\node (n4l) at (-3.5,6)  {$\YT{0.17 in}{}{{{1},{\overline{1}}},{{\overline{2}}}}$};
			\node (n3l) at (-2,8.5)  {$\YT{0.17 in}{}{{{1},{\overline{1}}},{{2}}}$};
			\node (n5l) at (-3.5,3.5)  {$\YT{0.17 in}{}{{{2},{\overline{1}}},{{\overline{2}}}}$};
			\node (n6l) at (-3.5,1)  {$\YT{0.17 in}{}{{{2},{\overline{1}}},{{\overline{1}}}}$};
			\node (n7) at (0,0)  {$\YT{0.17 in}{}{{{\overline{2}},{\overline{1}}},{{\overline{1}}}}$};
			
			\draw (0,12.5)--(-3.5,15.5);
			\draw (0,12.5)--(3.5,15.5);
			%		\draw (0,12.5)--(1,10);
			\draw (0,12.5)--(5,3.5);
			\draw (0,12.5)--(-5,7);
			%		\draw (0,12.5)--(4,8);
			\draw (0,12.5)--(-2,-.5);
			\draw (0,12.5)--(2,-.5);
			\draw (0,12.5)--(-5,12.5);
			\draw (0,12.5)--(5,12.5);
			%			\draw[->] [draw=red,  thick] ()--();
			%			\foreach \from/\to in {n0/n1r,n6l/n7,n3l/n4l,  n2l/n3ll,n1l/n2l,n3r/n4r,n4rr/n5r,n5r/n6r}
			%			\draw[->] [draw=red,  thick](\from) -- (\to);
			%%%			
			%			\foreach \from/\to in {n0/n1l,n1r/n2r,n2r/n3r,n3r/n4rr,n4r/n5r,n6r/n7,n3ll/n4l,n5l/n6l,n4l/n5l,n2l/n3l}
			%			\draw [->] [draw=blue,  thick](\from) -- (\to);
			\draw[->] [draw=red,  thick] (-2.5,0.7)--(-1,0.3);
			\draw[->] [draw=blue,  thick] (2.5,0.7)--(1,0.3);
			\draw[->] [draw=blue,  thick] (-3.5,2.5)--(-3.5,2);
			\draw[->] [draw=red,  thick] (3.5,2.5)--(3.5,2);
			\draw[->] [draw=blue,  thick] (-3.5,5)--(-3.5,4.5);
			\draw[->] [draw=blue,  thick] (2.5,5)--(3,4.5);
			\draw[->] [draw=red,  thick] (4.5,5)--(4,4.5);
			\draw[->] [draw=red,  thick] (-2.5,7.5)--(-3,7);
			\draw[->] [draw=blue,  thick] (-4.5,7.5)--(-4,7);
			\draw[<-] [draw=blue,  thick] (4.5,7)--(4,7.5);
			\draw[<-] [draw=red,  thick] (2.5,7)--(3,7.5);
			\draw[->] [draw=blue,  thick] (-3,10)--(-2.5,9.5);
			\draw[->] [draw=red,  thick] (-4,10)--(-4.5,9.5);
			\draw[->] [draw=blue,  thick] (3.5,10)--(3.5,9.5);
			\draw[->] [draw=red,  thick] (-3.5,12.5)--(-3.5,12);
			\draw[->] [draw=blue,  thick] (3.5,12.5)--(3.5,12);
			\draw[->] [draw=blue,  thick] (-1,14.2)--(-2.5,13.8);
			\draw[->] [draw=red,  thick] (1,14.2)--(2.5,13.8);
			\end{tikzpicture}
			%\end{array}
		}$
The crystal is split into pieces. Each piece is a Demazure atom and contains exactly one symplectic key tableau, so we can identify each part with the weight of that key tableau, a vector in the $B_2$-orbit of $(2,1)$. From the previous theorem we have that all tableaux in the same piece have the same right key.\linebreak
		One can check that $((1,\overline{2}))\mathfrak{U}=$ $\left\{\YT{0.15 in}{}{{{1},{\overline{2}}},{{2}}}, \YT{0.15 in}{}{{{1},{\overline{2}}},{{\overline{2}}}}\right\}=\hat{\mathfrak{B}}_{\lambda s_1s_2}$, for example.
	\end{multicols}
\end{ex}

\section{Lusztig involution, right and left keys}\label{SecLusztig}

Let $\mathfrak{B}^\lambda$ be the crystal of tableaux in $\mathcal{KN}(\lambda, n)$.
The type $C_n$ Lusztig involution $L:\mathfrak{B}^\lambda\rightarrow\mathfrak{B}^\lambda$ is the only involution such that, for all $i\in [n]$, $x\in \mathcal{KN}(\lambda, n)$:
(1) $((x)L)\text{wt}=((x)\text{wt})\omega_0=-(T)\text{wt}$, where $\omega_0$ is the longest element of $B_n$;
(2) $((x)L)e_i=((x)f_{i})L$ and $((x)L)f_i=((x)e_{i})L$; % where $i'$ is such that $(\alpha_i)\omega_0=-\alpha_{i'}$. 	
and (3) $((x)L)\varepsilon_i=(x)\varphi_{i}$ and $((x)L)\varphi_i=(x)\varepsilon_{i}$.%\end{enumerate}
%\end{definition}


The involution $L$ flips the crystal upside down.
The Sch\"utzenberger evacuation is a realization of Lusztig involution in type $A$. The algorithm below adapts it to KN tableaux. It sends a tableau $T\in \mathcal{KN}(\lambda,n)$ to $T^{\text{Ev}}\in \mathcal{KN}(\lambda,n)$, where $ (T)\text{wt}=-((T^{\text{Ev}})\text{wt})$.
\begin{alg}\label{rsi}\indent
	1. Let $(T)cr^\star$ be the word obtained by applying $\omega_0$ to the letters of $(T)cr$ and writing it backwards (or define $T^\#$ by $\pi$-rotating $T$ and applying $\omega_0$ to its entries).
	
	2. Insert $(T)cr^\star$ (or rectify $T^\#$). Define $T^{Ev}:=((T)cr^\star)P=$rectification of $T^\#$.
	%	\end{enumerate}
\end{alg}
%\begin{example}
Consider the KN tableau $T=\YT{0.15in}{}{
	{{1},{\overline{2}}},
	{{\overline{3}},{\overline{1}}},
	{{\overline{2}}}}$. 
Then, $w=(T)cr=\overline{2}\overline{1}1\overline{3}\overline{2}$ and $w^\star=23\overline{1}12$.
Then
$(w^\star)P=\YT{0.15in}{}{
	{{2},{2}},
	{{3},{3}},
	{{\overline{3}}}}$ which is the rectification of $T^\#=\SYT{0.15in}{}{
	{{2}},
	{{3},{1}},
	{{\overline{1}},{2}}}$. 



Let $w\in [\pm n]$. The connected component of the crystal $G_n$ that contains the word $w^\star $ is obtained applying $^\star$ to each vertex of the one containing $w$ and reverting arrows. They are  isomorphic  because they have the same highest weight, say $\lambda$.
Therefore $(w)P$, $(w^\star)P\in \mathfrak{B}^\lambda$ and have symmetric weights. $T^{Ev}$ is the only KN tableau with the same shape as $T$, and Knuth equivalent to $(T)cr^\star$.
The algorithm  is a realization of the type $C$ Lusztig involution.
It follows that  evacuation of the right key of a tableau is the left key of the evacuation of the same tableau. See \cite{santos2019symplectic} for the proof.

\begin{teo}\label{key&Lusz}
	Let T be a KN tableau and $^{Ev}$ the type $C$ Lusztig (Sch\"utzenberger) involution. Then
	$$(T)K_+\,^{Ev}=(T^{Ev})K_-.$$
\end{teo}

\acknowledgements{
I am grateful to C. Lenart, for his observations on the alcove path model, and to O. Azenhas, my Ph.D. advisor, for her help on the preparation of this paper. This work was partially supported by the Center for Mathematics of the  University of Coimbra - UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. It was also supported by FCT, through the grant PD/BD/142954/2018, under POCH funds, co-financed by the European Social Fund and Portuguese National Funds from MEC.
}

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