%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cylindric diagrams}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\input{NSTfpsac_sec2_subsec1.tex}	%	cylindric diagram
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Cylindric diagrams as posets}\label{SS:cylindric}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $(P,\le)$ be a poset.
For $x,y\in P$, define an \emph{interval} $[x,y]$ by
$$[x,y] = \{ z \in P \mid x \leq z \leq y\}.$$
We say that $y$ \emph{covers} $x$ if $[x,y]=\{x,y\}$ and \( x < y \).
%An interval is called a \emph{chain} if it is a totally ordered set.
%
%In the rest,
%we assume that a poset $P$ is \emph{locally finite}, namely,
%$\#[x,y]<\infty$ for any $x,y \in P$.

\begin{Def}
  Let $(P,\le)$ be a (possibly infinite) poset.
  A subset $J$ of $P$ is called an \emph{order filter} (resp. \emph{order ideal})
  if the following condition holds:
  $$
    x \in J,\ x \le y \implies y \in J
    \qquad (\text{resp. } x \in J,\ x \ge y \implies y \in J).
  $$
  The set of finite order ideals of \( P \) is denoted by \( \I(P) \)
  and we regard it
  as a poset with the inclusion relation.


\begin{comment}
  A subset $F$ of $P$ is called an \alert{order filter}
  if the following condition holds:
  $$x \in F,\ x \leqq y \implies y \in F.$$
  %An order filter $F$ is said to be \alert{non-trivial} if $F\neq \emptyset$ nor $F\neq P$.

  A subset $J$ of $P$ is called an \alert{order ideal}
  if the following condition holds:
  $$x \in J,\ x \geqq y \implies y \in J.$$
\end{comment}

%  An order filter (resp. order ideal) $J$ is said to be \alert{proper} if
%  $J \neq P$,
%  and it is said to be \alert{non-trivial} if $J \neq P$ nor $J \neq \emptyset$.

  %An order filter (resp. ideal $J$) is said to be \alert{non-trivial} if $F\neq \emptyset$ (resp. $J\neq \emptyset$) nor $F\neq P$ (resp. $J\neq P$).
  %
  %For $x\in P$, put
  %For $x_1,x_2,\dots\in P$, the set 
  %$\bigcup_iF_{x_i}$, where $F_x:=\{y\in P\mid x\geqq y\}$ 
  %is an order filter of $P$ and it is called the order filter
  %generated by $x_1,x_2,\dots$.
  %The condition (Y3) above means that   
  %$\sidiag$ is an order filter of $\Z^2$,
  %and a semi-infinite periodic diagram
  %is a $\Z\omega$-orbit of an order filter generated by finitely many elements.
\end{Def}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%poset
%periodic diagram
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




For $\peri \in \Z_{\geqq 1} \times \Z_{\leqq -1}$,
we let $\Z \peri$ denote the subgroup of (the additive group) $\Z^2$ generated by $\peri$,
and define the cylinder $\cyli_\peri$ by
$$
  \cyli_\peri = \Z^2 / \Z \peri.
$$
Let $\pi:\Z^2 \to \cyli_\peri$ be the natural projection.
The cylinder $\cyli_\peri$ inherits a $\Z^2$-module structure via $\pi$.
%The group $\Z^2$ acts on $\cyli_\peri$ by
%\begin{align*}
%\Z^2\times \cyli_\peri &\to \cyli_\peri\\
%(u,x)\ \ &\mapsto x+u:=\pi(\tilde x+u),
%\end{align*}
%where $\tilde x$ is any element of $\pi^{-1}(x)$.
%
%
%We regardas

Define a poset structure on   $\Z^2$ 
by 
%the following partial order
$$
(a,b) \leq (a', b') \iff \hbox{$a \geqq a'$ and $b \geqq b'$ 
as integers}.
$$
%
For $x,y \in \cyli_\peri$,
write $x \le y$ if there exists $\tilde{x}, \tilde{y} \in \Z^2$ such that
$\pi(\tilde{x})=x$, $\pi(\tilde{y})=y$ and $\tilde{x} \le \tilde{y}$.
%
It is not difficult to see the following:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}
  Let $\peri \in \Z_{\geqq 1} \times \Z_{\leqq -1}$.
  Then the relation $\leq$ on $\mathcal{C}_\omega$ is a partial order,
  and the projection $\pi$ is order preserving.
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the rest of this section,
we fix  $\omega \in \mathbb{Z}_{\geqq1} \times \mathbb{Z}_{\leqq -1}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{Def}
  %\begin{enumerate}
  %\begin{description}
  % \item{1.}
   \quad 1.\ 
   A non-trivial order filter of $\mathcal{C}_\omega$ is called a \emph{cylindric diagram}.
    
   % \item{2.}
   \medskip\noindent
   2.\ 
   A finite order ideal of a cylindric diagram is called a \emph{cylindric skew diagram}.
    
   \medskip\noindent
   %\item{3.}
   3.\ 
   A non-trivial order filter $\Theta$ of $\Z^2$ is called a 
   \emph{periodic diagram of period $\peri$} if $\Theta + \omega = \Theta$.
    %\end{description}
   %\end{enumerate}
\end{Def}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lem}\label{lem;CYD}
  %Let $\peri\in\Z_{\geqq1} \times \Z_{\leqq -1}$.
  %\begin{description}
 % \begin{enumerate}
   % \item[1.]
    \quad
    1.\ For a cylindric diagram $\cdiag$ in $\cyli_\peri$,
    the inverse image $\pi^{-1}(\cdiag)$ is a periodic diagram of period $\peri$.
    
    \medskip\noindent
    %\item[2.]
    2.\ For a periodic diagram $\Theta$ of period $\peri$,
    the image $\pi(\Theta)$ is a cylindric diagram in $\cyli_\peri$. 
  %\end{enumerate}
  %\end{description}
\end{lem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Fig. \ref{periodic diagram} indicates a periodic diagram of period $\peri=(4,-5)$.
The set consisting of colored cells is a fundamental domain with respect to the action of $\Z\peri$,
and it is in one to one correspondence with the associated cylindric diagram.
\begin{figure}[!ht]
  \begin{center}
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	\CL{(0,0)}{}	\CL{(1,0)}{}	\CL{(2,0)}{}	\CL{(3,0)}{}	\CL{(4,0)}{}	\CL{(5,0)}{}	\CL{(6,0)}{}	\CL{(7,0)}{}	\CL{(8,0)}{}	\CL{(9,0)}{}	\CL{(10,0)}{}	\CL{(11,0)}{}	\CL{(12,0)}{}	\CL{(13,0)}{}	\CL{(14,0)}{}	\CL{(15,0)}{}	\CL{(16,0)}{}
	\CL{(0,-1)}{}	\CL{(1,-1)}{}	\CL{(2,-1)}{}	\CL{(3,-1)}{}	\CL{(4,-1)}{}	\CL{(5,-1)}{}	\CL{(6,-1)}{}	\CL{(7,-1)}{}	\CL{(8,-1)}{}	\CL{(9,-1)}{}	\CL{(10,-1)}{}	\CL{(11,-1)}{}	\CL{(12,-1)}{}	\CL{(13,-1)}{}	\CL{(14,-1)}{}
	\color{black}
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	\CL[magenta]{(0,-3)}{}	\CL[magenta]{(1,-3)}{}	\CL[magenta]{(2,-3)}{}	\CL[magenta]{(3,-3)}{}	\CL[magenta]{(4,-3)}{}	\CL[magenta]{(5,-3)}{}	\CL[magenta]{(6,-3)}{}	\CL[magenta]{(7,-3)}{}	\CL[magenta]{(8,-3)}{}	\CL[magenta]{(9,-3)}{}	\CL[magenta]{(10,-3)}{}	\CL[magenta]{(11,-3)}{}
	\CL[magenta]{(0,-4)}{}	\CL[magenta]{(1,-4)}{}	\CL[magenta]{(2,-4)}{}	\CL[magenta]{(3,-4)}{}	\CL[magenta]{(4,-4)}{}	\CL[magenta]{(5,-4)}{}	\CL[magenta]{(6,-4)}{}	\CL[magenta]{(7,-4)}{}	\CL[magenta]{(8,-4)}{}	\CL[magenta]{(9,-4)}{}	\CL[magenta]{(10,-4)}{}	\CL[magenta]{(11,-4)}{}
	\CL[magenta]{(0,-5)}{}	\CL[magenta]{(1,-5)}{}	\CL[magenta]{(2,-5)}{}	\CL[magenta]{(3,-5)}{}	\CL[magenta]{(4,-5)}{}	\CL[magenta]{(5,-5)}{}	\CL[magenta]{(6,-5)}{}	\CL[magenta]{(7,-5)}{}	\CL[magenta]{(8,-5)}{}	\CL[magenta]{(9,-5)}{}
	\color{gray}
	\CL{(0,-6)}{}	\CL{(1,-6)}{}	\CL{(2,-6)}{}	\CL{(3,-6)}{}	\CL{(4,-6)}{}	\CL{(5,-6)}{}	\CL{(6,-6)}{}	\CL{(7,-6)}{}	\CL{(8,-6)}{}
	\CL{(0,-7)}{}	\CL{(1,-7)}{}	\CL{(2,-7)}{}	\CL{(3,-7)}{}	\CL{(4,-7)}{}	\CL{(5,-7)}{}	\CL{(6,-7)}{}
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	\CL{(0,-9)}{}	\CL{(1,-9)}{}	\CL{(2,-9)}{}	\CL{(3,-9)}{}	\CL{(4,-9)}{}
	\color{gray}
	\CL{(0,-10)}{}	\CL{(1,-10)}{}	\CL{(2,-10)}{}	\CL{(3,-10)}{}
	\CL{(0,-11)}{}	\CL{(1,-11)}{}
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	\linethickness{.8mm}
	\qbezier[20](2,1)(7,1)(12,1)
	\qbezier[4](-3,-4)(-2,-4)(-1,-4)
	\qbezier[4](-3,-8)(-2,-8)(-1,-8)
	\qbezier[4](0,-13)(1,-13)( 2,-13)
	\put(0,-2){\line(1,0){14}}
	\put(0,-6){\line(1,0){9}}
	\put(0,-10){\line(1,0){4}}
	\linethickness{.6mm}
	\qbezier[20](9,-6)(11.5,-6)(14,-6)
	\qbezier[16](14,-2)(14,-4)(14,-6)
	\linethickness{.4mm}
	\qbezier(9,-6)(10,-7)(11,-7)\put(11.5,-7){\makebox(0,0){$5$}}\qbezier(12,-7)(13,-7)(14,-6)
	\qbezier(14,-2)(15,-2.5)(15,-3)\put(15,-4){\makebox(0,0){$4$}}\qbezier(15,-5)(15,-5.5)(14,-6)
	\linethickness{.2mm}
	\qbezier(13,-5)(10.5,-7)(8,-9)
	\put(8,-9){\vector(-4,-3){0}}	\put(9,-9){\makebox(0,0)[l]{$(4,-5)$}}
%    \includegraphics[width=11cm]{content.eps}
    \end{picture}
%    \qquad
%    \includegraphics[width=7cm]{periodicDiagram.eps}
  \end{center}
  \caption{
    A periodic diagram of period $\peri=(4,-5)$.
  } %with respect to $\lm=(9,7,7,5)$.
  %\caption{Contents on cylindric diagram associated with $\lm=(5,4,4,2) \in \ptn_{4,4}$.}\label{fig1}
  \label{periodic diagram}
\end{figure}

\begin{comment}
\begin{Def}
  %Let $m,\ell\in\Z_{\geqq1}$.
  Let $m,\ell \in \Z_{\geqq1}$.
  A non-increasing sequence $\lambda=(\lambda_1,\dots,\lambda_m)$ of (possibly negative) integers is called a \alert{generalized partition of length $m$}.
  %and it
  %is said to be \alert{$\ell$-restricted} % generalized partition of length $m$}
  %if it satisfies %the following conditions:
  For $\peri=(m,-\ell) \in \Z_{\geqq1} \times \Z_{\leqq-1}$,
  we denote by $\ptn_{\peri}$ the set of %$\ell$-restricted 
  generalized partitions of length $m$ satisfying 
  $$
    %\lambda_1 \geqq \cdots \geqq \lambda_m,\quad
    \lambda_1-\lambda_m \leqq \ell.
  $$
\end{Def}
%\def\sinf{\frac{\infty}{2}}


%We call $\lm$ the \alert{fundamental domain} o $\perid{\lm}$.


For %a generalized partition  
$\lambda=(\lambda_1,\dots,\lambda_m) \in \ptn_\peri$,
% \in \P_{m,\ell}$.
we define
%
\begin{align*}
  \sem\lambda &= \{ (a,b) \in \mathbb{Z}^2 \mid 1 \leqq a \leqq m,\ b \leqq \lambda_a \},
  &
  \per{\lambda} &
  %=\per{\lm}_{(m,-\ell)}
  = \sem\lambda + \mathbb{Z} \peri,
  &
  \cyl\lm &
  %=\cyl\lm_{(m,-\ell)}
  =\pi(\per\lm).
\end{align*}
%
Note that  
$\sem\lambda = \per{\lambda} \cap ([1,m] \times \mathbb{Z})$ 
and $\sem\lm$ is a fundamental domain of $\per\lambda$ with respect to the action of
$\mathbb{Z} (m,-\ell)$. %(see \Cref{fig1}).
\end{comment}

\begin{comment}
  It is easy to see that $\per{\lm}$ is a periodic diagram of period  $\peri=(m,-\ell)$, and 
  that any diagram of period $(m,-\ell)$ is of the form $\per\lm$ for some 
  $\lm \in \ptn_{\peri}$.


%It is easy to see the fllowing:

%\begin{proposition}
%$\mathrm{(1)}$ 
If  $\lm\in \ptn_{\peri}$ then $\per{\lambda}$ is a periodic diagram of period  $\peri$
and $\cyl\lm$ is a cylindric diagram.
Moreover,
any periodic (resp. cylindric) diagram of period $\peri$
is of the form $\per\lambda$ (resp. $\cyl\lm$) for some $\lambda \in \ptn_{\peri}$.
\end{comment}




\begin{comment}
For a poset $P$ and its order filter $J$,
we denote the set-difference $P\setminus J$ also by $P/J$.
It is easy to see the following:
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{prop}\label{prop:skew}
  For a  subset $\xi$ of $\cyli_\peri$, the following conditions are equivalent $:$
  \begin{enumerate}
    \item
    $\xi$ is a proper order ideal of a cylindric diagram in $\cyli_\peri$.
    \item
    $\xi$ is a set-difference $\cdiag/\eta$ of
    two cylindric diagrams $\cdiag,\eta$ in $\cyli_\peri$ with $\cdiag\supset \eta$.
    \item
    $\xi$ is an intersection of a proper order ideal and 
    a proper order filter of $\cyli_\peri$.
    \item
    $\xi$ is a finite subset of $\cyli_\peri$
    and satisfies the following condition:
    $$x,y \in \xi \implies [x,y] \subset \xi.$$
    \item
    $\xi$ is a finite subset of $\cyli_\peri$
    and satisfies the following condition:
    $$x,x+(1,1) \in \xi \implies
    x+(0,1),x+(1,0) \in \xi\ \ (\text{the skew property})$$
  \end{enumerate}
\end{prop}
\begin{Def}
  A subset $\xi$ of $\cyli_\peri$ is called a \emph{cylindric skew diagram}
  if it satisfies one of the conditions (i)--(v) in \Cref{prop:skew}.
\end{Def}
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{comment}
\begin{Def}
  A finite order ideal $\xi$ of $\cyli_\peri$ is called a \alert{cylindric skew diagram}.
  We  denote the set of finite order ideals of $\cdiag$
  by $\I(\cdiag)$ and regard it
  as a poset with the inclusion relation.
%We will describe the poset structure of $(\I(\cdiag),\subseteq)$
%in terms of the semi-infinite word $w_\cdiag$.
\end{Def}
\end{comment}
%  Note that $\I(\cdiag)=\bigsqcup_{n=0}^\infty \I_n(\cdiag)$,
%  where we put 
%  $$\I_n(\cdiag)=\{\ideal\in\I(\cdiag)\mid |\ideal|=n\}.$$

\begin{figure}[!ht]
  \begin{center}
    \setlength{\unitlength}{3mm}
    \begin{picture}(14,13)(0,-12)
	\color{gray}
																							\CL{(11,0)}{}	\CL{(12,0)}{}	\CL{(13,0)}{}	\CL{(14,0)}{}	\CL{(15,0)}{}	\CL{(16,0)}{}
																							\CL{(11,-1)}{}	\CL{(12,-1)}{}	\CL{(13,-1)}{}	\CL{(14,-1)}{}
	\color{black}
																									\CL[magenta]{(8,-2)}{}	\CL[magenta]{(9,-2)}{}	\CL[magenta]{(10,-2)}{}	\CL[magenta]{(11,-2)}{}	\CL[magenta]{(12,-2)}{}	\CL[magenta]{(13,-2)}{}
																			\CL[magenta]{(6,-3)}{}	\CL[magenta]{(7,-3)}{}	\CL[magenta]{(8,-3)}{}	\CL[magenta]{(9,-3)}{}	\CL[magenta]{(10,-3)}{}	\CL[magenta]{(11,-3)}{}
																			\CL[magenta]{(6,-4)}{}	\CL[magenta]{(7,-4)}{}	\CL[magenta]{(8,-4)}{}	\CL[magenta]{(9,-4)}{}	\CL[magenta]{(10,-4)}{}	\CL[magenta]{(11,-4)}{}
																			\CL[magenta]{(6,-5)}{}	\CL[magenta]{(7,-5)}{}	\CL[magenta]{(8,-5)}{}	\CL[magenta]{(9,-5)}{}
	\color{gray}
							\CL{(3,-6)}{}		\CL{(4,-6)}{}		\CL{(5,-6)}{}		\CL{(6,-6)}{}		\CL{(7,-6)}{}		\CL{(8,-6)}{}
			\CL{(1,-7)}{}	\CL{(2,-7)}{}	\CL{(3,-7)}{}	\CL{(4,-7)}{}	\CL{(5,-7)}{}	\CL{(6,-7)}{}
			\CL{(1,-8)}{}	\CL{(2,-8)}{}	\CL{(3,-8)}{}	\CL{(4,-8)}{}	\CL{(5,-8)}{}	\CL{(6,-8)}{}
			\CL{(1,-9)}{}	\CL{(2,-9)}{}	\CL{(3,-9)}{}	\CL{(4,-9)}{}
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	\qbezier[20](2,1)(7,1)(12,1)
	\qbezier[4](-3,-4)(-2,-4)(-1,-4)
	\qbezier[4](-3,-8)(-2,-8)(-1,-8)
	\qbezier[4](0,-13)(1,-13)( 2,-13)
	\put(11,-2){\line(1,0){3}}
	\put(6,-6){\line(1,0){3}}
	\put(1,-10){\line(1,0){3}}
%	\linethickness{.6mm}
%	\qbezier[20](9,-6)(11.5,-6)(14,-6)
%	\qbezier[16](14,-2)(14,-4)(14,-6)
%	\linethickness{.4mm}
%	\qbezier(9,-6)(10,-7)(11,-7)\put(11.5,-7){\makebox(0,0){$5$}}\qbezier(12,-7)(13,-7)(14,-6)
%	\qbezier(14,-2)(15,-2.5)(15,-3)\put(15,-4){\makebox(0,0){$4$}}\qbezier(15,-5)(15,-5.5)(14,-6)
%	\linethickness{.2mm}
%	\qbezier(13,-5)(10.5,-7)(8,-9)
%	\put(8,-9){\vector(-4,-3){0}}	\put(9,-9){\makebox(0,0)[l]{$(4,-5)$}}
    \end{picture}
%    \includegraphics[width=8cm]{skewCylindricDiagram.eps}
  \end{center}
  \caption{A cylindric skew diagram.}\label{fig:skew diag}
\end{figure}










%\input{NSTfpsac_sec2_subsec2.tex}%standard tableau
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{NSTfpsac_sec2_subsec3.tex}%content bottom set
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Content map and bottom set} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\pdiag$ be a periodic diagram of period $\omega$.
Define the \emph{content map} %(or \emph{coloring map}) %(or \emph{residue map})
$$\con:\pdiag \to \Z$$ by $\con(a,b) = b-a.$
Put $\kappa=|\con(\omega)|$.
%
Let $\cdiag=\pi(\pdiag)$.
Since $\con(x + \omega) = \con(x)-\kappa$, 
the content map $\con$ induces the map
$$\cdiag\to \Z/\kappa\Z,$$
which we denote by the same symbol $\con$.
%(See \Cref{content}, where the number in each cell is the content with modulo 8.)
%(the $\con$ plays a role of ``d-complete coloring''.)
It is easy to show the following:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{prop:content}
  %Let $\cdiag$ be a cylindric diagram.
  For $x,y\in \cdiag$, the followings hold:
  %\begin{enumerate}
  %\begin{description}
    %\item[1.]
    
    
   \medskip\noindent 
   1.\ If  $\con(x) - \con(y) \equiv 0, \pm1 \mod \kappa$,
    then $x$ and $y$ are comparable.
%    \item
%    Let $[x,y]$ be a chain.
%If $\#[x,y] \leqq \kappa$ then $\con(z)$'s $(z \in [x,y])$ are distinct.
%If $\#[x,y] = \kappa+1$  then $\con(x)-\con(y)=\pm\kappa$.
%
%\item If $[x,y]$ is a $d_k$-interval, then $\con(x)=\con(y)$.
%
%\item 
 
   \medskip\noindent
    %\item[2.]
    2.\ If $x$ is covered by $y$, then $\con(x)-\con(y)\equiv\pm 1 \mod \kappa$.
  %\end{description}
  %\end{enumerate}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}
  By %\Cref{prop:skew}  and
  \Cref{prop:content},
  cylindric diagrams are infinite (locally finite)
  ``$\Z/\kappa\Z$-colored $d$-complete posets'' in the sense of \cite{Str2020, Str2021}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Let $i \in \Z/\kappa\Z$.
By \Cref{prop:content} (1),
the inverse image $\con^{-1}(i)$ is non-empty totally ordered subset of $\cdiag$.
Let $\cmin_i$ denote the minimum element in $\con^{-1}(i)$.


\begin{definition}
  %For a cylindric diagram $\cdiag$,
  Define the \emph{bottom set} $\Gamma$ of $\cdiag$ by
  $$
    \Gamma = \{ \cmin_i \mid i \in \Z/\kappa\Z\}.
  $$
  The yellowed cells indicate the elements of 
  bottom sets in each diagram of
  %Examples of bottom sets appear in 
  Fig. \ref{fig:content}
  %(the yellowed cells).
  %%%represent the bottom set.)
\end{definition}


\begin{comment}
\begin{figure}[!ht]
  \begin{center}
    \includegraphics[width=11cm]{content.eps}
  \end{center}
  \caption{Contents %on periodic diagram associated with
  %$\lm=(5,4,4,2) \in \ptn_{4,4}$.
  %Yellowed cells represent the bottom set.
  %The number in each cell is the content with modulo 8.
  }
  \label{content}
\end{figure}
\end{comment}
\smallskip
\begin{figure}[!ht]
  \setlength{\unitlength}{5mm}
  \begin{center}
 \hspace*{1cm}
  \begin{picture}(14,8)(0,-8)
	\color{gray}
	\CL{(0,0)}{8}	\CL{(1,0)}{0}	\CL{(2,0)}{1}	\CL{(3,0)}{2}	\CL{(4,0)}{3}	\CL{(5,0)}{4}	\CL{(6,0)}{5}	\CL{(7,0)}{6}	\CL{(8,0)}{7}	\CL{(9,0)}{8}	\CL{(10,0)}{0}	\CL{(11,0)}{1}	\CL[yellow]{(12,0)}{2}	\CL[yellow]{(13,0)}{3}	\CL[yellow]{(14,0)}{4}
	\CL{(0,-1)}{7}	\CL{(1,-1)}{8}	\CL{(2,-1)}{0}	\CL{(3,-1)}{1}	\CL{(4,-1)}{2}	\CL{(5,-1)}{3}	\CL{(6,-1)}{4}	\CL{(7,-1)}{5}	\CL{(8,-1)}{6}	\CL{(9,-1)}{7}	\CL{(10,-1)}{8}	\CL[yellow]{(11,-1)}{0}	\CL[yellow]{(12,-1)}{1}
	\color{black}
	\CL{(0,-2)}{6}	\CL{(1,-2)}{7}	\CL{(2,-2)}{8}	\CL{(3,-2)}{0}	\CL{(4,-2)}{1}	\CL{(5,-2)}{2}	\CL{(6,-2)}{3}	\CL{(7,-2)}{4}	\CL{(8,-2)}{5}	\CL[yellow]{(9,-2)}{6}	\CL[yellow]{(10,-2)}{7}	\CL[yellow]{(11,-2)}{8}
	\CL{(0,-3)}{5}	\CL{(1,-3)}{6}	\CL{(2,-3)}{7}	\CL{(3,-3)}{8}	\CL{(4,-3)}{0}	\CL{(5,-3)}{1}	\CL{(6,-3)}{2}	\CL{(7,-3)}{3}	\CL{(8,-3)}{4}	\CL[yellow]{(9,-3)}{5}
	\CL{(0,-4)}{4}	\CL{(1,-4)}{5}	\CL{(2,-4)}{6}	\CL{(3,-4)}{7}	\CL{(4,-4)}{8}	\CL{(5,-4)}{0}	\CL{(6,-4)}{1}	\CL[yellow]{(7,-4)}{2}	\CL[yellow]{(8,-4)}{3}	\CL[yellow]{(9,-4)}{4}
	\CL{(0,-5)}{3}	\CL{(1,-5)}{4}	\CL{(2,-5)}{5}	\CL{(3,-5)}{6}	\CL{(4,-5)}{7}	\CL{(5,-5)}{8}	\CL[yellow]{(6,-5)}{0}	\CL[yellow]{(7,-5)}{1}
	\color{gray}
	\CL{(0,-6)}{2}	\CL{(1,-6)}{3}	\CL{(2,-6)}{4}	\CL{(3,-6)}{5}	\CL[yellow]{(4,-6)}{6}	\CL[yellow]{(5,-6)}{7}	\CL[yellow]{(6,-6)}{8}
	\CL{(0,-7)}{1}	\CL{(1,-7)}{2}	\CL{(2,-7)}{3}	\CL{(3,-7)}{4}	\CL[yellow]{(4,-7)}{5}
	\color{black}
	\linethickness{.8mm}
	\qbezier[20](2,1)(7,1)(12,1)
	\qbezier[4](-3,-4)(-2,-4)(-1,-4)
	\qbezier[6](1,-9)(2.5,-9)( 4,-9)
	\put(0,-2){\line(1,0){12}}
	\put(0,-6){\line(1,0){7}}
%    \includegraphics[width=11cm]{content.eps}
  \end{picture}
  \hspace*{3cm}
  \begin{picture}(8,8)(0,-8)
	\color{gray}
	\CL{(0,0)}{0}	\CL{(1,0)}{1}	\CL{(2,0)}{2}	\CL{(3,0)}{3}	\CL{(4,0)}{0}	\CL{(5,0)}{1}	\CL{(6,0)}{2}	\CL[yellow]{(7,0)}{3}
	\CL{(0,-1)}{3}	\CL{(1,-1)}{0}	\CL{(2,-1)}{1}	\CL{(3,-1)}{2}	\CL{(4,-1)}{3}	\CL[yellow]{(5,-1)}{0}	\CL[yellow]{(6,-1)}{1}	\CL[yellow]{(7,-1)}{2}
	\color{black}
	\CL{(0,-2)}{2}	\CL{(1,-2)}{3}	\CL{(2,-2)}{0}	\CL{(3,-2)}{1}	\CL{(4,-2)}{2}	\CL[yellow]{(5,-2)}{3}
	\CL{(0,-3)}{1}	\CL{(1,-3)}{2}	\CL{(2,-3)}{3}	\CL[yellow]{(3,-3)}{0}	\CL[yellow]{(4,-3)}{1}	\CL[yellow]{(5,-3)}{2}
	\color{gray}
	\CL{(0,-4)}{0}	\CL{(1,-4)}{1}	\CL{(2,-4)}{2}	\CL[yellow]{(3,-4)}{3}
	\CL{(0,-5)}{3}	\CL[yellow]{(1,-5)}{0}	\CL[yellow]{(2,-5)}{1}	\CL[yellow]{(3,-5)}{2}
	\color{black}
	\linethickness{.8mm}
	\qbezier[8](2,1)(4,1)(6,1)
	\qbezier[4](-3,-3)(-2,-3)(-1,-3)
	\qbezier[4](1,-7)(2,-7)( 3,-7)
	\put(0,-2){\line(1,0){6}}
	\put(0,-4){\line(1,0){4}}
%    \includegraphics[width=11cm]{content.eps}
  \end{picture}
  \end{center}
  \caption{Contents  and bottom sets
  %on periodic diagram associated with
  %$\lm=(5,4,4,2) \in \ptn_{4,4}$.
  %Yellowed cells represent the bottom set.
  %The number in each cell is the content with modulo 8.
  }
  \label{fig:content}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input{NSTfpsac_sec2_subsec4_v2.tex}	%root system
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Roots, words and weights}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Root systems and affine Weyl groups of type \texorpdfstring{$A_{\kappa-1}^{(1)}$}{A(kappa-1)(1)}}
\label{sec:rootsystem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\kappa \in \Z_{\geqq2}$. 
In the rest, we often identify $\Z/\kappa\Z$ with $\{0,1,\dots,\kappa-1\}$.
%
We use the following notations:
\begin{align*}
	\h	& :\text{the cartan subalgebra of \( \widehat{\mathfrak{sl}_\kappa} \)},	\\
	\h^*	& :\text{the dual of $\h$},	\\
	\bra \ , \ \ket	&:\h^* \times \h \to \C : \text{the natrual pairing},	\\
	\Pi	&= \{ \alpha_0\,,\alpha_1\,,\cdots,\alpha_{\kappa-1} \} (\subset \h^*) : \text{the set of simple roots},	\\
	\Pi^\vee&= \{ \alpha_0^\vee,\alpha_1^\vee,\cdots,\alpha_{\kappa-1}^\vee \} (\subset \h)\,\,\, : \text{the set of simple coroots},
\end{align*}
\begin{comment}
Let $\h$ be a $(\kappa+1)$-dimensional vector space
and choose elements $\al^\vee_i\ (i\in\Z/\kappa\Z)$ and $d$
of $\h$ so that
$$\{\al^\vee_0,\al^\vee_1,\dots,\al^\vee_{\kappa-1},d\}$$ 
forms a basis for $\h$.
%
Let $\h^*$ be the dual space of $\h$.
%
Define elements 
%$\al_0,\al_1,\dots,\al_{\kappa-1}$ 
$\al_j\ (j\in\Z/\kappa\Z)$
and $\fdwt_0$ of $\h^*$ by
\begin{align*}
  &\bra \al_j,\al^\vee_i \ket = a_{ij},\quad
  \bra \fdwt_0,\al^\vee_i\ket = \delta_{i0}
  \quad (i,j\in\Z/\kappa\Z)
  %(i=0,1,\dots,\kappa-1)
  ,\\
  %\quad
  &\bra \al_j,d \ket = \delta_{j0},\quad
  \bra \fdwt_0,d \ket =0,
\end{align*}
where $\bra\cdot,\cdot\ket:\h^*\times\h\to\Z$ is the natural pairing
and
\end{comment}
The integers $a_{ij} \,\,\,(i,j \in \Z/\kappa\Z) $ are defined by
\begin{equation*}
  a_{ij}=
  \begin{cases}
    2 & \text{if $i=j$} \\
    -1 & \text{if $i-j= \pm 1$} \\
    0 & \text{otherwise}
  \end{cases}
  \quad
  \text{for $\kappa\geqq3$},
  \qquad
  a_{ij}=
  \begin{cases}
    2 & \text{if $i=j$} \\
    -2 & \text{if $i \neq j$}
  \end{cases}
  \quad
  \text{for $\kappa=2$.}
\end{equation*}
The pairing satisfies
\( \bra \alpha_j ,\, \alpha_i^\vee \ket = a_{ij} \).
\begin{comment}
%Then $\{\al_0,\al_1,\dots,\al_{\kappa-1},\fdwt_0\}$ forms a basis for $\h^*$.
%
Define $\fdwt_i \in \h^*$ ($i =1,2,\dots,\kappa-1$) by
$$
  \bra\fdwt_i,\al^\vee_j\ket = \delta_{ij},\quad \bra\fdwt_i,d\ket=0
  \quad (j\in\Z/\kappa\Z).
  %(j=0,1,\dots,\kappa-1)
$$
The weights $\fdwt_0,\fdwt_1,\dots,\fdwt_{\kappa-1}$ are called fundamental weights.
%
\end{comment}
Put $\delta = \al_0+\al_1+\cdots+\al_{\kappa-1}$
which is called the null root.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%	simple reflections and Weyl group
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For $i \in \Z/\kappa\Z
%[0,\kappa-1]
$,
define the simple reflection $s_i \in GL(\h^*)$  by
\begin{align*}
  &s_i(\zeta) = \zeta - \bra \zeta,\al^\vee_i \ket \al_i \quad (\zeta \in \h^*).
%\\
%&s_i(\eta^\vee) = \eta^\vee - \bra \al_i,\eta^\vee \ket \al^\vee_i \quad (\eta^\vee \in \h).
\end{align*}
%%
%It is called a simple reflection.
Define the \emph{affine Weyl group} $W$ of type $A_{\kappa-1}^{(1)}$
% $W$ be
as the subgroup of $GL(\h^*)$
generated by simple reflections: % $\{s_i \mid i \in [0,\kappa-1]\}$:
$$W=\langle s_i\mid i\in\Z/\kappa\Z\rangle.$$
%The group $W$ is called 
%
Define the action of $W$ on $\h$ by
\( s_i(h) = h - \bra \al_i,h \ket \al^\vee_i \quad (h \in \h) \).
The following is well-known:
%We remark that the group $W$ has the following relations:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
  The group $W$ has the following fundamental relations$:$
  \begin{align}
    s_i^2&=1, \label{eq:rel1} \\
    s_is_j&=s_js_i \quad (i-j \neq 0,\pm1), \label{eq:rel2} \\
    s_is_{i+1}s_i&=s_{i+1}s_is_{i+1}. \label{eq:rel3}
  \end{align}
\end{proposition}

We put 
\begin{align*}
%  \Pi &= \{\al_0,\al_1,\dots,\al_{\kappa-1}\},&
%  \Pi^\vee &= \{\al^\vee_0,\al^\vee_1,\dots,\al^\vee_{\kappa-1}\},
%  \\
  Q&=\left\{\sum_{i\in\Z/\kappa\Z}c_i \al_i\ \middle|\ c_i\in\Z\right\},&
  Q_+&=\left\{\sum_{i\in\Z/\kappa\Z}c_i \al_i\ \middle|\ c_i\in\Z_{\geqq 0}\right\}.
\end{align*}
The set $Q$ is called the root lattice.
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
Put
\begin{align*}
  R &= W\Pi\subset\h^*, & R^\vee &= W\Pi^\vee\subset\h. %\\
  %&\tilde{R}=R\sqcup\Z\delta \quad (\tilde{R}^\vee=R^\vee\sqcup\Z\delta^\vee).
\end{align*}
Then %$\tilde{R}$ is the affine root system, %(resp. affine coroot system),
$R$ (resp. $R^\vee$) is the set of real roots (resp. coroots) 
and $R\sqcup\Z\delta$ is the affine root system. %(resp. imaginary roots).
%
Define the set $R_+$ of positive real roots and the set $R_-$ of negative real roots by
$$R_+ =R\cap Q_+= \left\{ \sum_{i=0}^{\kappa-1} c_i \al_i \in R \mid c_i \in \Z_{\geqq0}\right\}, \quad
 R_-= \left\{ \sum_{i=0}^{\kappa-1} c_i \al_i \in R \mid c_i \in \Z_{\leqq0}\right\}.$$
For $\beta=\sum_{i=0}^{\kappa-1} k_i\al_i\in R,$
define $\beta^\vee=\sum_{i=0}^{\kappa-1} k_i\al^\vee_i \in R^\vee.$
Then the correspondence $\beta\mapsto \beta^\vee$ gives a bijection
$R\to R^\vee$.
%Similarly,
Define the set of positive (resp. negative) coroots $R^\vee_+$ (resp. $R^\vee_-$)
as the image of $R_+$ (resp. $R_-$) by this bijection.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For $w  \in W$, 
we define the \emph{length} $\ell(w)$ of $w$
as the smallest $r$ for which an expression 
\( w = s_{i_1}s_{i_2} \cdots s_{i_r} \in W\ \ (i_j \in \Z/\kappa\Z) \)
exists.
%and call the reduced expression.
An expression $w=s_{i_1}s_{i_2} \cdots s_{i_r}$ is said to be \emph{reduced}
 if $\ell(w)=r$.
%By $\RE(w)$, we denote the set of reduced expressions for $w$.
The set $ R(w)= R_+ \cap w R_- $ is called 
the \emph{inversion set} of $w$.
It is known for any reduced expression $w=s_{i_1}s_{i_2}\cdots s_{i_\ell}$
that $\ell(w)=|R(w)|$ and
\begin{equation*}
  R(w)=\{\al_{i_1}, s_{i_1}\al_{i_2},s_{i_1}s_{i_2}\al_{i_3},
  \dots,s_{i_1}s_{i_2}\cdots s_{i_{\ell-1}} \al_{i_\ell}\} .
\end{equation*}




\begin{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For $i,j\in\Z$ with $i<j$, we define
$$\al_{ij}=\sum_{i\leqq k\leqq j-1}\al_{\bar{k}},$$
where $\bar{k}=k\!\!\mod\kappa\Z\in\Z/\kappa\Z$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The followings are well-known:
\begin{align}
R_+&=\{\al_{ij}\mid i<j,\ j-i\notin \kappa\Z\}\label{eq:R+}
\\
&=\{\al_{ij}+k\delta\mid
1\leqq i<j\leqq\kappa,\ k\geqq 0\}\sqcup
\{-\al_{ij}+k\delta\mid
1\leqq i<j\leqq\kappa,\ k\geqq 1\},\label{eq:R+2}\\
R_-&=-R_+,\ \
%\{-\al_{ij}-k\delta\mid
%1\leqq i<j\leqq\kappa,\ k\geqq 0\}\sqcup
%\{\al_{ij}-k\delta\mid
%1\leqq i<j\leqq\kappa,\ k\geqq 1\},\label{eq:R-}\\
%&=\{-\al_{ij}\mid i<j,\ j-i\notin \kappa\Z\}\\
R=R_+\sqcup R_-.\notag
%\label{eq:R}
%=\{\al_{ij}+k\delta\mid
%1\leqq i\neq j\leqq \kappa,\ k\in\Z\}.\label{eq:R}
\end{align}
%where $\al_{ij}=\al_i+\al_{i+1}+\cdots+\al_{j-1}$ for $i<j$
%and $\al_{ij}=-\al_{ji}$ for $i>j$.
From the description of $R$ above, the following two lemmas  follow
easily  and they will be used later: 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{lem:addnull}
  %It holds that
  %
  If $\al\in R$, then $\al+k\delta\in R$ for all $k\in\Z$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{lem:moddelta}
  Let $\alpha \in R \sqcup \Z\delta$ and $\beta \in R$.
  %Then $\bra \alpha, \beta \ket = 2$ if and only if $\alpha \equiv \beta \mod \delta$.
  Then $\bra\alpha,\beta^\vee\ket=2$ if and only if $\alpha \equiv \beta \mod \delta$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{comment}





%\input{NSTfpsac_sec3_subsec1.tex}
\NP
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Colored hook length}\label{ss:hook}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Then, for the reduced semi-infinite word 
%$w_{\cdiag,\lex}$, we have

%Moreover, by \Cref{pr;hk=hkr}, we have
%In particular, t will be
%We will denote $R(w_{\cdiag,\lex})$ just by $R(w_\cdiag)$ below.


In this section, we will introduce colored hook length,
which is 
a key ingredient in this paper.
%
%Fix $\kappa, m,\ell\in\Z_{\geqq1}$ with $\kappa=m+\ell$
%and let $\cdiag$ be a cylindric diagram in $\cyli_{(m,-\ell)}$.
%
In the rest of this paper, we use the following notations:
$$\al(x)=\al_{\con(x)},\ \ s(x)=s_{\con(x)}\ \ \text{for } x\in \cdiag.$$
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{df:coloredhooklength}
  For $x\in \cdiag$, put
  \begin{align*}
    \arm(x)&=\{x+(0,k)\in\cdiag\mid k\in\Z_{\geqq1}\},	&
    \leg(x)&=\{x+(k,0)\in\cdiag\mid k\in\Z_{\geqq1}\},
  \end{align*}
  and define
  %The "multiset" 
  %\begin{definition}
  \begin{equation*}
    \hk(x)=\al(x)+\sum_{y\in\arm(x)}\al(y)+\sum_{y\in\leg(x)}\al(y).
  \end{equation*}
  %where $\al(x)=\al_{\con(x)}$.
  We call $\hk(x)$ the {\it colored hook length} at $x$.
  It is straightforward to see that the colored hook length is a positive real root \( \hk(x) \in R_+ \).
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% as before.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[!ht]
  \setlength{\unitlength}{5mm}
  \begin{center}
%    \includegraphics[width=13.5cm]{hook.eps}
    \begin{picture}(14,12)(0,-12)
	\color{gray}
	\CL{(0,0)}{}	\CL{(1,0)}{}	\CL{(2,0)}{}	\CL{(3,0)}{}	\CL{(4,0)}{}	\CL{(5,0)}{}	\CL{(6,0)}{}	\CL{(7,0)}{}	\CL{(8,0)}{}	\CL{(9,0)}{}	\CL{(10,0)}{}	\CL{(11,0)}{}	\CL{(12,0)}{}	\CL{(13,0)}{}	\CL{(14,0)}{}	\CL{(15,0)}{}	\CL{(16,0)}{}
	\CL{(0,-1)}{}	\CL{(1,-1)}{}	\CL{(2,-1)}{}	\CL{(3,-1)}{}	\CL{(4,-1)}{}	\CL{(5,-1)}{}	\CL{(6,-1)}{}	\CL{(7,-1)}{}	\CL{(8,-1)}{}	\CL{(9,-1)}{}	\CL{(10,-1)}{}	\CL{(11,-1)}{}	\CL{(12,-1)}{}	\CL{(13,-1)}{}	\CL{(14,-1)}{}
	\color{black}
	\CL{(0,-2)}{}	\CL{(1,-2)}{}	\CL{(2,-2)}{}	\CL{(3,-2)}{}	\CL{(4,-2)}{}	\CL{(5,-2)}{}	\CL{(6,-2)}{}	\CL{(7,-2)}{}	\CL{(8,-2)}{}	\CL{(9,-2)}{}	\CL{(10,-2)}{}	\CL{(11,-2)}{}	\CL{(12,-2)}{}	\CL{(13,-2)}{}
	\CL{(0,-3)}{}	\CL{(1,-3)}{}	\CL{(2,-3)}{}	\CL{(3,-3)}{}	\CL[red]{(4,-3)}{$x$}	\CL[cyan]{(5,-3)}{}	\CL[cyan]{(6,-3)}{}	\CL[cyan]{(7,-3)}{}	\CL[cyan]{(8,-3)}{}	\CL[cyan]{(9,-3)}{}	\CL[cyan]{(10,-3)}{}	\CL[cyan]{(11,-3)}{}
	\CL{(0,-4)}{}	\CL{(1,-4)}{}	\CL{(2,-4)}{}	\CL{(3,-4)}{}	\CL[green]{(4,-4)}{}	\CL{(5,-4)}{}	\CL{(6,-4)}{}	\CL{(7,-4)}{}	\CL{(8,-4)}{}	\CL{(9,-4)}{}	\CL{(10,-4)}{}	\CL{(11,-4)}{}
	\CL{(0,-5)}{}	\CL{(1,-5)}{}	\CL{(2,-5)}{}	\CL{(3,-5)}{}	\CL[green]{(4,-5)}{}	\CL{(5,-5)}{}	\CL{(6,-5)}{}	\CL{(7,-5)}{}	\CL{(8,-5)}{}	\CL{(9,-5)}{}
	\color{gray}
	\CL{(0,-6)}{}	\CL{(1,-6)}{}	\CL{(2,-6)}{}	\CL{(3,-6)}{}	\CL[green]{(4,-6)}{}	\CL{(5,-6)}{}	\CL{(6,-6)}{}	\CL{(7,-6)}{}	\CL{(8,-6)}{}
	\CL{(0,-7)}{}	\CL{(1,-7)}{}	\CL{(2,-7)}{}	\CL{(3,-7)}{}	\CL[green]{(4,-7)}{}	\CL{(5,-7)}{}	\CL{(6,-7)}{}
	\CL{(0,-8)}{}	\CL{(1,-8)}{}	\CL{(2,-8)}{}	\CL{(3,-8)}{}	\CL[green]{(4,-8)}{}	\CL{(5,-8)}{}	\CL{(6,-8)}{}
	\CL{(0,-9)}{}	\CL{(1,-9)}{}	\CL{(2,-9)}{}	\CL{(3,-9)}{}	\CL[green]{(4,-9)}{}
	\color{gray}
	\CL{(0,-10)}{}	\CL{(1,-10)}{}	\CL{(2,-10)}{}	\CL{(3,-10)}{}
	\CL{(0,-11)}{}	\CL{(1,-11)}{}
	\color{black}
	\linethickness{.8mm}
	\qbezier[20](2,1)(7,1)(12,1)
	\qbezier[4](-3,-4)(-2,-4)(-1,-4)
	\qbezier[4](0,-13)(1,-13)( 2,-13)
	\put(0,-2){\line(1,0){14}}
	\put(0,-6){\line(1,0){9}}
	\put(0,-10){\line(1,0){4}}
%    \includegraphics[width=11cm]{content.eps}
    \end{picture}
  \end{center}
  \caption{ The sets {\color{green}$\arm(x)$} and {\color{cyan}$\leg(x)$} for $x$ in the cylindric diagram.}
  \label{fig:coloredhook}
\end{figure}




\begin{comment}
For $x\in \mathcal{C}_{(m,-\ell)}\setminus \cdiag$, we set $\hk(x)=0$ for convenience. 
%\end{definition}
It is easy to see that for $x\in\cdiag$
$$\hk(x-(0,\ell))=\hk(x-(m,0))=\hk(x)+\delta$$
and
\begin{equation}\label{eq:hk=al}
\hk(x)=\al_{ij}\ \text{for some integers }i<j.
\end{equation}
% and thus $\hk(x)\in R_+$ by \eqref{eq:R+}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example} (See \Cref{fig:coloredhook}.)\ 
  Let $\peri=(4,-5)$. Then $\lm=(5,3,3,1) \in \ptn_\peri$.
  For a cell $x=\pi (2,-4)\in \cyl\lm$, we have $\con (x)=3 +9\Z\in \Z/9\Z$.
  The colored hook length at $x$ is 
  \begin{align*}
    \hk(x)
    &=\al_{-6}+(\al_{-5}+\al_{-4}+\al_{-3}+\al_{-2}+\al_{-1}+\al_0+\al_1)\\
    &+(\al_{-7}+\al_{-8}+\al_{-9}+\al_{-10}+\al_{-11}+\al_{-12})\\
    &=\al_3+(\al_4+\al_5+\al_6+\al_7+\al_8+\al_0+\al_1)+(\al_2+\al_1+\al_0+\al_8+\al_7+\al_6)\\
    &=\delta +\al_0+\al_1+\al_6+\al_7+\al_8,
  \end{align*}
  which can be expressed as 
  $\hk(x)=\al_{-12, 2}$.
%(Note thta  $\con(8,-4)=-12$ and
%$\con(2,3)=1$ in the periodic diagram $\per\lm$.)
\end{example}
\end{comment}


%%%%%%%%%%%%%%%%%%%%%
\begin{remark}
 % \begin{description}
 %   \item[1.]
 \quad 1.\ For $x\in\cdiag$, the ``multiset''
    $\mathrm{H}(x):=\{x\}\sqcup \arm(x)\sqcup \leg(x)$
    is a cylindric analogue of the hook at $x$.
    %and $\hk(x)$ is thought as a colored version of hook length.
    
   \medskip
   \noindent
   %\item[2.]
   2.\ A conjectural  hook formula 
    concerning the number of standard tableaux on cylindric skew diagrams is proposed
    in \cite{SuzToy2022}, where the hook length at $x\in \cdiag$ is given by 
    $|H(x)|=|\arm(x)|+|\leg(x)|+1$.
  %\end{description}
\end{remark}



%\input{NSTfpsac_sec3_subsec2.tex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Predominant weights and hooks}\label{sec:predom}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{definition}\label{df:predom}
  We take an integral weight $\predom_{\cdiag}\in \h^*$ such that
  \begin{gather}\label{eq:opo}
    \bra \predom_{\cdiag} ,\, \alpha_i^\vee \ket
    =
    \begin{cases}
      1 & \hbox{if }\cmin_i\in \btm_{\mathrm{max}} \\
      -1 & \hbox{if }\cmin_i\in \btm_{\mathrm{min}} \\
      0 & \hbox{otherwise}.
    \end{cases}% \notag
  \end{gather}
%\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



Note that maximal and minimal elements are lined up alternatively in $\btm$.
This implies that the weight $\predom_\cdiag$ is predominant, namely,
$\bra \predom_\cdiag,\al^\vee \ket\geqq -1$ for all $\al^\vee\in R^\vee_+$.
%
\begin{comment}
Note that 
$$ \bra \predom_\cdiag,\al^\vee\ket =-1 \iff
%$ if and only if
|\Supp (\al)\cap \btmmax|-|\Supp (\al) \cap \btmmin|=-1.$$
for $\al\in R_+$ with $N(\al)=0$.
\end{comment}
Define 
$$\DS(\predom_\cdiag)=\{\al\in R_+\mid \bra\predom_\cdiag,\al^\vee\ket=-1\}.$$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}[{\cite[Proposition 28]{NST2024}}]\label{thm:hkinR+}
  The correspondence $x\mapsto \hk(x)$ gives a bijection
  $$\hk:\cdiag\to \DS(\predom_\cdiag).$$
  %Moreover it holds that
  %$$\im \hk=\{\al\in R_+\mid \bra\predom_\cdiag,\al^\vee\ket=-1\}.$$
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%\input{NSTfpsac_sec3_subsec3.tex}
%\input{NSTfpsac_sec3_subsec3_v2.tex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Weyl group elements and their inversion sets}%{Weyl group elements associated with standard tableaux and inversion sets}
%\label{sec:inversion}


\begin{definition}
  Let $\zeta\in \h^*$ be an integral weight.
 % \begin{description}
   
   \medskip\noindent
    %\item[1]
    1.\ An element $w$ of $W$ is 
    said to be $\zeta$-{\it pluscule} if
    \( \bra \zeta,\al^\vee\ket=-1 \) for all \( \al\in R(w) \).

    \medskip\noindent
    %\item[2]
    2.\ An element $w$ of $W$ is 
    said to be $\zeta$-{\it minuscule} if
    \( \bra \zeta,\al^\vee\ket=1 \) for all \( \al\in R(w^{-1}) \).
  %\end{description}
\end{definition}

\begin{remark}
  %\begin{description}
   % \item[1.]
    \quad 1.\ An element $w\in W$ is $\zeta$-pluscule if and only if 
    $w$ is $(w^{-1}\zeta)$-minuscule.

   \medskip\noindent
    %\item[2.]
    2.\ An element $w \in W$ is said to be \emph{fully commutative} if any reduced 
    expression of $w$ can be obtained from any other by using only the relations \eqref{eq:rel2}.
    It is known that if $w$ is $\zeta$-minuscule for some integral weight $\zeta$
    then $w$ is fully commutative (\cite{Stem1996}).
  %\end{description}
\end{remark}

\begin{proposition}
  We symbolically put 
  \( R(w_\cdiag) := \bigcup_{w : \text{$\predom_\cdiag$-pluscule}} R(w) \).
  Then we have
  \begin{equation*}
    \DS(\predom_\cdiag) = R(w_\cdiag)
    %\bigcup_{w : \text{$\predom_\cdiag$-%pluscule}} R(w).
  \end{equation*}
\end{proposition}
%We often symbolically write \( R(w_\cdiag) := 
%\bigcup_{w : \text{$\predom_\cdiag$-pluscule}}
%R(w) \).
The set $R(w_\cdiag)$ can be thought as the ``inversion set'' associated with
the semi-infinite word \( w_\cdiag \).

We denote by \( \rbleq \) the weak right Bruhat order, namely,
\begin{equation*}
  v \rbleq w \iff \ell(v)+\ell(v^{-1}w)=\ell(w), \qquad(v,w \in W).
\end{equation*}
\begin{prop}\label{prop:order_ideal_w}
  Let \( v, w \in W \).
  If \( w \) is \( \predom_\cdiag \)-pluscule and \( v \rbleq w \),
  then \( v \) is also \( \predom_\cdiag \)-pluscule.
\end{prop}
We denote by \( [e,w_\cdiag) \) the set of $\predom_\cdiag$-pluscule elements.
\Cref{prop:order_ideal_w} justifies
using the notation \( [e,w_\cdiag) \).% forms an order ideal of \( (W,\rbleq) \).
%--- corrected :  the use > using









