\newcommand\pring{f}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Poset structure of cylindric diagrams}\label{sec:1stresult}\label{sec:inversion set}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%






Recall that $Q$ denote the root lattice: $Q=\bigoplus_{i\in\Z/\kappa\Z}\Z \al_i$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
  Define the partial order $\oord$ on $Q$ by
  \begin{equation*}
    \alpha \oord \beta \iff
    \beta-\alpha \in Q_+ = \bigoplus_{i\in\Z/\kappa\Z} \Z_{\geqq0} \al_i
    %=\left\{\sum_{i\in\Z/\kappa\Z}k_i \al_i\mid k_i\in\Z_{\geqq0}
    %\ (i\in\Z/\kappa\Z)\right\}.
    %\text{ is a sum of elements in }\Pi
  \end{equation*}
  The order $\oord$ is called the \emph{ordinary order}.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%
The restriction of the ordinary order defines a poset structure on $R(w_\cdiag)=\DS(\predom_\cdiag)$.

Let $\cdiag$ be a cylindric diagram in $\cyli_\peri$ with $|\peri|=\kappa$. 
We have introduced a poset structure on $\cdiag$
and also  have seen that the map $\hk$ gives a bijection 
between $\cdiag$ and $R(w_\cdiag)$.
Remark that this is not a poset isomorphism as seen in the following example:
%Note that $(R(w_\cdiag),\oord)$  is not equivalent to $(\cdiag,\leq)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}\label{ex:ord}
  In the cylindric diagram \( \cdiag \) in Fig.\, \ref{fig:content} (right),
  the cells \( x=\pi(3,3) \) and \( y=\pi(2,4) \) are incomparable.
  On the other hand, $\hk(x)=\delta+\al_2$ and $\hk(y)=\al_1+\al_2+\al_3$,
  and hence $\hk(x)-\hk(y)=\al_0+\al_2$.
  This implies $\hk(y)\oord\hk(x)$. 
\end{example}
%%%%%%%%%%%%%%%%%
%\end{remark}











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Partial orders on $\DS(\predom_\cdiag)=R(w_\cdiag)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will introduce a modified ordinary order $\neword$, 
%which is
%will turn out to be 
for which we will have
$(\cdiag,\leq)\cong (R(w_\cdiag),\neword)$.




Let $\btm=\{b_i\mid i\in\Z/\kappa\Z\}$
be the bottom set of $\cdiag$,
where $b_i$ is the element such that $\con(b_i)=i$.
Let $\btmmax$ (resp. $\btmmin$) denote the set of maximal (resp. minimal)
elements in $\btm$.

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{df:Deltaord}
  Define
  \[ \Pi_\cdiag = \Pi_\cdiag^0 \sqcup \Pi_\cdiag^\mathrm{arm} \sqcup \Pi_\cdiag^\mathrm{leg}. \]
  %
  Here,
  \begin{align*}\label{eq:delta}
    \Pi_{\cdiag}^0 
    &=\{\al(x) \mid x\in\btm\setminus (\btmmax\sqcup \btmmin)\},\\
    %\{ \alpha_i \in \Pi \mid \bra \predom_{\cdiag},\alpha^\vee_i \ket = 0 \}, \\
    %
    \Pi_{\cdiag}^\mathrm{arm} 
    &=\left\{\al(x)+\sum_{y\in \arm(x)}\al(y)\ \middle|\  x\in\btmmax\right\},\\
    %\{ \al[i,j] \middle\mid \bra\predom_\cdiag,\al^\vee_i\ket=1,\ \bra\predom_\cdiag,\al^\vee_j\ket=-1,\ 
    %\bra\predom_\cdiag,\al^\vee_k\ket=0\ \forall k \in ((i,j))\}, \\
    %
    \Pi_{\cdiag}^\mathrm{leg} 
    &=\left\{\al(x)+\sum_{y\in \leg(x)}\al(y)\ \middle|\ x\in\btmmax\right\}.
    %\{ \al[i,j] \mid \bra\predom_\cdiag,\al^\vee_i\ket=-1,\ \bra\predom_\cdiag,\al^\vee_j\ket=1,\ 
    %\bra\predom_\cdiag,\al^\vee_k\ket=0\ \forall k \in ((i,j))\}.
  \end{align*}
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%It is easy to see that $\bra\zeta_\cdiag^\circ,\al^\vee\ket=0$ for
%$\al\in \Pi_\cdiag$.
Note that %$\al[i,j]$ may be the null root and 
$\Pi_\cdiag \subset R_+ \sqcup \Z_{\geqq0}\delta$.

\begin{example}
  For the cylindric diagram described in Fig. \ref{fig:content} (left), we have
  \begin{align*}
    &\Pi_\cdiag^0=\{\al_3, \al_5, \al_7\},\\
    &\Pi_\cdiag^\mathrm{arm} =\{\al_6+\al_7+\al_8, \al_2+\al_3+\al_4, \al_0+\al_1\},\\
    &\Pi_\cdiag^\mathrm{leg} =\{\al_4+\al_5+\al_6, \al_1+\al_2, \al_0+\al_8\}.
  \end{align*}
\end{example}
\begin{comment}
\begin{example}
  Let $\lm=(n)$ and $\peri=(1,-n+1)$.
  Then, for the corresponding cylindric diagram $\cyl\lm$, we have
  \begin{align*}
    &\Pi_{\cyl\lm}^0=\{\al_1,\al_2,\dots,\al_{n-2}\},\\
    &\Pi_{\cyl\lm}^\mathrm{arm} =\{\delta\},\\
    &\Pi_{\cyl\lm}^\mathrm{leg} =\{\al_0+\al_{n-1}\}.
  \end{align*}
\end{example}
\end{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\label{df:neword}
  Define the partial order $\neword$ on $R(w_{\cdiag})$ by
  %as the transitive closure of the relations
  \begin{equation}\label{eq:neword}
    \alpha \neword \beta \iff
    \beta-\alpha \in\sum_{\gamma\in\Pi_\cdiag}\Z_{\geqq0}\gamma
    =\left\{
    \sum_{\gamma\in \Pi_\cdiag}
    k_\gamma \gamma \ \middle| \ k_\gamma\in\Z_{\geqq0}
    \ (\forall \gamma \in\Pi_\cdiag)\right\}.
    %\text{ is a sum of elements in }\Pi_\cdiag
  \end{equation}
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thm}[{\cite[Theorem 47]{NST2024}}]\label{th:cylord}
  The map 
  $$\hk: (\cdiag,\leq)\to (R(w_\cdiag),\neword)$$
  is a poset isomorphism.
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %Let $x,y\in\cdiag$. Then
  %$$x\leq y\ \Longleftrightarrow\  \hk(x)\neword \hk(y).$$
  %Let $\al,\beta\in R(w_\cdiag)$. Then
  %$$\al\neword \beta\Longleftrightarrow \al\tcord\beta$$
\end{thm}

\begin{definition}
  Let $\tcord$ denote the transitive closure of the relations
  \begin{equation}\label{eq:ord3}
    \alpha \tcord
    %\hspace{-1.2mm} \raisebox{0.3mm}{$\cdot$} 
    \,  \beta \ \text{whenever}\ \beta-\alpha \in \Pi_{\cdiag}.
  \end{equation}
\end{definition}
\begin{proposition}\label{prop:tcord}
  The partial order $\neword$ on $R(w_\cdiag)$ coincides with $\tcord$.
\end{proposition}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
  Let $\lex : \cdiag \to \Z_{\geqq1} $ be a standard tabuleau (or a linear extension), namely
  \( \lex \) be an order preserving bijection.
  Define a partial order $\preceq_\lex$ on $\Z_{\geqq1}$
  as the transitive closure of the relations
  \begin{align*}
    &\hbox{$a \preceq_\lex b$ whenever $a \leqq b$ and}
    \hbox{ either
    $s_{i_a}s_{i_b}=s_{i_b}s_{i_a}$
    or 
    $i_a =i_b$.}
  \end{align*}
  where $i_k=\con(\lex^{-1}(k))$ for $k\in\Z_{\geqq1}$.
  The poset $(\Z_{\geqq1},\preceq_\lex)$ is called the \emph{heap} of $w_{\cdiag}$.  
%(\cite{Stem2001}).
\end{definition}
%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{prop:cdiagandheap}
  Let $\cdiag$ be a cylindric diagram and 
  $\lex$ a standard tableau on $\cdiag$.
  %\smallskip\noindent
  %$\mathrm{(1)}$ 
  Then, the map $\lex:\cdiag\to\Z_{\geqq 1}$ gives a poset isomorphism  
  %between 
  $$(\cdiag,\le)\cong (\Z_{\geqq1},\preceq_\lex).$$
  %
  %\smallskip
\end{proposition}
The posets $(\Z_{\geqq1}, \preceq_\lex)$ are 
thought as semi-infinite analogue of heaps introduced by Stembridge \cite{Stem1996}.
Stembridge also introduced the heap order on the inversion sets.
%%%%%%%%%%%%%%%%%%%%%%%%%






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We treat a slightly modified version of heap order by Nakada \cite{Nak2008}.
% by Nakada. 
%In classical case. 
%Several orders in $R(w_\cdiag[n])$ which interpolate $(R(w_\cdiag[n])$
%and $\cdiag$ have been proposed (\cite{Stem1996}, Nakada).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}
  Define  a partial order $\hpord$ on $\DS(\predom_\cdiag)=R(w_{\cdiag})$
  as the transitive closure of the relations
  $$
    \hbox{$\alpha \hpord
    \beta$ whenever $\alpha \oord \beta$ and 
    $\bra\alpha,\beta^\vee\ket\neq 0$}.
    %\hbox{$\gam(a) \hpord_\lex \gam(b)$ whenever $a \leqq b$ and $\bra\gam(a),\gam(b)^\vee\ket\neq 0$}.
  $$
  %The order $\hpord_{\lex}$ is called the \alert{heap order} on $R(w_{\cdiag})=\DS{\predom_\cdiag}$.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{prop:cdiagandheap2}
  The partial order $\hpord$ on 
  $\DS(\predom_\cdiag)=R(w_{\cdiag})$
  %%%$\DS(\predom_\cdiag)$ 
  coincides with $\neword$.
\end{proposition}

In summary,  we get the following poset isomorphisms:
\begin{itemize}
  \item
  \( (\cdiag,\le) \stackrel{\lex}{\rightarrow} (\Z_{\geqq1},\hpord_\lex) \).
  \item
  \( (\cdiag,\le) \stackrel{\hk}{\rightarrow}  (R(w_\cdiag),\neword) = (R(w_\cdiag),\tcord) = (R(w_\cdiag),\hpord) \).
\end{itemize}
%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%


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


A poset isomorphism \( \cdiag \simeq R(w_\cdiag) \) (\Cref{th:cylord}) induces
a poset isomorphism \( \I(\cdiag) \simeq \I(R(w_\cdiag)) \).
\begin{proposition}
  Let \( w \) be a \( \predom_\cdiag \)-pluscule element.
  Then the inversion set $R(w)$ is a finite order ideal of \( (R(w_\cdiag),\neword) \).
\end{proposition}
\begin{thm}[{\cite[Theorem 58]{NST2024}}]\label{thm:ideal}
%  The poset \( (\I(\theta), \subset) \) is order-isomorphic to
%  \( ([e,w_\cdiag),\rbleq) \).
  The map 
  $ \Psi: ([e,w_{\cdiag}),\rbleq) \to (\I(R(w_\cdiag)),\subset) \simeq(\I(\cdiag),\subset) $
  given by $\Psi(w) = R(w) $ is a poset isomorphism.
%  The map 
%  $$\Phi: (\I(\cdiag),\subset) \to ([e,w_{\cdiag}),\rbleq)$$
%  given by $\Phi(\ideal) = w_{\ideal}$ is a poset isomorphism.
\end{thm}










