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\articlenumber{75}
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%% define your title in the usual way
\title[Order structure of shapes]{Order structure of shapes of predominant integral weights and cylindric Young diagrams}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%\author{Kento NAKADA}
\author{Kento Nakada \thanks{\href{mailto:nakada@okayama-u.ac.jp}{nakada@okayama-u.ac.jp}. Partially supported by Grant KAKENHI(B) 18H01435}\addressmark{1}}

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\address{\addressmark{1} Okayama University, Graduate School of Education, Okayama-shi, Japan}

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\received{\today}

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%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{
  In this paper, we give a generalization of d-complete posets to ``infinite'' d-complete posets.
  Our infinite d-complete posets are realized as a subset of coroots on the Kac--Moody Lie algebra,
  and has two different order relations in general.
  One is the ordinary order on coroots.
  The other is called heap order.
  We give a sufficient condition for both orders to coincide.
  As an application,
  we give a multivariate hook formula for cylindric Young diagrams.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{
  Dans ce papier, nous donnons une g\'en\'eralisation de posets d-complet \`a posets d-complet ``infini''.
  De nos posets d-complets infinis se rendent compte comme un sous-ensemble de coroots sur l'alg\`ebre du Kac--Moody Lie et ont deux relations de l'ordre diff\'erentes dans le g\'en\'eral.
  On est l'ordre ordinaire sur coroots.
  L'autre est appel\'e l'ordre du tas.
  Nous donnons une condition suffisante pour les deux ordonne de co\"incider.
  Comme une application, nous donnons une formule du crochet du multivariate pour cylindric Young diagrammes.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{d-complete posets, heap order}



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\maketitle
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\excludecomment{proof}
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\renewcommand{\d}{\mathrm{d}}

\section{ Introduction }\label{sec:1}

Since the end of 1980's,
$\d$-complete posets, or minuscule elements equivalently,
 have been studied in combinatorics with the other regions in mathematics,
\cite{Carrell,ProctorI,StembridgeI,CHF,NOarXiv} and so on.
In these studies, the order structure of $\d$-complete poset plays important rolls.
In \cite{Carrell,StembridgeI,CHF},
 $\d$-complete posets are realized by certain subsets of real coroot systems for Kac--Moody Lie algebras.
In such a situation, the partial order of a $\d$-complete poset is defined two different ways:
One is the ordinary coroot order (\cref{sec:2}), and the other the heap order (see \cref{sec:3} for details).
In \cite{StembridgeI}, J.~R.~Stembridge proved the following theorem:
\begin{thm}[Stembridge \cite{StembridgeI}]
  These two partial orders coincide with each other in a $\d$-complete poset.
\end{thm}
%Precise terminologies and statement are shown in section 3.
\noindent
One purpose of the present paper is to give another proof of Stembridge's theorem (as \cref{cor:order.isom2}).

Recently,
a generalization of $\d$-complete posets to ``infinite $d$-complete posets'' was studied in
\cite{Infpdiw,StrayerarXiv}.
In \cite{CHF}, the author defined the notion of \emph{predominant} integral weights.
Predominant integral weights are certain integral weights (see \cref{sec:3}),
which genera\-lize minuscule elements in Weyl groups \cite{Carrell,ProctorI}.
Furthermore, \emph{finite} predominant integral weights bijectively correspond to minuscule elements \cite{CHF}.


The main purpose of the present paper is to study
 the \emph{infinite} predominant integral weights.
Typical examples of infinite predominant integral weights appear
if the under\-ly\-ing Dynkin diagram is one in the following list:
\begin{itemize}[wide,labelwidth=0pt,leftmargin=0pt]
  \item[(A)]
  affine Dynkin diagrams (see \cite{Infpdiw} for details):
 {\small
  \begin{equation*}
    A_n^{(1)}(n\geq1),\ B_n^{(1)}(n\geq3),\ C_n^{(1)}(n\geq2),\ D_n^{(1)}(n\geq4),\ E_6^{(1)},\ E_7^{(1)},\ D_n^{(2)}(n\geq3),\ A_{2n-1}^{(2)}(n\geq3).
  \end{equation*}
  }
  \item[(B)]
  infinite Dynkin diagrams (see \cite{Kac,MP} for details):
%  \begin{equation*}
    \( A_\infty, A_{+\infty}, B_\infty, C_\infty, D_\infty \).
%  \end{equation*}
\end{itemize}
The infinite predominant integral weights over affine Dynkin diagrams are determined in \cite{Infpdiw}.
Predominant integral weights over the infinite Dynkin diagram \( A_\infty \) are closely related to
combinatorics of plane partitions (e.g. MacMahon identity \cite{QHFfpsac}).

In the present paper,
we define the \emph{shape} \( \shape{\lambda} \) of a predominant integral weight \( \lambda \),
which is a certain subset of positive real coroots.
We consider two kinds of partial orders over the shape.
One is the \emph{ordinary order} of coroots (\cref{sec:2}).
The other is a certain order called the \emph{heap order} (\cref{sec:3}),
 which is a generalization of the notion defined by J.~Stembridge in \cite{StembridgeI}. 
In the present paper,
 we study a relation between the ordinary order and the heap order,
 and give a sufficient condition
 for both orders to coincide (\cref{cor:order.isom}).



%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%%%%%%%%% Sec 2
%
%
%
%
\section{ Preliminaries for root system and coroot system }\label{sec:2}

Let \( A=(a_{i,j})_{i,j \in I } \) be a
 (not necessarily symmetrizable) Cartan matrix 
 of a Kac--Moody Lie algebra \cite{Kac,MP}.
We denote the set of real numbers by \( \RR \).
Let \( \h \) be an \( \RR \) -vector space
 and \( \h^{*} \) the dual space of \( \h \)
 and \( \langle , \rangle : \h^{*} \times \h \rightarrow \RR \) the canonical bilinear form.
We suppose
 the existence of linearly independent subsets
 \( \Pi := \set{ \alpha_i }{ i \in I } \subset \h^{*} \)
 and \( \Pi^\vee := \set{ \alpha_i^{\vee} }{ i \in I } \subset \h \)
 such that \( \langle \alpha_j , \alpha_i^\vee \rangle = a_{i,j} \).
An element \( \lambda \in \h^{*} \) is said to be an \emph{integral weight} if
\begin{equation*}
  \langle \lambda ,\, \alpha_i^{\vee} \rangle \in \ZZ , \quad i \in I
  .
\end{equation*}
The set of integral weights is denoted by \( P \).
For each \( i \in I \), we define the \emph{simple reflection} \( s_i \in GL( \h^{*} ) \) by:
\begin{equation*}
  s_i : \lambda \mapsto \lambda - \langle \lambda ,\, \alpha_i^{\vee} \rangle \alpha_i
  , \quad \lambda \in \h^{*}
  .
\end{equation*}
The group \( W \) generated by \( \set{ s_i }{ i \in I } \) is called the \emph{Weyl group},
 which acts on \( \h \) by:
% satisfying the relations \( s_i^2 = ( s_i s_j )^{m_{i,j}} = 1 \) ,
% where ,for \( i \neq j \) , \( m_{i,j}=2,3,4,6, \aut \infty \) ,
% according to whether \( a_{i,j} a_{j,i} =0,1,2,3, \aut \geq 4 \) .
\begin{equation*}
  \langle w( \lambda ) , w( h ) \rangle = \langle \lambda , h \rangle,
  \quad w \in W , \lambda \in \h^{*} , h \in \h
  .
\end{equation*}
We define the \emph{real root system} (resp. \emph{real coroot system})
 by \( \rroot := W \Pi \) (resp. \( \rroot^\vee := W \Pi^\vee \)).
The \emph{dual} \( \beta^\vee \in \rroot^\vee \) of a root \( \beta \in \rroot \) is defined by
\begin{equation*}
  \beta^\vee = w (\alpha_i^\vee) , \qquad (\beta = w(\alpha_i), w \in W, \alpha_i \in \Pi).
\end{equation*}
This is independent from choice of \( w \in W \) and \( \alpha_i \in \Pi \).
In general, we have
\begin{equation*}
  (w(\beta))^\vee = w(\beta^\vee), \qquad (\beta \in \rroot, w \in W).
\end{equation*}
The map \( \rroot \ni \beta \mapsto \beta^\vee \in \rroot^\vee \) is a bijection.
For \( R \subseteq \rroot \), we put \( R^\vee := \set{ \beta^\vee }{ \beta \in R } \subseteq \rroot^\vee \).
For each \( \beta \in \rroot \), we define \( s_{\beta} \in W \) by:
\begin{equation*}
  s_{\beta}( \lambda ) = \lambda - \langle \lambda ,\, \beta^\vee \rangle \beta, \quad \lambda \in \h^{*} ,
  \quad \text{ or, equivalently, by } \quad
\end{equation*}
 or, equivalently, by
We denote:
\begin{equation*}
  Q^\vee := \bigoplus_{i \in I} \ZZ \alpha_i^\vee ,
  \qquad
  \et
  \qquad
  Q_+^\vee := \bigoplus_{i \in I} \NN \alpha_i^\vee ,
\end{equation*}
where \( \NN \) is the set of non-negative integers.
We have \( \rroot^\vee \subseteq Q^\vee \).
Put \( \rroot_+^\vee := \rroot^\vee \cap Q_+^\vee \), and \( \rroot_-^\vee := \rroot^\vee \cap (-Q_+^\vee) \).
We have \( \rroot^\vee = \rroot_+^\vee \amalg \rroot_-^\vee \).
For \( \beta^\vee,\gamma^\vee \in \rroot^\vee \),
we denote \( \beta^\vee \leq \gamma^\vee \) if
\begin{equation*}
  \gamma^\vee - \beta^\vee \in Q_+^\vee .
\end{equation*}
This order is called the \emph{ordinary order} of coroots.

For each \( w \in W \), we define a set \( \inv{w} \, ( \subseteq \rroot_+ ) \) by:
\begin{equation*}
  \inv{w} := \set{ \gamma \in \rroot_+ }{ w^{-1}(\gamma)<0 }
  .
\end{equation*}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
%
%
%%%%%%%%%% Sec 3
%
%
%
%

\section{ Pre-dominant Integral Weights, Shapes, and \( \lambda \)-Paths  }\label{sec:3}


\begin{Def}
  An integral weight \( \lambda \) is \emph{pre-dominant} if
  \begin{equation*}
    \langle \lambda , \beta^{\vee} \rangle \geq -1 \, , \quad \beta^\vee \in \rroot_+^\vee .
  \end{equation*}
  The set of pre-dominant integral weights is denoted by \( \pdiw \).
\end{Def}

\begin{Def}\label{Def:shape}
  For \( \lambda \in \pdiw \),
  we define a set \( \shape{\lambda} \) by
  \begin{equation*}
    \shape{\lambda} := \set{ \beta^\vee \in \rroot_+^\vee }{ \langle \lambda , \beta^\vee \rangle =-1 }
  \end{equation*}
  We call the set \( \shape{\lambda} \) the \emph{shape of \( \lambda \)}.
  A pre-dominant integral weight \( \lambda \) is said to be \emph{finite} (resp. \emph{infinite})
  if \( \# \shape{\lambda} < \infty \) (resp. \( \# \shape{\lambda} = \infty \)).
\end{Def}
Set \( \diag{\lambda} := \set{ \beta \in \rroot_+ }{ \langle \lambda ,\, \beta^\vee \rangle = -1 } \) as in  \cite{CHF}.
We note that \( \bigl( \diag{\lambda} \bigr)^\vee = \shape{\lambda} \).




\begin{Def}
  Let \( \beta^\vee , \gamma^\vee \in \rroot^\vee \).
  We denote \( \beta^\vee \horder{} \gamma^\vee \) if
  \( \beta^\vee < \gamma^\vee \) and \( \langle \gamma ,\, \beta^\vee \rangle \geq 1 \).
\end{Def}




\begin{Def}\label{Def:horder}
  Let \( \lambda \in \pdiw \).
  The reflective and transitive closure of the restriction of the relation \( \horder{} \) to \( \shape{\lambda} \) is denoted by \( \preceq \).
  Namely,
  for \( \beta^\vee ,\gamma^\vee \in \shape{\lambda} \),
  we denote \( \beta^\vee \preceq \gamma^\vee \) if there exists \( \beta_1^\vee , \cdots ,\beta_l^\vee \in \shape{\lambda} \) such that
  \begin{equation*}
    \beta^\vee \horder{} \beta_1^\vee \horder{} \cdots \horder{} \beta_l^\vee \horder{} \gamma^\vee ,
  \end{equation*}
  or \( \beta^\vee = \gamma^\vee \).
  The partial order \( \preceq \) is called the \emph{heap order over \( \shape{\lambda} \)}.
\end{Def}

\begin{Rem}
  For \( \beta^\vee ,\gamma^\vee \in \shape{\lambda} \),
  if \( \beta^\vee \preceq \gamma^\vee \), then \( \beta^\vee \leq \gamma^\vee \).
  However, in general, the ordinary order does not coincide with the heap order over a shape \( \shape{\lambda} \).
  We see such examples in \cref{Ex:bad4,Ex:bad3} later.
\end{Rem}



\begin{lem}[\cite{CHF}]\label{move}
  Let \( \lambda \in \pdiw \) and \( \beta \in \diag{\lambda} \).
  Then we have:
  \begin{enumerate}
    \item
    \( s_{\beta}( \lambda ) \in \pdiw \).
    \item
    \( \diag{s_{\beta}( \lambda )} = s_{\beta} \left( \diag{\lambda} \setminus \inv{s_{\beta}} \right) \).
  \end{enumerate}
\end{lem}
As a corollary, we get:
\begin{cor}\label{cor:smove}
  Let \( \lambda \in \pdiw \) and \( \alpha_i^\vee \in \shape{\lambda} \cap \Pi^\vee \).
  Then we have:
  \begin{enumerate}
    \item
    \( s_i \lambda \in \pdiw \).
    \item
    The map \( s_i : \shape{\lambda} \setminus \{ \alpha_i^\vee \} \ni \beta^\vee \mapsto s_i\beta^\vee \in \shape{s_i\lambda} \)
    is an order isomorphism on \( \preceq \).
  \end{enumerate}
\end{cor}
\begin{proof}
  (1) \quad
  This follows from \cref{move}(1).

  (2) \quad
  By \cref{move}(2),
  the map \( s_i : \shape{\lambda} \setminus \{ \alpha_i^\vee \} \Mapsto \shape{s_i\lambda} \) is a bijection.
  Since \( \langle \lambda ,\, \beta^\vee \rangle = \langle \lambda ,\, \gamma^\vee \rangle = \langle \lambda ,\, \alpha_i^\vee \rangle = 1 \),
  we have \( \gamma^\vee \neq \beta^\vee \pm \alpha_i^\vee \).
  Hence, by \cref{lem:horder.isom},
  the map \( s_i : \shape{\lambda} \setminus \{ \alpha_i^\vee \} \Mapsto \shape{s_i\lambda} \) is an order isomorphism on \( \preceq \).
\end{proof}

\begin{Def}\label{Def:SPath}
  Let \( \lambda \in \pdiw \).
  Let \( l \in \NN \).
% be a nonnegative integer.
  A sequence  \( \mathcal{B} =( \alpha_{i_1} , \alpha_{i_2} , \cdots , \alpha_{i_l} ) \) of simple roots
  is said to be a \emph{simple \( \lambda \)-path} if
  \begin{equation*}
    \alpha_{i_p}^\vee \in \shape{ s_{i_{p-1}} \cdots s_{i_{1}} (\lambda) } \, , \quad 1 \leq p \leq l .
  \end{equation*}
\end{Def}



As mentioned above,
in general, the ordinary coroot order and the heap order do not coincide.
Hereafter, we consider the conditions for these two orders to coincide.
For this purpose, we introduce several concepts: bad pairs, bad quartets, and bad triplets.


\subsection{ Bad pairs }

\begin{Def}
  Let \( \lambda \in \pdiw \).
  Let \( \beta^\vee , \gamma^\vee \in \shape{\lambda} \).
  A pair \( ( \beta^\vee , \gamma^\vee ) \) is called
  a \emph{bad pair in \( \shape{\lambda} \)} if we have
  \( \beta^\vee \not\heap \gamma^\vee \) and \( \beta^\vee \leq \gamma^\vee \).
\end{Def}
We can see typical examples of bad pairs in \cref{Ex:bad4,Ex:bad3}.





\newcommand{\arr}[1]{{\begin{array}{c}#1\end{array}}}
\newcommand{\alp}[4]{{#1\alpha_0^\vee+#2\alpha_1^\vee+#3\alpha_2^\vee+#4\alpha_3^\vee}}
\subsection{ Bad quartets }
\begin{Def}
  Let \( \lambda \in \pdiw \).
  Let \( \delta_1^\vee,\delta_2^\vee,\beta_1^\vee,\beta_2^\vee \in \shape{\lambda} \).
  A quartet \( ( \delta_1^\vee,\delta_2^\vee;\beta_1^\vee,\beta_2^\vee ) \)
  is said to be a \emph{bad quartet in \( \shape{\lambda} \)}
  if the following conditions hold:
  \begin{enumerate}
    \item
    \( \delta_j^\vee \horder{} \beta_{j'}^\vee \quad (j,j' =1,2) \),
    \item
    \( \beta_1^\vee \) and \( \beta_2^\vee \) are incomparable on \( \preceq \),
    \item
    \( \langle \delta_1 ,\, \delta_2^\vee \rangle = 0 \).
  \end{enumerate}
\end{Def}


\begin{Ex}\label{Ex:bad4}
  Let \( \Pi = \{ \alpha_0,\alpha_1,\alpha_2,\alpha_3 \} \) be a simple root system such that
  \begin{equation*}
    \MATd{
      \langle \alpha_0 ,\, \alpha_0^\vee \rangle & \langle \alpha_0 ,\, \alpha_1^\vee \rangle & \langle \alpha_0 ,\, \alpha_2^\vee \rangle & \langle \alpha_0 ,\, \alpha_3^\vee \rangle \\
      \langle \alpha_1 ,\, \alpha_0^\vee \rangle & \langle \alpha_1 ,\, \alpha_1^\vee \rangle & \langle \alpha_1 ,\, \alpha_2^\vee \rangle & \langle \alpha_1 ,\, \alpha_3^\vee \rangle \\
      \langle \alpha_2 ,\, \alpha_0^\vee \rangle & \langle \alpha_2 ,\, \alpha_1^\vee \rangle & \langle \alpha_2 ,\, \alpha_2^\vee \rangle & \langle \alpha_2 ,\, \alpha_3^\vee \rangle \\
      \langle \alpha_3 ,\, \alpha_0^\vee \rangle & \langle \alpha_3 ,\, \alpha_1^\vee \rangle & \langle \alpha_3 ,\, \alpha_2^\vee \rangle & \langle \alpha_3 ,\, \alpha_3^\vee \rangle
    }
    =
    \MATd{
       2 & -1 &  0 & -1 \\
      -1 &  2 & -1 &  0 \\
       0 & -1 &  2 & -1 \\
      -1 &  0 & -1 &  2
    }
  \end{equation*}
  \begin{minipage}{7cm}
  of type \( A_3^{(1)} \).
  Put
  \begin{equation*}
    \lambda := -\omega_0 + \omega_2 ,
  \end{equation*}
  where \( \omega_i \) denotes a fundamental weight.
  Then \( \lambda \in \pdiw \)
  and the shape \( \shape{\lambda} \) is depicted in Figure 1. %\cref{fig:bad4}.
  We notice that the shape \( \shape{\lambda} \) contains a bad quartet
  \begin{equation*}
    \begin{split}
    & (\alpha_0^\vee+\alpha_1^\vee,\alpha_3^\vee+\alpha_0^\vee;
    \\& \alpha_3^\vee+\alpha_0^\vee+\alpha_1^\vee,\alpha_3^\vee+2\alpha_0^\vee+\alpha_1^\vee+\alpha_2^\vee) .
    \end{split}
  \end{equation*}
  Note that
  \begin{equation*}
    (\alpha_3^\vee+\alpha_0^\vee+\alpha_1^\vee,\alpha_3^\vee+2\alpha_0^\vee+\alpha_1^\vee+\alpha_2^\vee)
  \end{equation*}
  is a bad pair.
  \end{minipage}
  \begin{minipage}{9cm}
  \xybox{
	(-25,100) *{\arr{\ \ \ \ \ }}="3223" ,										( 25,100) *{\arr{\ \ \ \ \ }}="3322" ,
	(-25, 80) *{}="L2u" ,			(0, 90) *[F:<8pt>]{\arr{\alp3222}}="3222" ,					( 25, 80) *{}="R2u" ,
	(-25, 80) *{}="L2d" ,			(0, 70) *[F:<8pt>]{\arr{\alp{2}{2}{}{2}}}="2212" ,				( 25, 80) *{}="R2d" ,
	(-25, 60) *[F:<8pt>]{\arr{\alp{2}{}{}{2}}}="2112" ,									( 25, 60) *[F:<8pt>]{\arr{\alp{2}{2}{}{}}}="2211" ,
	(-25, 40) *{}="L1u" ,			(0, 50) *[F:<8pt>]{\arr{\alp{2}{}{}{}}}="2111" ,				( 25, 40) *{}="R1u" ,
	(-25, 40) *{}="L1d" ,			(0, 30) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee+\alpha_3^\vee}}="1101" ,	( 25, 40) *{}="R1d" ,
	(-25, 20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_3^\vee}}="1001" ,							( 25, 20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee}}="1100" ,
						(0, 10) *[F:<8pt>]{\arr{\alpha_0^\vee}}="1000" ,
	{"3222" \ar@{-} "3322"},	{"3222" \ar@{-} "3223"},
	{"2212" \ar@{-} "L2d"},		{"L2d" \ar@{-}@/^40pt/ "3223"},	{"2212" \ar@{-} "R2u"},		{"R2u" \ar@{-}@/_40pt/ "3322"},
	{"2112" \ar@{-}@/^40pt/ "L2u"},	{"L2u"   \ar@{-} "3222"},	{"2211" \ar@{-}@/_40pt/ "R2d"},	{"R2d" \ar@{-} "3222"},	
	{"2112" \ar@{-} "2212"},	{"2211" \ar@{-} "2212"},
	{"2111" \ar@{-} "2112"},	{"2111" \ar@{-} "2211"},
	{"1101" \ar@{-} "L1u"},		{"L1u" \ar@{-}@/^40pt/ "2112"},	{"1101" \ar@{-} "R1d"},		{"R1d" \ar@{-}@/_40pt/ "2211"},
	{"1001" \ar@{-}@/^40pt/ "L1d"},	{"L1d"   \ar@{-} "2111"},	{"1100" \ar@{-}@/_40pt/ "R1u"},	{"R1u" \ar@{-} "2111"},
	{"1001" \ar@{-} "1101"},	{"1100" \ar@{-} "1101"},
	{"1000" \ar@{-} "1001"},	{"1000" \ar@{-} "1100"}
  }
    \addtocounter{FIG}{1}
    {\bf Figure \theFIG:}
    { Hasse diagram of heap order of \( \shape{\lambda} \) }
    \label{fig:bad4}
  \end{minipage}
\end{Ex}


\begin{prop}\label{prop:bad4.1}
  Let \( \lambda \in \pdiw \).
  If \( \shape{\lambda} \) contains a bad quartet,
  then there exists a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \) such that
  \( \shape{s_{i_l} \cdots s_{i_1}\lambda} \) contains a bad pair.
\end{prop}
\begin{proof}
  Let \( (\delta_1^\vee,\delta_2^\vee;\beta_1^\vee,\beta_2^\vee) \) be a bad quartet in \( \shape{\lambda} \).
  Put \( \eta_1^\vee := -s_{\beta_1}s_{\beta_2} \delta_1^\vee \), \( \eta_2^\vee := -s_{\beta_1}s_{\beta_2} \delta_2^\vee \).
  Then, by \cref{lem:ref2}, we have
  \begin{equation*}
    \beta_1^\vee \horder{} \eta_1^\vee ,
    \beta_2^\vee \horder{} \eta_1^\vee ,
    \beta_1^\vee \horder{} \eta_2^\vee , \et
    \beta_2^\vee \horder{} \eta_2^\vee .
  \end{equation*}
  Put
  \begin{equation*}
    S := \set{ \gamma^\vee \in \shape{\lambda} }{ \gamma^\vee \prec \beta_1^\vee \aut \gamma^\vee \prec \beta_2^\vee } .
  \end{equation*}
  Then we have \( S \in \SShape{\lambda} \).
  Take a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \) to \( S \).
  Put
  \begin{equation*}
    \begin{split}
    \lambda'        &:= s_{i_l} \cdots s_{i_1} \lambda ,\\
    \end{split}
    \quad
    \begin{split}
    {\beta'_1}^\vee &:= s_{i_l} \cdots s_{i_1} \beta_1^\vee ,\\
    {\beta'_2}^\vee &:= s_{i_l} \cdots s_{i_1} \beta_2^\vee ,\\
    \end{split}
    \quad
    \begin{split}
    {\eta'_1}^\vee  &:= s_{i_l} \cdots s_{i_1} \eta_1^\vee ,\\
    {\eta'_2}^\vee  &:= s_{i_l} \cdots s_{i_1} \eta_2^\vee .
    \end{split}
  \end{equation*}
  Then
  \( {\beta'_1}^\vee , {\beta'_2}^\vee \) are simple coroots.
  Put \( \ep_1^\vee := - s_{\eta'_1} s_{\eta'_2} {\beta'_1}^\vee \),
  \( \ep_2^\vee := - s_{\eta'_1} s_{\eta'_2} {\beta'_2}^\vee \).
  Then we have \( \ep_1^\vee = {\eta'_1}^\vee + {\eta'_2}^\vee - {\beta'_1}^\vee \)
  and \( \ep_2^\vee = {\eta'_1}^\vee + {\eta'_2}^\vee - {\beta'_2}^\vee \).
  Comparing the coefficients of \( \ep_1^\vee \) and \( \ep_2^\vee \) in \( {\beta'_1}^\vee \) and \( {\beta'_2}^\vee \),
  \( \ep_1^\vee \) and \( \ep_2^\vee \) are incomparable on \( \leq \).
  In particular,
  \( \ep_1^\vee \) and \( \ep_2^\vee \) are incomparable on \( \preceq \).
  Hence \( ({\eta'_1}^\vee,{\eta'_2}^\vee;\ep_1^\vee,\ep_2^\vee) \) is a bad quartet in \( \shape{\lambda'} \).
  Put \( \mu := s_{\beta'_2} \lambda' \).
  Then we have
  \( s_{\beta'_2} \ep_1^\vee , s_{\beta'_2} \ep_2^\vee \in \shape{\mu} \).
  \( s_{\beta'_2}\ep_1^\vee \) and \( s_{\beta'_2}\ep_2^\vee \) are incomparable on \( \preceq \).
  Since
  \begin{equation*}
    s_{\beta'_2}\ep_1^\vee
    =
    {\eta'_1}^\vee + {\eta'_2}^\vee - {\beta'_1}^\vee - 2{\beta'_2}^\vee
    <
    {\eta'_1}^\vee + {\eta'_2}^\vee - {\beta'_2}^\vee
    =
    s_{\beta'_2}\ep_2^\vee ,
  \end{equation*}
  \( (s_{\beta'_2}\ep_1^\vee, s_{\beta'_2}\ep_2^\vee) \) is a bad pair in \( \shape{\mu} \).
\end{proof}

\begin{prop}\label{prop:bad4inf}
  If the shape \( \shape{\lambda} \) contains a bad quartet,
  then \( \lambda \) is infinite.
\end{prop}
\begin{proof}
  Let \( (\beta_1^\vee,\beta_2^\vee;\gamma_1^\vee,\gamma_2^\vee) \) be a bad quartet in \( \shape{\lambda} \).
  Put \( \delta_1^\vee := -s_{\beta_1}s_{\beta_2}(\gamma_1^\vee) \) and \( \delta_2^\vee := -s_{\beta_1}s_{\beta_2}(\gamma_2^\vee) \).
  Then it is straightforward to see that
  \( (\gamma_1^\vee,\gamma_2^\vee;\delta_1^\vee,\delta_2^\vee) \) is also a bad quartet in \( \shape{\lambda} \).
  Hence, \( \lambda \) is infinite.
\end{proof}






\renewcommand{\alp}[3]{{#1\alpha_0^\vee+#2\alpha_1^\vee+#3\alpha_2^\vee}}
%\newpage
\subsection{ Bad triplets }

\begin{Def}
  Let \( \lambda \in \pdiw \).
  Let \( \delta^\vee,\beta_1^\vee,\beta_2^\vee \in \shape{\lambda} \).
  A triplet \( ( \delta^\vee;\beta_1^\vee,\beta_2^\vee ) \)
  is said to be a \emph{bad triplet in \( \shape{\lambda} \)}
  if the following conditions hold:
  \begin{enumerate}
    \item
    \( \delta^\vee \horder{} \beta_{j'}^\vee \quad (j' =1,2) \),
    \item
    \( \beta_1^\vee \) and \( \beta_2^\vee \) are incomparable on \( \preceq \),
    \item
    \( \langle \delta ,\, \beta_{j'}^\vee \rangle = 2 \quad (j' =1,2) \).
  \end{enumerate}
\end{Def}

\begin{Ex}\label{Ex:bad3}
  Let \( \Pi = \{ \alpha_0,\alpha_1,\alpha_2 \} \) be a simple root system such that
  \begin{equation*}
    \MATc{
      \langle \alpha_0 ,\, \alpha_0^\vee \rangle & \langle \alpha_0 ,\, \alpha_1^\vee \rangle & \langle \alpha_0 ,\, \alpha_2^\vee \rangle \\
      \langle \alpha_1 ,\, \alpha_0^\vee \rangle & \langle \alpha_1 ,\, \alpha_1^\vee \rangle & \langle \alpha_1 ,\, \alpha_2^\vee \rangle \\
      \langle \alpha_2 ,\, \alpha_0^\vee \rangle & \langle \alpha_2 ,\, \alpha_1^\vee \rangle & \langle \alpha_2 ,\, \alpha_2^\vee \rangle
    }
    =
    \MATc{
       2 & -1 &  0 \\
      -2 &  2 & -2 \\
       0 & -1 &  2 
    }
  \end{equation*}
  \begin{minipage}{6.5cm}
  of type \( D_3^{(2)} \).
  Put
  \begin{equation*}
    \lambda := -\omega_0 + \omega_2 ,
  \end{equation*}
  where \( \omega_i \) denotes a fundamental weight.
  Then \( \lambda \in \pdiw \)
  and the shape \( \shape{\lambda} \) is depicted in Figure 2.%\cref{fig:bad3}.
  We notice that the shape \( \shape{\lambda} \) contains the bad triplet
 {\small
\[
    (\alpha_0^\vee+\alpha_1^\vee;\alpha_0^\vee+2\alpha_1^\vee,2\alpha_0^\vee+2\alpha_1^\vee+\alpha_2^\vee) .
\]
}
  Note that
\[
    (\alpha_0^\vee+2\alpha_1^\vee,2\alpha_0^\vee+2\alpha_1^\vee+\alpha_2^\vee)
\]
  is a bad pair.
  \end{minipage}
  \begin{minipage}{9.5cm}
  \hspace{0.2in}
  \xybox{
	(-25,100) *{\arr{\\ \ \ \ \ }}="3223" ,
	( 25,100) *{\arr{\ \ \ \ \ }}="3322" ,
	(-25, 80) *{}="L2u" ,			(0, 90) *[F:<8pt>]{\arr{\alp342}}="3222" ,					( 25, 80) *{}="R2u" ,
						(0, 70) *[F:<8pt>]{\arr{\alp{2}{4}{}}}="2212" ,					( 25, 80) *{}="R2d" ,
	(-25, 60) *[F:<8pt>]{\arr{\alp{2}{3}{}}}="2112" ,					
	(-25, 40) *{}="L1u" ,			(0, 50) *[F:<8pt>]{\arr{\alp{2}{2}{}}}="2111" ,					( 25, 40) *{}="R1u" ,
	(-25, 40) *{}="L1d" ,			(0, 30) *[F:<8pt>]{\arr{\alpha_0^\vee+2\alpha_1^\vee}}="1101" ,	( 25, 40) *{}="R1d" ,
																( 25, 20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee}}="1100" ,
						(0, 10) *[F:<8pt>]{\arr{\alpha_0^\vee}}="1000" ,
	{"3222" \ar@{-} "3322"},
	{"2112" \ar@{-}@/^40pt/ "L2u"},	{"L2u"   \ar@{-} "3222"},	{"2212" \ar@{-} "R2u"},		{"R2u" \ar@{-}@/_40pt/ "3322"},
	{"2112" \ar@{-} "2212"},
	{"2111" \ar@{-} "2112"},
	{"1101" \ar@{-} "L1u"},		{"L1u" \ar@{-}@/^40pt/ "2112"},	{"1100" \ar@{-}@/_40pt/ "R1u"},	{"R1u" \ar@{-} "2111"},
	{"1100" \ar@{-} "1101"},
	{"1000" \ar@{-} "1100"}
  }
    \addtocounter{FIG}{1}
    {\bf Figure \theFIG:\,} \label{fig:bad3}{ Hasse diagram of heap order of \( \shape{\lambda} \). }
   
  \end{minipage}
\end{Ex}


\begin{prop}\label{prop:bad3.1}
  Let \( \lambda \in \pdiw \).
  If \( \shape{\lambda} \) contains a bad triplet,
  then there exists a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \) such that
  \( \shape{s_{i_l} \cdots s_{i_1}\lambda} \) contains a bad pair.
\end{prop}
\begin{proof}
  Let \( (\delta^\vee;\beta_1^\vee,\beta_2^\vee) \) be a bad triplet in \( \shape{\lambda} \).
  Put \( \eta^\vee := -s_{\beta_1}s_{\beta_2} \delta^\vee \).
  Then, by \cref{lem:ref2}, we have
  \begin{equation*}
    \beta_1^\vee \horder{} \eta^\vee,
    \beta_2^\vee \horder{} \eta^\vee ,
    \et
    \langle \eta ,\, \beta_1^\vee \rangle = \langle \eta ,\, \beta_2^\vee \rangle = 2.
  \end{equation*}
  Put
  \begin{equation*}
    S := \set{ \gamma^\vee \in \shape{\lambda} }{ \gamma^\vee \prec \beta_1^\vee \aut \gamma^\vee \prec \beta_2^\vee } .
  \end{equation*}
  Then we have \( S \in \SShape{\lambda} \).
  Take a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \) to \( S \).
  Put
  \begin{equation*}
    \begin{split}
    \lambda'        &:= s_{i_l} \cdots s_{i_1} \lambda ,\\
    {\beta'_1}^\vee &:= s_{i_l} \cdots s_{i_1} \beta_1^\vee ,\\
    {\beta'_2}^\vee &:= s_{i_l} \cdots s_{i_1} \beta_2^\vee ,\\
    {\eta'}^\vee    &:= s_{i_l} \cdots s_{i_1} \eta^\vee .
    \end{split}
  \end{equation*}
  Then
  \( {\beta'_1}^\vee , {\beta'_2}^\vee \) are simple coroots.
  Put \( \ep_1^\vee := -s_{\beta'_1} {\eta'}^\vee \),
  \( \ep_2^\vee := -s_{\beta'_2} {\eta'}^\vee \).
  Then we have \( \ep_1^\vee = 2{\eta'}^\vee - {\beta'_1}^\vee \)
  and \( \ep_2^\vee = 2{\eta'}^\vee - {\beta'_2}^\vee \).
  Comparing the coefficients of \( \ep_1^\vee \) and \( \ep_2^\vee \) in \( {\beta'_1}^\vee \) and \( {\beta'_2}^\vee \),
  \( \ep_1^\vee \) and \( \ep_2^\vee \) are incomparable on \( \leq \).
  In particular,
  \( \ep_1^\vee \) and \( \ep_2^\vee \) are incomparable on \( \preceq \).
  Hence \( ({\eta'}^\vee;\ep_1^\vee,\ep_2^\vee) \) is a bad triplet in \( \shape{\lambda'} \).
  Put \( \mu := s_{\beta'_2} \lambda' \).
  Then we have
  \( s_{\beta'_2} \ep_1^\vee , s_{\beta'_2} \ep_2^\vee \in \shape{\mu} \).
  \( s_{\beta'_2}\ep_1^\vee \) and \( s_{\beta'_2}\ep_2^\vee \) are incomparable on \( \preceq \).
  Since
  \begin{equation*}
    s_{\beta'_2}\ep_1^\vee
    =
    2{\eta'}^\vee - {\beta'_1}^\vee - 2{\beta'_2}^\vee
    <
    2{\eta'}^\vee - {\beta'_2}^\vee
    =
    s_{\beta'_2}\ep_2^\vee ,
  \end{equation*}
  \( (s_{\beta'_2}\ep_1^\vee, s_{\beta'_2}\ep_2^\vee) \) is a bad pair in \( \shape{\mu} \).
\end{proof}

\begin{prop}\label{prop:bad3inf}
  If the shape \( \shape{\lambda} \) contains a bad triplet,
  then \( \lambda \) is infinite.
\end{prop}
\begin{proof}
  Let \( (\delta^\vee;\beta_1^\vee,\beta_2^\vee) \) be a bad triplet in \( \shape{\lambda} \).
  Put \( \delta_1^\vee := -s_{\beta_1}s_{\beta_2}(\delta^\vee) \).
  Then, by \cref{lem:ref1},
  we have \( \langle \delta,\,\delta_1^\vee \rangle = \langle \delta_1,\,\delta^\vee \rangle = 2 \et \delta^\vee < \delta_1^\vee \).
  By \cref{lem:value_inf}, \( \lambda \) is infinite.
\end{proof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%%%%%%%%%% Sec 4
%
%
%
% 

\section{ Main theorem }\label{sec:4}
\begin{thm}\label{thm:bad}
  Let \( \lambda \in \pdiw \).
  Then the following two conditions are equivalent:
%  to each other:
  \begin{enumerate}
    \item
    There exists a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \,\,\,(l \geq0) \) such that
    \( \shape{s_{i_l} \cdots s_{i_1}\lambda} \) contains a bad pair.
    \item
    \( \shape{\lambda} \) contains a bad quartet or a bad triplet.
  \end{enumerate}
\end{thm}

\noindent We note that we cannot replace condition (1) in \cref{thm:bad} with the simpler condition (3) below:
  \begin{enumerate}
    \item[3.]
    the shape \( \shape{\lambda} \) contains a bad pair.
  \end{enumerate}

\renewcommand{\alp}[4]{{#1\alpha_0^\vee+#2\alpha_1^\vee+#3\alpha_2^\vee+#4\alpha_3^\vee}}

  Take \( \lambda \in \pdiw \) in \cref{Ex:bad4}.
  The ordinary coroot order of the shape \( \shape{\lambda} \) is depicted in \cref{fig:bad4ord1}.
  Note that the pair \( (\alpha_3^\vee+\alpha_0^\vee+\alpha_1^\vee, \alpha_3^\vee+2\alpha_0^\vee+\alpha_1^\vee+\alpha_2^\vee) \) is a bad pair.
  The poset \( (\shape{\lambda};\leq) \) is not order-isomorphic to the poset \( (\shape{\lambda};\preceq) \).
  By \( \alpha_0^\vee \in \shape{\lambda} \), we have \( s_0\lambda \in \pdiw \).
  The ordinary coroot order of the shape \( \shape{s_0\lambda} \) is depicted in \cref{fig:bad4ord2}.
  The shape \( \shape{s_0\lambda} \) contains no bad pairs.
  In general, for \( \alpha_i^\vee \in \shape{\lambda} \),
   the poset \( (\shape{\lambda} \setminus \{\alpha_i^\vee\}; \leq) \) is not order-isomorphic to
   the poset \( (\shape{s_i\lambda} ; \leq) \), in contrast to \cref{cor:smove}.
  However, the shape \( \shape{s_i\lambda} \) still contains bad quartets.
Hence, we cannot replace the condition 1 in \cref{thm:bad} with the condition 3.
\setcounter{figure}{\theFIG}
\begin{figure}[h!]
\begin{minipage}{8cm}
\hspace{0.2in}
 \resizebox{9cm}{!}{
    \xybox{
	(-25,100) *{\arr{\ \ \ \ \ }}="3223" ,										( 25,100) *{\arr{\ \ \ \ \ }}="3322" ,
	(-25,80) *{}="L2u" ,			(0, 90) *[F:<8pt>]{\arr{\alp3222}}="3222" ,					( 25,80) *{}="R2u" ,
	(-25,80) *{}="L2d" ,			(0, 70) *[F:<8pt>]{\arr{\alp{2}{2}{}{2}}}="2212" ,				( 25,80) *{}="R2d" ,
	(-25,60) *[F:<8pt>]{\arr{\alp{2}{}{}{2}}}="2112" ,									( 25,60) *[F:<8pt>]{\arr{\alp{2}{2}{}{}}}="2211" ,
	(-25,40) *{}="L1u" ,			(0, 50) *[F:<8pt>]{\arr{\alp{2}{}{}{}}}="2111" ,				( 25,40) *{}="R1u" ,
	(-25,40) *{}="L1d" ,			(0, 30) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee+\alpha_3^\vee}}="1101",	( 25,40) *{}="R1d" ,
	(-25,20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_3^\vee}}="1001" ,								( 25,20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee}}="1100" ,
						(0, 10) *[F:<8pt>]{\arr{\alpha_0^\vee}}="1000" ,
	{"3222" \ar@{-} "3322"},	{"3222" \ar@{-} "3223"},
			{"2212" \ar@{-} "3222"},
	{"2112" \ar@{-} "2212"},	{"2211" \ar@{-} "2212"},
	{"2111" \ar@{-} "2112"},	{"2111" \ar@{-} "2211"},
			{"1101" \ar@{-} "2111"},
	{"1001" \ar@{-} "1101"},	{"1100" \ar@{-} "1101"},
	{"1000" \ar@{-} "1001"},	{"1000" \ar@{-} "1100"}
    }
    }
%    \addtocounter{FIG}{1}
%    {\bf Figure \, \theFIG:\,}
    \caption
    { Hasse diagram of ordinary coroot order of \( \shape{\lambda} \) }
    \label{fig:bad4ord1}
\end{minipage}
\hspace*{-7mm}
\begin{minipage}{8cm}
   \resizebox{9cm}{!}{
    \xybox{
	(-25,110) *{\arr{\ \ \ \ \ }}="xxxx" ,
	(-25,100) *{\arr{\ \ \ \ \ }}="2223" ,										( 25,100) *{\arr{\ \ \ \ \ }}="2322" ,
	(-25,80) *{}="L2u" ,			(0, 90) *[F:<8pt>]{\arr{\alp{}222}}="1222" ,					( 25,80) *{}="R2u" ,
	(-25,80) *{}="L2d" ,			(0, 70) *[F:<8pt>]{\arr{\alp{2}{2}{}{2}}}="2212" ,				( 25,80) *{}="R2d" ,
	(-25,60) *[F:<8pt>]{\arr{\alp{}{}{}{2}}}="1112" ,									( 25,60) *[F:<8pt>]{\arr{\alp{}{2}{}{}}}="1211" ,
	(-25,40) *{}="L1u" ,			(0, 50) *[F:<8pt>]{\arr{\alpha_1^\vee+\alpha_2^\vee+\alpha_3^\vee}}="0111" ,				( 25,40) *{}="R1u" ,
	(-25,40) *{}="L1d" ,			(0, 30) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee+\alpha_3^\vee}}="1101",	( 25,40) *{}="R1d" ,
	(-25,20) *[F:<8pt>]{\arr{\alpha_3^\vee}}="0001" ,								( 25,20) *[F:<8pt>]{\arr{\alpha_1^\vee}}="0100" ,
	{"1222" \ar@{-} "2322"},	{"1222" \ar@{-} "2223"},
	{"2212" \ar@{-}@(ru,d) "2322"},	{"2212" \ar@{-}@(lu,d) "2223"},
	{"1112" \ar@{-}@(u,ld) "1222"},	{"1211" \ar@{-}@(u,rd) "1222"},
	{"1112" \ar@{-} "2212"},	{"1211" \ar@{-} "2212"},
	{"0111" \ar@{-} "1112"},	{"0111" \ar@{-} "1211"},
	{"1101" \ar@{-}@(lu,d) "1112"},	{"1101" \ar@{-}@(ru,d) "1211"},
	{"0001" \ar@{-}@(u,ld) "0111"},	{"0100" \ar@{-}@(u,rd) "0111"},
	{"0001" \ar@{-} "1101"},	{"0100" \ar@{-} "1101"},
    }
    }
%    \addtocounter{FIG}{1}
%    {\bf Figure \, \theFIG:\,}
    \caption
    { Hasse diagram of ordinary coroot order of \( \shape{s_0\lambda} \) }
    \label{fig:bad4ord2}
\end{minipage}
\end{figure}


As a corollary of \cref{thm:bad}, we get:
\begin{cor}\label{cor:order.isom}
  Let \( \lambda \in \pdiw \).
  If the shape \( \shape{\lambda} \) contains no bad quartets and no bad triplets,
  then the ordinary order coincides with the heap order over the shape \( \shape{\lambda} \).
\end{cor}

By \cref{prop:bad4inf,prop:bad3inf}, we get:
\begin{prop}
  If the shape \( \shape{\lambda} \) contains a bad quartet or a bad triplet,
  then we have \( \# \shape{\lambda} = \infty \).
\end{prop}
Immediately we get:
\begin{cor}\label{cor:order.isom2}
  If \( \lambda \in \pdiw \) is finite,
  then the ordinary order coincides with the heap order over the shape \( \shape{\lambda} \).
\end{cor}

Furthermore, we can get:
\begin{cor}\label{type:Aff}
  Suppose that the underlying Dynkin diagram \( \Gamma \) is of affine type,
  listed as (A) in the introduction.
  Denote by \( W \) the Weyl group.
  Let \( \lambda \) be an infinite predominant integral weight.
  Then the following conditions are equivalent to each other:
  \begin{enumerate}
    \item
    the shape \( \shape{\lambda} \) contains no bad quartets and no bad triplets.
    \item
    we have either
    \begin{enumerate}
      \item
      \( \Gamma \) is of type \( A_{l-1}^{(1)} \) for some \( l \geq 2 \) and
      there exists \( \mu \in \pdiw \) such that
      \( \lambda \in W\mu \) and
      \( \langle \mu,\, \alpha_i^\vee \rangle = \CASE{ 1 & \num i=0 \\ -1 & \num i= 1 \,\,\,(\resp l-1) \\ 0 & \num 2 \leq i \leq l-1 \,\,\,(\resp 1 \leq i \leq l-2) } \); \quad or
      \item
      \( \Gamma \) is of type \( C_l^{(1)} \) for some \( l \geq 2 \).
    \end{enumerate}
  \end{enumerate}
\end{cor}
\begin{cor}\label{type:Inf}
  Suppose that the underlying Dynkin diagram \( \Gamma \) is of infinite type,
  listed as (B) in the introduction.
  Let \( \lambda \) be an infinite predominant integral weight.
  Then the shape \( \shape{\lambda} \) contains no bad quartets and no bad triplets.
\end{cor}


\subsection{ Case of type $ A_n^{(1)}$ }

In this subsection,
 we assume \cref{thm:bad}
 and suppose that the underlying Dynkin diagram is of type \( A_n^{(1)} \).
The purpose of this subsection is to prove the following theorem:
\begin{thm}\label{type:AffA}
  Denote by \( W \) the Weyl group.
  Let \( \lambda \) be an infinite predominant integral weight.
  Then the following conditions are equivalent to each other:
  \begin{enumerate}
    \item
    \( \lambda \in W(\omega_1-\omega_0) \) or \( \lambda \in W(\omega_n-\omega_0) \).
    \item
    for any \( \mu \in W\lambda \),
     the ordinary order coincides with the heap order over the shape \( \shape{\mu} \).
  \end{enumerate}
\end{thm}

First, we review several results in \cite{Infpdiw}.
Let the index set \( I \) be \( \{ 0,1,\cdots,n \} \)
and set the cartan matrix \( A = (a_{ij}) \) be
\begin{equation*}
  a_{ij} =
  \begin{cases}
     2 & i=j	\\
    -1 & i-j \equiv 1,n \pmod{n+1}	\\
     0 & i-j \not\equiv 0,1,n \pmod{n+1}.
  \end{cases}
\end{equation*}
Denote by \( P \) the set of integral weights.
For \( i \in I \), set
\begin{equation*}
  P(i,0;I) := \set{ \lambda \in P }{ \langle \lambda ,\, \alpha_k^\vee \rangle = \delta_{i,k}-\delta_{0,k}, \text{ for each \( k \in I \)} }
\end{equation*}
\newcommand{\Psig}{P_\mathrm{sig}}
Denote by \( \Psig \) the set of integral weight \( \lambda \) such that
\begin{equation*}
  \forall \beta \in \Phi; \langle \lambda ,\, \beta^\vee \rangle = 1,0,-1.
\end{equation*}
\begin{thm}[{\cite[Theorem 5.15]{Infpdiw}}]
  \begin{equation*}
    \Psig = \coprod_{i=0}^n W \cdot P(i,0;I)
  \end{equation*}
\end{thm}
\newcommand{\Pinf}{{P^\mathrm{inf}_{\geq-1}}}
Denote by \( \Pinf \) the set of infinite predominant integral weights.
\begin{prop}[{\cite[Proposition 5.8]{Infpdiw}}]
  \begin{equation*}
    \Psig = P(0,0;I) \sqcup \Pinf.
  \end{equation*}
\end{prop}
Since the set \( P(0,0;I) \) is closed under the action of the Weyl group \( W \), we get
\begin{cor}\label{cor:type:AffA}
  \begin{equation*}
    \Pinf = \coprod_{i=1}^n W \cdot P(i,0;I).
  \end{equation*}
\end{cor}
For simplicity, we denote \( W \cdot P(i,0;I) = W(\omega_i-\omega_0) \).

Now, we give a proof of \cref{type:AffA}.
\begin{proof}
  \( (1) \ArrR (2) \):\quad
  Let \( \mu \in W\lambda \).
  Then, by (1),
  we have \( \mu \in W(\omega_1-\omega_0) \) or \( \mu \in W(\omega_n-\omega_0) \).
  We can assume \( \mu \in W(\omega_n-\omega_0) \) without loss of generality.
  Take \( \nu \in P(n,0;I) \) such that
  \begin{equation*}
    \nu = w \mu \text{ for some } w \in W.
  \end{equation*}
  We have
  \begin{equation*}
    \shape{\nu} = \set{ n\sum_{i\in I} \alpha_i^\vee + \sum_{i=0}^k \alpha_i^\vee }{ n \geq 0 \et 0 \leq k \leq n-1 }.
  \end{equation*}
  It is straightforward to see that \( \shape{\nu} \) contains no bad quartets and no bad triplets.
  Hence, so is \( \mu \).
  Therefore, by \cref{thm:bad}, the ordinary order coincides with the heap order over the shape \( \shape{\mu} \).

  \( (2) \ArrR (1) \):\quad
  Suppose \( \lambda \in W(\omega_i-\omega_0) \) for some \( 2 \leq i \leq n-1 \).
  Take \( \nu \in P(i,0;I) \) such that
  \begin{equation*}
    \nu = w \lambda \text{ for some } w \in W.
  \end{equation*}
  It is straightforward to see that
  \(
    (\alpha_n^\vee+\alpha_0^\vee , \alpha_0^\vee+\alpha_1^\vee;
    \alpha_n^\vee+\alpha_0^\vee+\alpha_1^\vee,
    2\alpha_0^\vee+\alpha_1^\vee+\cdots+\alpha_n^\vee)
  \) is a bad quartet in \( \shape{\nu} \).
\end{proof}



\section{ Application of main theorem }
We give an application of main theorem to the hook length formula.
In this section, we suppose that the underlying Dynkin diagram is of type \( A_{l-1}^{(1)} \).
Let \( \lambda \in P_{\geq-1} \).

\renewcommand{\alp}[5]{#1\alpha_0^\vee\!+\!#2\alpha_1^\vee\!+\!#3\alpha_2^\vee\!+\!#4\alpha_3^\vee\!+\!#5\alpha_4^\vee}
\newcommand{\dia}[1]{
  \put(1,0){\line(-1,1){1}}  \put(1,0){\line(-1,-1){1}}  \put(-1,0){\line(1,1){1}}  \put(-1,0){\line(1,-1){1}}
  \put(0,0){\makebox(0,0){#1}}
}
\begin{Ex}\label{Ex:cylin}
  Let \( l = 5 \) and
  \( \lambda = -\omega_0 + \omega_2 - \omega_3 + \omega_4 \).
  Then the heap order of the shape \( \shape{\lambda} \) is depicted below:
  
\hspace{-0.3in}
 \resizebox{\textwidth}{!}{
  \xybox{
	(-45,100) *{}="-3" ,	(-15,100) *{}="-1" ,	(15,100) *{}="+1" ,	(45,100) *{}="+3" ,(75,100) *{}="+5" ,
	(-45,100) *{}="NW" ,	(105,100) *{}="NE" ,
	(-45, 10) *{}="SW" ,	(105, 10) *{}="SE" ,
	(-30, 90) *[F:<8pt>]{\arr{\alp{2}{2}{2}{2}{}}}="22221" ,								(60, 90) *[F:<8pt>]{\arr{\alp{2}{2}{}{2}{2}}}="22122" ,
															(45, 80) *[F:<8pt>]{\arr{\alp{2}{}{}{2}{2}}}="21122" ,
																		(90, 70) *[F:<8pt>]{\arr{\alp{2}{2}{}{}{}}}="22111" ,
							(15, 60) *[F:<8pt>]{\arr{\alp{}{}{}{2}{}}}="11121" ,				(75, 60) *[F:<8pt>]{\arr{\alp{2}{}{}{}{}}}="21111" ,
			(-15, 40) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee+\alpha_3^\vee+\alpha_4^\vee}}="11011" ,	(45, 40) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee+\alpha_2^\vee+\alpha_3^\vee}}="11110" ,
	(-30, 30) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_3^\vee+\alpha_4^\vee}}="10011" ,
							(15, 20) *[F:<8pt>]{\arr{\alpha_0^\vee+\alpha_1^\vee}}="11000" ,
					(0, 10) *[F:<8pt>]{\arr{\alpha_0^\vee}}="10000" ,							(90, 10) *[F:<8pt>]{\arr{\alpha_3^\vee}}="00010" ,
	(-45, 80) *{}="L4" ,	(105, 80) *{}="R4" ,
	(-45, 60) *{}="L3" ,	(105, 60) *{}="R3" ,
	(-45, 40) *{}="L2" ,	(105, 40) *{}="R2" ,
	(-45, 20) *{}="L1" ,	(105, 20) *{}="R1" ,
	{"22221" \ar@{-} "-3"},		{"22221" \ar@{-} "-1"},		{"22122" \ar@{-} "+3"},		{"22122" \ar@{-} "+5"},
	{"21122" \ar@{-} "+1"},		{"21122" \ar@{-} "22122"},
	{"22111" \ar@{-} "22122"},	{"22111" \ar@{-} "R4"},		{"L4" \ar@{-} "22221"},
	{"11121" \ar@{-} "22221"},	{"11121" \ar@{-} "21122"},	{"21111" \ar@{-} "21122"},	{"21111" \ar@{-} "22111"},
	{"11011" \ar@{-} "L3"},		{"R3" \ar@{-} "22111"},		{"11011" \ar@{-} "11121"},	{"11110" \ar@{-} "11121"},	{"11110" \ar@{-} "21111"},
	{"10011" \ar@{-} "L2"},		{"R2" \ar@{-} "21111"},		{"10011" \ar@{-} "11011"},
	{"11000" \ar@{-} "11011"},	{"11000" \ar@{-} "11110"},
	{"10000" \ar@{-} "10011"},	{"10000" \ar@{-} "11000"},	{"00010" \ar@{-} "11110"},	{"00010" \ar@{-} "R1"},	{"L1" \ar@{-} "10011"},
	{"NW" \ar@{--} "SW"},	{"NE" \ar@{--}|\hole "SE"}
  }
  }
  The west end and the east end are connected cylindrically.
  This shape is order isomorphic to the cylindric Young diagram depicted below:
  \begin{center}
    \setlength{\unitlength}{.8cm}
    \begin{picture}(15,6)(-5,-1)
	\put(-4,4){\dia{1}}		\put(-2,4){\dia{3}}		\put(0,4){\dia{0}}		\put( 2,4){\dia{2}}		\put( 4,4){\dia{4}}		\put( 6,4){\dia{1}}		\put( 8,4){\dia{3}}		\put(10,4){\dia{0}}
			\put(-3,3){\dia{2}}		\put(-1,3){\dia{4}}		\put( 1,3){\dia{1}}		\put( 3,3){\dia{3}}		\put( 5,3){\dia{0}}		\put( 7,3){\dia{2}}		\put( 9,3){\dia{4}}
	\put(-4,2){\dia{1}}		\put(-2,2){\dia{3}}		\put(0,2){\dia{0}}		\put( 2,2){\dia{2}}		\put( 4,2){\dia{4}}		\put( 6,2){\dia{1}}		\put( 8,2){\dia{3}}		\put(10,2){\dia{0}}
			\put(-3,1){\dia{2}}		\put(-1,1){\dia{4}}		\put( 1,1){\dia{1}}		\put( 3,1){\dia{3}}		\put( 5,1){\dia{0}}
	\put(-4,0){\dia{1}}		\put(-2,0){\dia{3}}		\put(0,0){\dia{0}}
	\linethickness{2mm}
	\thicklines
	\put(0,0){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}	\put(-1,2){\line(1,1){1}}	\put( 0,3){\line(-1,1){1}}}
	\put(5,1){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}	\put(-1,2){\line(1,1){1}}}
	\put(10,2){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}}
    \end{picture}

    \addtocounter{FIG}{3}
    {\bf Figure \, \theFIG:\,}~{ corresponding cylindric Young diagram }
    \label{fig:Cyl}
  \end{center}
  The entries of cells are contents.
\end{Ex}

According to \cite{CHF}, we define the hook at \( \beta^\vee \in \shape{\lambda} \):
\begin{Def}
  For \( \beta^\vee \in \shape{\lambda} \), we put
  \begin{equation*}
    \Hook{\lambda}{\beta^\vee} := \shape{\lambda} \cap \inv{s_\beta}^\vee .
  \end{equation*}
  The set \( \Hook{\lambda}{\beta^\vee} \) is called the \emph{hook at \( \beta^\vee \in \shape{\lambda} \)}.
  See also \cref{Ex:hook}.
\end{Def}
Then we can define a unique coloring \( c :\shape{\lambda} \Mapsto \{0,1,\cdots,l-1\} \) with
\begin{equation*}
  \sum_{\gamma^\vee \in \Hook{\lambda}{\beta^\vee}} \alpha_{c(\gamma^\vee)} = \beta
\end{equation*}
for any \( \beta^\vee \in \shape{\lambda} \).
  See also \cref{Ex:hook}.

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  \qbezier[21](0,0)(-.5,-.5)(-1,-1)
}
\newcommand{\shdia}[1]{
  \put(1,0){\line(-1,1){1}}  \put(1,0){\line(-1,-1){1}}  \put(-1,0){\line(1,1){1}}  \put(-1,0){\line(1,-1){1}}
  \put(.95,.05){\shline}  \put(.90,.10){\shline}
  \put(.85,.15){\shline}  \put(.80,.20){\shline}
  \put(.75,.25){\shline}  \put(.70,.30){\shline}
  \put(.65,.35){\shline}  \put(.60,.40){\shline}
  \put(.55,.45){\shline}  \put(.50,.50){\shline}
  \put(.45,.55){\shline}  \put(.40,.60){\shline}
  \put(.35,.65){\shline}  \put(.30,.70){\shline}
  \put(.25,.75){\shline}  \put(.20,.80){\shline}
  \put(.15,.85){\shline}  \put(.10,.90){\shline}
  \put(.05,.95){\shline}
  \put(0,0){\makebox(0,0){#1}}
}

\begin{Ex}\label{Ex:hook}
  In the shape of \cref{Ex:cylin},
   the hook \( \Hook{\lambda}{\beta^\vee} \) at the coroot
  \begin{equation*}
    \beta^\vee = \alp{2}{}{}{2}{2}
  \end{equation*}
   is depeicted as the set of boxes with shadow below.
  \begin{center}
    \setlength{\unitlength}{.8cm}
    \begin{picture}(15,6)(-5,-1)
	\put(-4,4){\dia{1}}		\put(-2,4){\dia{3}}		\put(0,4){\dia{0}}		\put( 2,4){\shdia{2}}		\put( 4,4){\dia{4}}		\put( 6,4){\dia{1}}		\put( 8,4){\dia{3}}		\put(10,4){\dia{0}}
			\put(-3,3){\dia{2}}		\put(-1,3){\dia{4}}		\put( 1,3){\shdia{1}}		\put( 3,3){\shdia{3}}		\put( 5,3){\dia{0}}		\put( 7,3){\dia{2}}		\put( 9,3){\dia{4}}
	\put(-4,2){\dia{1}}		\put(-2,2){\dia{3}}		\put(0,2){\shdia{0}}		\put( 2,2){\dia{2}}		\put( 4,2){\shdia{4}}		\put( 6,2){\dia{1}}		\put( 8,2){\dia{3}}		\put(10,2){\dia{0}}
			\put(-3,1){\dia{2}}		\put(-1,1){\shdia{4}}		\put( 1,1){\dia{1}}		\put( 3,1){\dia{3}}		\put( 5,1){\shdia{0}}
	\put(-4,0){\dia{1}}		\put(-2,0){\shdia{3}}		\put(0,0){\dia{0}}
	\linethickness{2mm}
	\thicklines
	\put(0,0){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}	\put(-1,2){\line(1,1){1}}	\put( 0,3){\line(-1,1){1}}}
	\put(5,1){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}	\put(-1,2){\line(1,1){1}}}
	\put(10,2){\put(-1,0){\line(-1,1){1}}	\put(-2,1){\line(1,1){1}}}
    \end{picture}

    \addtocounter{FIG}{1}
    {\bf Figure \, \theFIG:\,}~{ corresponding cylindric Young diagram }
    \label{fig:Cyl}
  \end{center}
  Furthermore, we notice that
  \begin{equation*}
    \sum_{\gamma^\vee \in \Hook{\lambda}{\beta^\vee}} \alpha_{c(\gamma^\vee)}
    =
    \alpha_3 + \alpha_4 + \alpha_0 + \alpha_1 + \alpha_2 + \alpha_3 + \alpha_4 + \alpha_0
    =
    \beta .
  \end{equation*}
\end{Ex}
According to R.~P.~Stanley, we define \( \shape{\lambda} \)-partitions:
\begin{Def}
  A map \( \pi : \shape{\lambda} \Mapsto \NN = \{ 0,1,2,\cdots \} \) is said to be a \emph{\( \shape{\lambda} \)-partition} if the following two conditions hold:
  \begin{enumerate}
    \item
    if \( \beta^\vee, \gamma^\vee \in \shape{\lambda} \) satisfy \( \beta^\vee \horder{} \gamma^\vee \),
    then we have \( \pi(\beta^\vee) \geq \pi(\gamma^\vee) \).
    \item
    the number of elements \( \beta^\vee \in \shape{\lambda} \) with \( \pi(\beta^\vee) > 0 \) is finite.
  \end{enumerate}
  The set of \( \shape{\lambda} \)-partitions is denoted by \( \Part{\shape{\lambda}} \).
  Let \( x_0, x_1, \cdots, x_{l-1} \) be indeterminates.
  For \( \pi \in \Part{\shape{\lambda}} \),
  we define a monomial \( x^\pi \) by
  \begin{equation*}
    x^\pi := \prod_{\beta^\vee \in \shape{\lambda}} x_{c(\beta^\vee)}^{\pi(\beta^\vee)} .
  \end{equation*}
  For \( \beta = \sum_{i \in I} c_i\alpha_i \in Q_+ = \bigoplus_{i \in I} \NN \alpha_i \),
  we define a monomial \( x^\beta \) by
  \begin{equation*}
    x^\beta := \prod_{i \in I} x_i^{c_i} = x_0^{c_0}x_1^{c_1}\cdots x_{l-1}^{c_{l-1}} .
  \end{equation*}
\end{Def}


Then, we get the following result:
\begin{thm}
  Let \( \lambda \) be an infinite predominant integral weight over the root system of type \( A_{l-1}^{(1)} \).
  Then the generating function of \( \shape{\lambda} \)-partitions is decomposed as:
  \begin{equation*}
    \sum_{\pi \in \Part{\shape{\lambda}}} x^\pi = \prod_{n=1}^\infty \frac{1}{1-x^{n\delta}} \cdot \prod_{\beta \in \diag{\lambda}} \frac{1}{1-x^\beta} ,
  \end{equation*}
  where
  \( \delta \) denotes the null root: \( \delta = \sum_{i=0}^{l-1} \alpha_i \),
\end{thm}
The proof of the above theorem is given by induction on heap order.




%\acknowledgements{
%  The author would like to thank Dr. Takeshi Suzuki.
%}


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\end{document}
