\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}
\begin{fullversion}
  The proof of \cref{thm:bad} is given in subsection \ref{section:proof}.
\end{fullversion}

We note that we cannot replace the 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 Corollary \ref{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}
    \xybox{
	(-25,100) *{\arr{\text{　　　　　}}}="3223" ,										( 25,100) *{\arr{\text{　　　　　}}}="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*{-10mm}
\begin{minipage}{8cm}
    \xybox{
	(-25,110) *{\arr{\text{　　　　　}}}="xxxx" ,
	(-25,100) *{\arr{\text{　　　　　}}}="2223" ,										( 25,100) *{\arr{\text{　　　　　}}}="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 Proposition \ref{prop:bad4inf} and \ref{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}


\begin{fullversion}

\input{Heap_affine}


\subsection{ A proof of \cref{thm:bad} }\label{section:proof}
Now, we give a proof of \cref{thm:bad}.

  \( \lqq (2) \ArrR (1) \rqq \) follows from Proposition \ref{prop:bad4.1} and \ref{prop:bad3.1}.
  We show \( \lqq (1) \ArrR (2) \rqq \) by three steps.

  [STEP1] \quad
  Let \( (\beta^\vee,\gamma^\vee) \) be a bad pair in \( \shape{\lambda} \).
  Then \( \beta^\vee,\gamma^\vee \) are incomparable on \( \preceq \) in the shape \( \shape{\lambda} \).
  Put
  \begin{equation*}
    S := \set{ \delta^\vee \in \shape{\lambda} }{ \delta^\vee \prec \gamma^\vee }.
  \end{equation*}
  Then we have \( S \in \SShape{\lambda} \) with \( \beta^\vee,\gamma^\vee \not\in S \).
  Take a simple \( \lambda \)-path \( (\alpha_{i_1},\cdots,\alpha_{i_l}) \) to \( S \).
  Then we have \( s_{i_l} \cdots s_{i_1} \beta^\vee, s_{i_l} \cdots s_{i_1} \gamma^\vee \in \shape{s_{i_l} \cdots s_{i_1} \lambda} \)
  and \( s_{i_l} \cdots s_{i_1} \gamma^\vee \) is a simple coroot.
  Since \( s_{i_l} \cdots s_{i_1} \beta^\vee, s_{i_l} \cdots s_{i_1} \gamma^\vee \) is incomparable on \( \leq \),
  there exists an index \( k \in \{ 1,2,\cdots,l \} \) such that
  \begin{itemize}
    \item
    \( (s_{i_{k-1}} \cdots s_{i_1} \beta^\vee, s_{i_{k-1}} \cdots s_{i_1} \gamma^\vee) \) is a bad pair in \( \shape{s_{i_{k-1}} \cdots s_{i_1} \lambda} \).
    \item
    \( (s_{i_k} s_{i_{k-1}} \cdots s_{i_1} \beta^\vee, s_{i_k} s_{i_{k-1}} \cdots s_{i_1} \gamma^\vee) \) are incomparable on \( \leq \) in \( \shape{s_{i_k} s_{i_{k-1}} \cdots s_{i_1} \lambda} \).
  \end{itemize}


  [STEP2] \quad
  In the STEP2, we rewrite
  \begin{equation*}
    \begin{split}
    \mu &:= s_{i_{k-1}} \cdots s_{i_1} \lambda \\
    \beta^\vee &:= s_{i_{k-1}} \cdots s_{i_1} \beta^\vee \\
    \gamma^\vee &:= s_{i_{k-1}} \cdots s_{i_1} \gamma^\vee \\
    i &:= i_k .
    \end{split}
  \end{equation*}
  Now we have
  \begin{itemize}
    \item
    \( \alpha_i^\vee \in \shape{\mu} \) and \( \beta^\vee \neq \alpha_i^\vee \).
    \item
    \( (\beta^\vee, \gamma^\vee) \) is a bad pair in \( \shape{\mu} \).
    \item
    \( (s_i \beta^\vee, s_i \gamma^\vee) \) are incomparable on \( \leq \) in \( \shape{s_i \mu} \).
  \end{itemize}
  We show \( \shape{\mu} \) contains a bad quartet or a bad triplet.

  By Proposition \ref{prop:bad},
  we have \( \alpha_i^\vee \norder \beta^\vee \).
  By Lemma \ref{lem:incomp}, we have \( \langle \alpha_i,\, \gamma^\vee \rangle \geq 2 \).
\begin{comment}
%  Since \( s_i\beta^\vee \) and \( s_i\gamma^\vee \) are \( \leq \)-incomparable,
%  \( s_i\beta^\vee \) and \( s_i\gamma^\vee \) are \( \preceq \)-incomparable.
%  Hence,
%  \( \beta^\vee \) and \( \gamma^\vee \) are \( \preceq \)-incomparable.


%  \( \alpha_i^\vee  \horder{} \beta^\vee \).

%  and \( s_i(\beta^\vee) \incomp s_i(\gamma^\vee) \)
%  in \( \shape{s_i(\lambda)} \).

  Let \( \beta^\vee = \sum_{j \in I} b_j \alpha_j^\vee \,\,(b_j \geq 0) \)
  and \( \gamma^\vee = \sum_{j \in I} c_j \alpha_j^\vee \,\,(c_j \geq 0) \).
  Since \( \beta^\vee \leq \gamma^\vee \), 
  we have \( b_j \leq c_j \) for each \( j \in I \).
  Since \( s_i(\beta^\vee) , s_i(\gamma^\vee) \) are incomparable on \( \leq \) in \( \shape{s_i\mu} \),
  we have \( \langle \alpha_i ,\, \gamma^\vee \rangle \geq 1 \).
  Hence, we have
%  \( 1 \leq 
  \begin{equation*}
    \begin{split}
    1
    &\leq
    \langle \alpha_i ,\, \gamma^\vee \rangle
    =
    \langle \alpha_i ,\, \gamma^\vee \rangle
    - \langle \alpha_i ,\, \beta^\vee \rangle
    =
    \left\langle \alpha_i ,\, \sum_{j \in I} (c_j-b_j)\alpha_j^\vee \right\rangle
    \\&=
    2(c_i-b_i)
    + \sum_{j \in I\,\,j \neq i} (c_j-b_j)\langle \alpha_i ,\, \alpha_j^\vee \rangle
    \leq 2(c_i-b_i).
    \end{split}
  \end{equation*}
  Therefore, we have \( b_i < c_i \).
  Furthermore, since \( s_i(\beta^\vee) \incomp s_i(\gamma^\vee) \),
  \( s_i(\beta^\vee) = \beta^\vee = \sum_{j \in I} b_j\alpha_j^\vee \),
  and
  \( s_i(\gamma^\vee)
   = \gamma^\vee - \langle \alpha_i ,\, \gamma^\vee \rangle \alpha_i^\vee
   = (c_i - \langle \alpha_i ,\, \gamma^\vee \rangle) \alpha_i^\vee + \sum_{j \in I,j\neq i} c_j\alpha_j^\vee \),
  we have
  \( b_i > c_i - \langle \alpha_i ,\, \gamma^\vee \rangle \).
  Hence, we have \( \langle \alpha_i ,\, \gamma^\vee \rangle \geq 2 \).
\end{comment}

  By Corollary \ref{cor:S2}, we have
  \( \delta_1^\vee , \delta^\vee_2 \in \shape{\mu} \) such that
  \( \alpha_i^\vee \horder{} \delta_j^\vee \horder{} \beta^\vee
     \,\, (j=1,2) \) and
  \( \alpha_i^\vee+\beta^\vee = \delta_1^\vee+\delta_2^\vee \).
  Since \( \langle \alpha_i,\, \gamma^\vee \rangle \geq 2 \),
  we have \( \langle \gamma,\, \alpha_i^\vee \rangle \geq 1 \).
  Applying Lemma \ref{lem:ref0} to \( \beta, \delta_j, \gamma \),
  we have \( \langle \gamma,\, \delta_j^\vee \rangle = 0,1 \).
  Since \( 0 = \langle \gamma ,\, \beta^\vee \rangle = \langle \gamma,\, \delta_1^\vee+\delta_2^\vee-\alpha_i^\vee \rangle \),
  we have \( \langle \gamma,\, \alpha_i^\vee \rangle = \langle \gamma,\, \delta_1^\vee \rangle + \langle \gamma,\, \delta_2^\vee \rangle \).
  Hence, one of the following three cases holds:
  \begin{enumerate}
    \item
    \( \langle \gamma,\, \delta_1^\vee \rangle = 0 \) and \( \langle \gamma,\, \delta_2^\vee \rangle = 1 \).
    \item
    \( \langle \gamma,\, \delta_1^\vee \rangle = 1 \) and \( \langle \gamma,\, \delta_2^\vee \rangle = 1 \).
    \item
    \( \langle \gamma,\, \delta_1^\vee \rangle = 1 \) and \( \langle \gamma,\, \delta_2^\vee \rangle = 0 \).
  \end{enumerate}

  %\newpage
  (Case (1)) \quad
  Since
  \begin{equation*}
    \langle \alpha_i,\, \ep^\vee \rangle
    =
    \langle \alpha_i,\, \beta^\vee+\gamma^\vee-\delta_2^\vee \rangle
    =
    \langle \alpha_i,\, \gamma^\vee \rangle -1,
  \end{equation*}
  and \( \langle \alpha_i,\, \gamma^\vee \rangle \geq 2 \),
  we have \( \langle \alpha_i,\, \ep^\vee \rangle \geq 1 \).
  Hence, we have \( \langle \ep,\, \alpha_i^\vee \rangle \geq 1 \).
  We have
  \begin{equation*}
    \langle \ep,\, \alpha_i^\vee \rangle
    =
    \langle -s_\gamma s_\beta(\delta_2),\, \alpha_i^\vee \rangle
    =
    \langle \delta_2,\, \beta^\vee \rangle  \langle \beta,\, \alpha_i^\vee \rangle
    + \langle \delta_2,\, \gamma^\vee \rangle  \langle \gamma,\, \alpha_i^\vee \rangle
    - \langle \delta_2,\, \alpha_i^\vee \rangle
    =
    \langle \delta_2,\, \gamma^\vee \rangle -1,
  \end{equation*}
  \( \langle \delta_2,\, \gamma^\vee \rangle = 1 \aut 2 \) by Lemma \ref{lem:ref2} for \( \beta^\vee, \gamma^\vee,\delta_2^\vee \),
  and \( \langle \alpha_i,\, \ep^\vee \rangle \geq 1 \),
  we have \( \langle \delta_2,\, \gamma^\vee \rangle = 2 \).

  \begin{equation*}
    \langle \delta_1,\, \delta_2^\vee \rangle
    =
    \langle -s_\beta s_i \delta_2,\, \delta_2^\vee \rangle
    =
    \langle \delta_2,\, \beta^\vee \rangle  \langle \beta,\, \delta_2^\vee \rangle
    + \langle \delta_2,\, \alpha_i^\vee \rangle  \langle \alpha_i,\, \delta_2^\vee \rangle
    - \langle \delta_2,\, \delta_2^\vee \rangle
    =
    \langle \delta_2,\, \beta^\vee \rangle -1 .
  \end{equation*}
  Now, we consider the following two subcases:
  \begin{enumerate}
    \item[(1A)]
    \( \langle \delta_1,\, \delta_2^\vee \rangle = 1 \).
    \item[(1B)]
    \( \langle \delta_1,\, \delta_2^\vee \rangle = 0 \).
  \end{enumerate}

  (Subcase (1A)) \quad
  Since \( \langle \delta_1,\, \delta_2^\vee \rangle = 1 \),
  we have \( \langle \delta_2,\, \beta^\vee \rangle = 2 \).
  Hence, \( (\delta_2^\vee;\beta^\vee,\gamma^\vee) \) is a bad triplet.

  (Subcase (1B)) \quad
  Since \( \langle \delta_1,\, \delta_2^\vee \rangle = 0 \),
  we have \( \langle \delta_2,\, \beta^\vee \rangle = 1 \).

  Now, we note that \( 2\delta_2^\vee - \gamma^\vee = -s_{\delta_2}\gamma^\vee \in \rroot \).
  We consider the following two subsubcases:
  \begin{enumerate}
    \item[(1Ba)]
    \( 2\delta_2^\vee - \gamma^\vee > 0 \).
    \item[(1Bb)]
    \( 2\delta_2^\vee - \gamma^\vee < 0 \).
  \end{enumerate}

  (Subsubcase (1Ba)) \quad
  Put
  \begin{equation*}
    \begin{split}
    \eta_3^\vee &:= \delta_1^\vee - \alpha_i^\vee = s_i \delta_1^\vee \in \rroot_+^\vee , \\
    \eta_2^\vee &:= \alpha_i^\vee \in \rroot_+^\vee , \\
    \eta_1^\vee &:= \gamma^\vee - \delta_1^\vee - \alpha_i^\vee = s_i s_{2\delta_2^\vee-\gamma^\vee} \delta_2^\vee \in \rroot_+^\vee , \\
    \eta_0^\vee &:= 2\delta_2^\vee - \gamma^\vee \in \rroot_+^\vee .
    \end{split}
  \end{equation*}
  Then, it is straightforward to see
  \begin{equation*}
    \MATd{
    \langle \eta_3 ,\, \eta_3^\vee \rangle & \langle \eta_3 ,\, \eta_2^\vee \rangle & \langle \eta_3 ,\, \eta_1^\vee \rangle & \langle \eta_3 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_2 ,\, \eta_3^\vee \rangle & \langle \eta_2 ,\, \eta_2^\vee \rangle & \langle \eta_2 ,\, \eta_1^\vee \rangle & \langle \eta_2 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_1 ,\, \eta_3^\vee \rangle & \langle \eta_1 ,\, \eta_2^\vee \rangle & \langle \eta_1 ,\, \eta_1^\vee \rangle & \langle \eta_1 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_0 ,\, \eta_3^\vee \rangle & \langle \eta_0 ,\, \eta_2^\vee \rangle & \langle \eta_0 ,\, \eta_1^\vee \rangle & \langle \eta_0 ,\, \eta_0^\vee \rangle
  }
  =
  \MATd{
     2 & -1 &  0 &  0 \\
    -1 &  2 & -1 &  0 \\
     0 & -1 &  2 & -2 \\
     0 &  0 & -1 &  2
  } ,
  \end{equation*}
  which is a cartan matrix of type \( B_4 \).
  We have
  \( \gamma^\vee = 2 \eta_2^\vee + 2 \eta_1^\vee + \eta_0^\vee \) and
  \( \beta^\vee = \eta_3^\vee + \eta_2^\vee + \eta_1^\vee + \eta_0^\vee \).
  This contradicts \( \beta^\vee < \gamma^\vee \).

  (Subsubcase (1Bb)) \quad
  Put
  \begin{equation*}
    \begin{split}
    \eta_3^\vee &:= \delta_1^\vee - \alpha_i^\vee = s_i \delta_1^\vee \in \rroot_+^\vee , \\
    \eta_2^\vee &:= \alpha_i^\vee \in \rroot_+^\vee , \\
    \eta_1^\vee &:= \delta_2^\vee - \alpha_i^\vee = s_i \delta_2^\vee \in \rroot_+^\vee , \\
    \eta_0^\vee &:= \gamma^\vee - 2\delta_2^\vee \in \rroot_+^\vee .
    \end{split}
  \end{equation*}
  Then, it is straightforward to see
  \begin{equation*}
    \MATd{
    \langle \eta_3 ,\, \eta_3^\vee \rangle & \langle \eta_3 ,\, \eta_2^\vee \rangle & \langle \eta_3 ,\, \eta_1^\vee \rangle & \langle \eta_3 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_2 ,\, \eta_3^\vee \rangle & \langle \eta_2 ,\, \eta_2^\vee \rangle & \langle \eta_2 ,\, \eta_1^\vee \rangle & \langle \eta_2 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_1 ,\, \eta_3^\vee \rangle & \langle \eta_1 ,\, \eta_2^\vee \rangle & \langle \eta_1 ,\, \eta_1^\vee \rangle & \langle \eta_1 ,\, \eta_0^\vee \rangle
    \\
    \langle \eta_0 ,\, \eta_3^\vee \rangle & \langle \eta_0 ,\, \eta_2^\vee \rangle & \langle \eta_0 ,\, \eta_1^\vee \rangle & \langle \eta_0 ,\, \eta_0^\vee \rangle
  }
  =
  \MATd{
     2 & -1 &  0 &  0 \\
    -1 &  2 & -1 &  0 \\
     0 & -1 &  2 & -2 \\
     0 &  0 & -1 &  2
  } ,
  \end{equation*}
  which is a cartan matrix of type \( B_4 \).
  We have
  \( \gamma^\vee = 2 \eta_2^\vee + 2 \eta_1^\vee + \eta_0^\vee \) and
  \( \beta^\vee = \eta_3^\vee + \eta_2^\vee + \eta_1^\vee \).
  This contradicts \( \beta^\vee < \gamma^\vee \).

  %\newpage
  (Case (2)) \quad
  Since \( \langle \gamma,\, \delta_1^\vee \rangle = \langle \gamma,\, \delta_2^\vee \rangle = 1 \),
  we have \( \delta_1^\vee , \delta_2^\vee \horder{} \gamma^\vee \).
  Since
  \begin{equation*}
    \begin{split}
    \langle \delta_1,\, \delta_2^\vee \rangle
    &=
    \langle -s_\beta s_i \delta_2,\, \delta_2^\vee \rangle
    =
    \langle \delta_2,\, \beta^\vee \rangle  \langle \beta,\, \delta_2^\vee \rangle
    + \langle \delta_2,\, \alpha_i^\vee \rangle  \langle \alpha_i,\, \delta_2^\vee \rangle
    - \langle \delta_2,\, \delta_2^\vee \rangle
    \\&=
    \langle \delta_2,\, \beta^\vee \rangle + \langle \delta_2,\, \alpha_i^\vee \rangle - 2
    =
    0 ,
    \end{split}
  \end{equation*}
  Hence, \( (\delta_1^\vee,\delta_2^\vee;\beta^\vee,\gamma^\vee) \) is a bad quartet.

  (Case (3)) \quad
  This is similarly proved as Case (1).

  Thus, the shape \( \shape{\mu} \) contains a bad quartet or a bat triplet.

  [STEP3] \quad
  Let \( B \) be a bad quartet or a bad triplet contained in \( \shape{\mu} \),
  Since \( \mu = s_{i_l} \cdots s_{i_1} \lambda \),
  the shape \( \shape{\lambda} \) contains a bad quartet or a bad triplet \( s_{i_1} \cdots s_{i_l} B \).

  This completes the proof of \cref{thm:bad}

\end{fullversion}

\section{ Application of main theorem }
We give an application of main theorem to 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:

  \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}{1}
    {\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}.

\newcommand{\shline}{
  \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.

