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

In this subsection,
 we assume Theorem \ref{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{Infpdiw}Theorem 5.15]
  \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{Infpdiw}Proposition 5.8]
  \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 Theorem \ref{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 Theorem \ref{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}

