\renewcommand{\d}{\mathrm{d}}
\section{ Introduction }

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 has shown the following theorem:
\begin{thm}[Stembridge \cite{StembridgeI}]
  These two partial orders coincide with each other in $\d$-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 the Stembridge's theorem (as \cref{cor:order.isom2}).

Recently,
generalization of $\d$-complete posets to ``infinite $d$-complete posets'' are 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 generalize 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 underlying Dynkin diagram is one in the following list:
\begin{itemize}
  \item[(A)]
  affine Dynkin diagrams (see \cite{Infpdiw} for details):
  \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},\cite{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 on 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}).

\begin{fullversion}

{\bf \cref{cor:order.isom}}.
  If the shape \( \shape{\lambda} \) of a predominant integral weight \( \lambda \) contains
  no bad quartets and no bad triplets,
  then the ordinary order coincides with the heap order over the shape \( \shape{\lambda} \).

As corollaries, we get:
% we get the following three corollaries:

{\bf \cref{cor:order.isom2}}.
  If a predominant integral weight \( \lambda \) is finite, then
  the ordinary order coincides with the heap order over the shape \( \shape{\lambda} \).

{\bf \cref{type:AffA}}.
  Suppose that the underlying Dynkin diagram is of type \( A_n^{(1)} \).
  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}
  \begin{itemize}
    \item[(A)]
    affine Dynkin diagram of type \( A_1^{(1)} \) or \( A_2^{(1)} \).
    \begin{equation*}
      A_1^{(1)},A_2^{(1)},\, C_n^{(1)}(n\geq2), E_6^{(1)}, E_7^{(1)} .
    \end{equation*}
    \item[(B)]
    infinite Dynkin diagrams ( see \cite{Kac},\cite{MP} for details):
    \begin{equation*}
      A_\infty, A_{+\infty}, B_\infty, C_\infty, D_\infty .
    \end{equation*}
  \end{itemize}
  the ordinary order coincides with the heap order over the shape \( \shape{\lambda} \).
\end{fullversion}










