\usepackage[notref]{showkeys}

\DeclareMathOperator{\LCM}{LCM}

As examples of spin models, we know only Potts models \cite{Jo:pac,J},

From Remark~\ref{0902skype},

\begin{equation}        \label{potts}
\begin{split}
(u^2+u^{-2})^2=|X|\text{ if $|X|\geq 2$,}\\
u^4=1 \text{ if $|X|=1$.}
\end{split}
\end{equation}

%%%%% Lemma:5.01
\[
\mu(W) \mid \max\{ 2m^2,16 \}.
\]

\DeclareMathOperator{\LCM}{LCM}

 \cite{KMW}


\begin{equation}        \label{nshsm}
W =
   \begin{array}{r@{}l}
      & 
         \begin{array}{cc}
            \rule[-2ex]{0cm}{4ex}\hspace{4em} \mbox{\scriptsize $X_{0}$} & 
             \hspace{8em} \mbox{\scriptsize $X_{1}$}
         \end{array}
      \\
      \begin{array}{l}
        \mbox{\scriptsize $X_{0}$} \\
        \mbox{} \\
        \mbox{\scriptsize $X_{1}$} \\
        \mbox{}
      \end{array}
      & 
      \left(
        \begin{array}{cc}
        \left(
        \begin{array}{rr}
        1 & 1 \\
        1 & 1
        \end{array}
        \right) \otimes A &
        \left(
        \begin{array}{rr}
        1 & -1 \\
        -1 & 1
        \end{array}
        \right) \otimes \xi H \\
        \left(
        \begin{array}{rr}
         -1 & 1 \\
        1 & -1
        \end{array}
        \right) \otimes \xi H^T &
        \left(
        \begin{array}{rr}
        1 & 1 \\
        1 & 1
        \end{array}
        \right) \otimes A
        \end{array}
      \right),
  \end{array}
\end{equation}



%%%%% Lemma:2.3
\begin{align*}
&(W^T W^{-1})(\alpha,\beta) 
\\ &= 
%\sum_{z \in X} W^T(\alpha,z) W^{-1}(z,\beta) 
%\nexteq 
n^{-1} \sum_{z \in X} W(z, \alpha) W(\beta,z)^{-1} 
%\nexteq 
%n^{-1} \sum_{i,\ell \in \ZZ_m} \sum_{x \in Y} 
%\frac{W\bigl((i,\ell,x),(i_1,\ell_1,x_1)\bigr)}{
%W\bigl((i_2,\ell_2,x_2),(i,\ell,x)\bigr)} 
\nexteq 
n^{-1} \sum_{i,\ell \in \ZZ_m} \sum_{x \in Y} 
\frac{S_{i,i_1}(\ell,\ell_1) T_{i,i_1}(x,x_1)}{S_{i_2,i}(\ell_2,\ell) T_{i_2,i}(x_2,x)} 
\nexteq 
n^{-1} \sum_{i,\ell \in \ZZ_m} 
\frac{\eta^{(\ell-\ell_1)(i-i_1)}}{\eta^{(\ell_2-\ell)(i_2-i)}}
\sum_{x \in Y} \frac{\eta^{i-i_1}T_{i_1,i}(x_1,x)}{T_{i_2,i}(x_2,x)} 
\nexteq 
n^{-1} \sum_{i \in \ZZ_m} 
\eta^{-\ell_1(i-i_1)-\ell_2(i_2-i)+(i-i_1)} 
\sum_{\ell \in \ZZ_m} \eta^{\ell(i_2-i_1)} 
\sum_{x \in Y} \frac{T_{i_1,i}(x_1,x)}{T_{i_2,i}(x_2,x)} 
\nexteq 
mn^{-1} \delta_{i_1,i_2}
\sum_{i \in \ZZ_m} 
\eta^{-\ell_1(i-i_1)-\ell_2(i_1-i)+(i-i_1)} 
\sum_{x \in Y} \frac{T_{i_1,i}(x_1,x)}{T_{i_2,i}(x_2,x)} 
\nexteq 
mn^{-1} \delta_{i_1,i_2} 
\sum_{i\in \ZZ_m} \eta^{(i_1-i)(\ell_1-\ell_2-1)} \sum_{x \in Y} \frac{T_{i_1,i}(x_1,x)}{T_{i_1,i}(x_2,x)} 
\nexteq mn^{-1} \delta_{i_1,i_2} \cdot m\delta_{\ell_1,\ell_2+1} \cdot r\delta_{x_1,x_2} 
\nexteq \delta_{i_1,i_2}\delta_{\ell_1,\ell_2+1}\delta_{x_1,x_2}.
\end{align*}


\begin{align*}
W^T(\alpha, \beta) &= W(\beta,\alpha) 
%\nexteq 
%S_{i_2,i_1}(\ell_2,\ell_1) T_{i_2,i_1}(x_2, x_1) 
\nexteq 
\eta^{(\ell_2-\ell_1)(i_2-i_1)} T_{i_2,i_1}(x_2, x_1), \\
(RW)(\alpha, \beta) &= 
\bigl( (I_m \otimes z_m \otimes I_r)W \bigr)
((i_1,\ell_1,x_1), (i_2,\ell_2,x_2))
\nexteq
W((i_1,\ell_1-1,x_1), (i_2,\ell_2,x_2)) 
%\nexteq 
%S_{i_1,i_2}(\ell_1-1,\ell_2)T_{i_1,i_2}(x_1, x_2) 
\nexteq 
\eta^{(\ell_1-1-\ell_2)(i_1-i_2)} T_{i_1,i_2}(x_1, x_2)
\nexteq 
\eta^{(\ell_2-\ell_1)(i_2-i_1)}\eta^{-(i_1-i_2)} 
T_{i_1,i_2}(x_1, x_2).
\end{align*}