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%                    by Camille Jaret
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\usepackage{tikz}
\usetikzlibrary{positioning}

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\title{Introduction to Coxeter groups}
\author{Camille Jaret}
\date{15/05/2026}

\begin{document}

\geotitle
\section{Coxeter systems}
\geosection
\subsection{An example : Dihedral groups}
\begin{frame}
  \frametitle{\subsecname}

  \begin{definition}
    Let  $D_{n}$, called \textbf{dihedral group of order $2n$}
    be the group of all transformations leaving a regular n-gon unchanged.\\
  \end{definition} 
  \vfill
  $D_{n}$ can be given a Coxeter group struture with :
  \vfill
  \begin{itemize}
    \item Coxeter matrix : 
    \vfill
    % Colours from the theme can be used like any other colour :
    $$MC(D_{n}, S_{n})  = \begin{pmatrix}
    1 & \textcolor{accent}{n} \\                      % <--- Here (accent)
    \textcolor{accent}{n} & 1 \\
    \end{pmatrix}$$
    \item Coxeter graph :
    \vfill 
    \begin{center}
      \begin{tikzpicture}[
        node/.style={circle, draw=accent!100, fill=bg1!40, very thick, minimum size=10mm},   % <--- And here (bg1)
        ]
        \node[node] (s1)                      {$s_{1}$};
        \node[node] (s2) [right=of s1]   {$s_{2}$};

        \draw[-] (s1.east) -- (s2.west);
        \draw (s1) -- (s2) node [midway, fill=white] {$\textcolor{accent}{n}$};
      \end{tikzpicture}
    \end{center}
  \end{itemize}
\end{frame}

\end{document}
  
