\[
  \bigl(W,\ R,\ I,\ \varphi : R \times I \longrightarrow W\bigr)
\]

\[
  R \overset{\Psi}{\longrightarrow} \text{Syst. gén. admissibles de } W
  \text{ paramétrés par le diagramme de Dynkin } I
\]

\[
  \operatorname{Aut}(\Sigma) \overset{\sim}{\longrightarrow}
  \operatorname{Aut}(\Phi(\Sigma))
\]

\[
  W \overset{\alpha}{\longrightarrow} \operatorname{Aut}(\Phi(\Sigma))
\]

\[
  \alpha(w) = \bigl(\operatorname{int}(w),\ w_R,\ \mathrm{id}_I\bigr)
\]

\[
  w_R(w'r) = \operatorname{int}(w)(w')\ w_R\,r
\]

\[
  \varphi\bigl(w_R(r),\ i\bigr) = \operatorname{int}(w)\,\varphi(r,i)
\]

\[
  (\ast\ast) \qquad \varphi(u_R\,r,\ i) = u_W\,\varphi(r,i)
  \qquad \text{i.e.} \qquad
  \varphi(r,i) = u_W^{-1}\,\varphi(u_R\,r,\ i)
\]

\[
  u_R(w\,r_0) = u_W(w)\,a\,r_0
  \qquad \text{où } \struck{\ill{}}\ u_R(r_0) = a\,r_0
\]

\[
  \varphi\bigl(u_W(w)\,a\,r_0,\ i\bigr) = u_W\bigl(\varphi(w\,r_0,\ i)\bigr)
\]

\[
  a\,\varphi(r_0,i)\,a^{-1} = u\,\varphi(r_0,i)
  \qquad \forall\, i \in I
\]

\[
  u_R(w\,r_0) = (a\,w\,a^{-1})(a\,r_0) = a(w\,r_0) = a_R(w\,r_0)
\]

\[
  \mathrm{Rep}'(I) \Longrightarrow \mathrm{Bij}\bigl([1,\ell],\ I\bigr)
  \longrightarrow \text{l'ens. des ordres totaux sur l'ens. } I
  \text{ (de card. } \ell) .
\]

\[
  \sigma_{i'}\bigl(r,\ u : [1,\ell] \overset{\sim}{\to} I\bigr)
  = (r,\ u \circ \tau_{i'}),
  \qquad 0 \leqslant i' \leqslant \ell-1
\]

\[
  \sigma_{\ell-1}\bigl(r,\ u : [1,\ell] \longrightarrow I\bigr)
  = \bigl(\varphi(r,\ u(\ell))\cdot r,\ u\bigr)
\]

\[
  \sigma_i\,\sigma_{\ell-1}(r,u)
  = \sigma_i\bigl(\varphi(r,u(\ell))\cdot r,\ u\bigr)
  = \bigl(\varphi(r,u(\ell))\cdot r,\ u \circ \tau_{i+1}\bigr)
\]

\[
  \sigma_{\ell-1}\,\sigma_i(r,u)
  = \sigma_{\ell-1}\bigl(r,\ u \circ \tau_{i+1}\bigr)
  = \bigl(\varphi\bigl(r,\ (u \circ \tau_{i+1})(\ell)\bigr)\cdot r,\
    u \circ \tau_{i+1}\bigr)
\]

\[
  \sigma_{\ell-1}\,\sigma_{\ell-2}(r,u)
  = \sigma_{\ell-1}\bigl(r,\ u \circ \tau_{\ell-1}\bigr)
  = \bigl(\varphi\bigl(r,\ (u \circ \tau_{\ell-1})(\ell)\bigr)\cdot r,\
    u \circ \tau_{\ell-1}\bigr)
\]

\[
  \pi(r,u) = \bigl(\varphi(r,\ u(\ell-1))\cdot r,\ u \circ \tau_{\ell-1}\bigr)
\]

\[
  \pi^2(r,u) = \struck{\ill{}}\
  \bigl(\varphi(r',\ u'(\ell-1))\cdot r',\ u' \circ \tau_{\ell-1}\bigr)
\]

\[
  = \bigl(\varphi(w\cdot r,\ u(\ell))\cdot w\,r,\ u\bigr)
\]

\[
  \pi^2(r,u)
  = \bigl(w\,\varphi(r,u(\ell))\,w^{-1}\,w\,r,\ u\bigr)
  = \bigl(w\,\varphi(r,u(\ell))\,r,\ u\bigr)
\]

\[
  = \bigl(\varphi(r,u(\ell-1))\,\varphi(r,u(\ell))\,r,\ u\bigr)
\]

\[
  \pi^3(r,u)
  = \bigl(\varphi(r'',\ u(\ell-1))\,r'',\ u \circ \tau_{\ell-1}\bigr)
  \quad \struck{cqf}
\]

\[
  \pi^3(r,u) = \bigl(\varphi(r,u(\ell-1))\,\varphi(r,u(\ell))\,
    \varphi(r,u(\ell-1))\cdot r,\ u \circ \tau_{\ell-1}\bigr)
\]

\[
  \pi^4(r,u)
  = \bigl(\varphi(r''',\ u'(\ell-1))\,r''',\ u' \circ \tau_{\ell-1}\bigr)
  = \bigl(w'''\,\varphi(r,u(\ell))\,r,\ u\bigr)
  = \Bigl[\bigl(\varphi(r,u(\ell-1))\,\varphi(r,u(\ell))\bigr)^2 r,\ u\Bigr]
\]

\[
  \varphi(r,\ u(\ell-1)) = a, \qquad \varphi(r,\ u(\ell)) = b .
\]

\[
  \begin{cases}
    \pi^{2n}(r,u) = \bigl((ab)^n\,r,\ u\bigr) \\[2pt]
    \pi^{2n+1}(r,u) = \bigl(((ab)^n a)\,r,\ u \circ \tau_{\ell-1}\bigr)
  \end{cases}
\]

\[
  \pi^i(r,u) = (r,u)
  \iff
  \begin{cases}
    i \equiv 0 \ (2), \text{ i.e. } i = 2n \\[2pt]
    (ab)^n = 1
  \end{cases}
\]

\[
  \pi^{2n+1}(r,u)
  = \bigl(\varphi(r'_n,\ u'(\ell-1))\cdot r'_n,\ u' \circ \tau_{\ell-1}\bigr)
  = \struck{\ill{}}\ \bigl((ab)^{n+1}\,r,\ u\bigr)
\]

\[
  \pi^{2n+2}(r,u)
  = \bigl(\varphi(r_{n+1},\ u(\ell-1))\cdot r_{n+1},\
    u \circ \tau_{\ell-1}\bigr)
\]

\[
  \pi^i(r,u) = (r,u) \quad \forall\, (r,u) \in R \times \mathrm{Rep}(I)
  \iff
  \begin{cases}
    i \equiv 0\ (2), \text{ i.e. } i = 2n \\[2pt]
    \bigl(\varphi(r,u(\ell-1))\,\varphi(r,u(\ell))\bigr)^n = 1
    \quad \forall\, r, u
  \end{cases}
\]

\[
  \begin{cases}
    \text{ordre de } \pi = \sigma_{\ell-2}\,\sigma_{\ell-1} = 2\nu \\[2pt]
    \text{où } \nu = \text{le ppcm des } n_{ij}
      \ (i, j \in I,\ i \neq j)
  \end{cases}
\]

\[
      \operatorname{Aut}(C_{p,q})
      \overset{\sim}{\longleftarrow}
      \widehat{\Gamma}\big/\bigl(\rho_s^{\,p},\ \rho_f^{\,q}\bigr)
      \overset{\text{déf}}{=} \Gamma_{p,q}
    \]

\[
      (3,3) \quad (3,4) \quad (4,3) \quad (3,5) \quad (5,3)
    \]

\[
      \text{tétraèdre} \quad \text{cube} \quad \text{octaèdre} \quad
      \text{dodécaèdre} \quad \text{icosaèdre}
    \]

\[
      (4,4), \quad (3,6), \quad (6,3)
    \]

\[
      \text{pavage carré} \quad \text{pavage \ill{}} \quad
      \text{pavage triangulaire}
    \]