\[
\mathfrak{G}_n = \Bigl\{ \sigma_0,\dots,\sigma_n \Bigm| \sigma_0^2=\dots=\sigma_n^2=1,\quad \sigma_i\sigma_j=\sigma_j\sigma_i \ \text{si}\ \begin{cases} j\geq i+2 \\ 0\leq i,j\leq n \end{cases} \Bigr\}
\]

\[
\begin{cases}
p_1 = \text{ordre de } \sigma_0\sigma_1 \\
\dots \\
p_n = \text{ordre de } \sigma_{n-1}\sigma_n
\end{cases}
\qquad \text{\emph{NB} les $p_i$ peuvent être infinis !}
\]

\[
3\leq p_i\leq +\infty \qquad \text{pour } 1\leq i\leq n .
\]

\[
(\sigma_{i-1}\sigma_i)^{p_i}=1 .
\]

\[
p_* = (p_1,\dots,p_n)\in\bigl([3,+\infty]\cap\mathbb{N}\bigr)^n ,
\]

\[
\sigma_I\neq 1 \ \text{dans}\ \Gamma_{p_*}
\]

\[
\underline{R}_{p_*} = \mathfrak{G}_n/\Gamma_{p_*}
\]

\[
H\subset K \Longrightarrow \mathfrak{G}_K\subset\mathfrak{G}_H ,
\qquad
\underbrace{\underline{R}/\mathfrak{G}_K}_{=D_K} \longrightarrow \underbrace{\underline{R}/\mathfrak{G}_H}_{=D_H} .
\]

\[
D_{\{i\}} = D_i = \text{ens. des $i$-facettes \add{du polyèdre régulier $\Pi$ défini par $\underline{R}$}}.
\]

\[
  \varphi_i : D_i \longrightarrow \text{ens. des sous-espaces affines de dim.~$i$ dans $E$} \qquad (0\leq i\leq n)
  \]

\[
  \forall g\in\mathrm{Aut}(\Pi),\ \exists!\ \bar g\in\mathrm{Aff}(E)\ \text{tel que}\quad \bar g\circ\varphi_i=\varphi_i\circ g_{D_i}\quad \forall\, 0\leq i\leq n
  \]

\[
\varphi_{\underline{R}} : \underline{R} \longrightarrow \struck{\mathrm{Rep\,aff}(E)}\ \mathrm{Drap.aff}_n(E)
\]

\[
\mathfrak{G} \xrightarrow{\ \varphi_{\mathfrak{G}}\ } \mathrm{Aut\,aff}\,E
\]

\[
\begin{cases}
s_0 = f_0 \\
s_1 = \sigma_0 f_0 \\
s_2 = \sigma_1 s_1 \\
\dots \\
s_i = \sigma_{i-1} s_{i-1} \\
\dots \\
s_{n+1} = \sigma_n s_n
\end{cases}
\]

\[
\sigma_i(f_i) = \sigma_i\,\mathrm{Env}(f_{i-1},s_i) = \mathrm{Env}\bigl(\underbrace{\sigma_i f_{i-1}}_{f_{i-1}},\ \underbrace{\sigma_i s_i}_{\in f_i}\bigr)\subset f_i
\]

\[
\sigma_k s_{i+1}=s_{i+1}\quad\text{pour } n\geq k>i+1,\ \text{i.e.}\ \sigma_k\sigma_i s_i=\sigma_i s_i ,
\]

\[
\sigma_k\sigma_i s_i = \sigma_i\underbrace{\sigma_k s_i}_{=s_i}\ (\text{car } k>i+1) = \sigma_i s_i ,\quad \text{OK.}
\]

\[
\begin{cases}
\{s_i\}_{0\leq i\leq n+1}\ \text{base affine fibre par fibre} \\
f_i=\mathrm{Env}(s_0,\dots,s_i) \quad \text{si } 0\leq i\leq n+1 \quad (\text{posant } f_{n+1}=E) \\
\sigma_k s_i = s_i \quad \text{si } \struck{k}\ 0\leq i<k\leq n \\
\sigma_i s_i = s_{i+1} \quad 0\leq i\leq n
\end{cases}
\]

\[
\sigma_j f_i = f_i \quad\text{si } j>i ,
\]

\[
\sigma_j f_i = f_i \quad\text{si } j<i .
\]

\[
\begin{cases}
\sigma_i^2=1 \quad \struck{\ill{}} & 0\leq i\leq n \\
\sigma_i\sigma_j=\sigma_j\sigma_i & \text{si } 0\leq i\ \ill{} \\
(\sigma_{i-1}\sigma_i)^{p_i}=1 & 1\leq i\ \ill{}
\end{cases}
\]

\[
e_i = s_i - s_0 \in V, \quad 1\leq i\leq n+1 \ \text{base de } V, \qquad s_0 \ \text{\emph{origine} de } E
\]

\[
\begin{cases}
\sigma_0 s_0 = s_1 \\
\sigma_0 s_1 = s_0 \\
\sigma_0 s_2 = s_2^0 \in f_2\smallsetminus f_1 \\
\sigma_0 s_3 = s_3^0 \in f_3\smallsetminus f_2 \\
\dots \\
\sigma_0 s_{n+1} = s_{n+1}^0 \in f_{n+1}\smallsetminus f_n
\end{cases}
\qquad
\begin{aligned}
s_2^0 &= s_0+\lambda_2^{01}e_1+\lambda_2^{02}e_2 \\
s_3^0 &= s_0+\lambda_3^{01}e_1+\lambda_3^{02}e_2+\lambda_3^{0\,\uncertain{3}}e_3 \\
&\dots \\
s_{n+1}^0 &= s_0+\lambda_{n+1}^{01}e_1+\lambda_{n+1}^{02}e_2+\dots+\lambda_{n+1}^{0,n+1}e_{n+1}
\end{aligned}
\]

\[
\begin{cases}
\sigma_1 s_0 = s_0 \\
\sigma_1 s_1 = s_2 \\
\sigma_1 s_2 = s_1 \\
\sigma_1 s_3 = s_3^1 \in f_3\smallsetminus f_2 \\
\sigma_1 s_4 = s_4^1 \in f_4\smallsetminus f_2 \\
\dots \\
\sigma_1 s_{n+1} = s_{n+1}^1 \in f_{n+1}\smallsetminus f_n
\end{cases}
\qquad
\begin{cases}
\sigma_2 s_0 = s_0 \\
\sigma_2 s_1 = s_1 \\
\sigma_2 s_2 = s_3 \\
\sigma_2 s_3 = s_2 \\
\sigma_2 s_4 = s_4^2 \in f_4\smallsetminus f_3 \\
\dots \\
\sigma_2 s_{n+1} = s_{n+1}^2 \in f_{n+1}\smallsetminus f_n
\end{cases}
\]

\[
\begin{cases}
\sigma_i s_0 = s_0 \\
\sigma_i s_1 = s_1 \\
\dots \\
\sigma_i s_{i-1} = s_{i-1} \\
\sigma_i s_i = s_{i+1} \\
\sigma_i s_{i+1} = s_i \\
\sigma_i s_{i+2} = s_{i+2}^i \in f_{i+2}\smallsetminus f_{i+1} \\
\dots \\
\sigma_i s_{n+1} = s_{n+1}^i \in f_{n+1}\smallsetminus f_n
\end{cases}
\quad 0\leq i\leq \struck{\ill{}}
\qquad
\begin{cases}
\sigma_n s_0 = s_0 \\
\sigma_n s_1 = s_1 \\
\dots \\
\sigma_n s_{n-1} = s_{n-1} \\
\sigma_n s_n = s_{n+1} \\
\sigma_n s_{n+1} = s_n
\end{cases}
\]

\[
\begin{aligned}
\sigma_0 s_2 &= s_2^0\ (\in f_2) = s_0+\lambda_{01}e_1+\lambda_{02}e_2 \\
\sigma_1 s_3 &= s_3^1\ (\in f_3) = s_0+\lambda_{11}e_1+\lambda_{12}e_2+\lambda_{13}e_3 \\
&\dots \\
\sigma_{n-1} s_{n+1} &= s_{n+1}^{n-1}\ (\in f_{n+1}) = s_0+\lambda_{n-1,1}e_1+\lambda_{n-1,2}e_2+\dots+\lambda_{n-1,n+1}e_{n+1}
\end{aligned}
\]

\[
\begin{pmatrix}
\lambda_{01} & \lambda_{02} & 0 & 0 & \cdots & 0 \\
\lambda_{11} & \lambda_{12} & \lambda_{13} & 0 & \cdots & 0 \\
\cdots & & & & & \\
\lambda_{n-1,1} & \lambda_{n-1,2} & \lambda_{n-1,3} & \lambda_{n-1,4} & \cdots & \lambda_{n-1,n+1}
\end{pmatrix}
\]

\[
2+3+\dots+(n+1) = \frac{(n+2)(n+1)}{2}-1 = \frac{n(n+3)}{2}
\]

\[
\sigma_i s_j = \sigma_i\bigl(\sigma_{j-1}\,\struck{\sigma_{j-2}\cdots\sigma_{i+2}\,s_{i+2}}\bigr) = (\sigma_{j-1}\cdots\sigma_{i+2})\,\sigma_i s_{i+2}
\qquad j\geq i+3\ \text{i.e. } j-1\geq i+2
\]

\[
= (\sigma_{j-1}\sigma_{j-2}\cdots\sigma_{i+2})\,\underbrace{s_{i+2}^i}_{\in f_{i+2}} \qquad s_{i+2}^i = s_0+\lambda_{i1}e_1+\lambda_{i2}e_2+\dots+\lambda_{i,i+2}e_{i+2}
\]

\[
= s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j
\]

\[
\boxed{\ \sigma_i s_j = (s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1})+\lambda_{i,i+2}e_j \quad\text{si } j\geq i+2\ }
\]

\[
\sigma_i e_j =
\begin{cases}
e_j & \text{si } j<i \\
e_{i+1} & \text{si } j=i \\
e_i & \text{si } j=i+1 \\
\lambda_{i1}e_1+\lambda_{i2}e_2+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j & \text{si } j\geq i+2
\end{cases}
\qquad (i\geq 1)
\]

\[
\det(\sigma_i|_V) = -(\lambda_{i,i+2})^{n-i} =
\begin{cases}
-1 & \text{si } n-i \text{ pair} \quad \text{ok} \\
-\lambda_{i,i+2} & \text{si } n-i \text{ impair}
\end{cases}
\]

\[
\sigma_0 e_j =
\begin{cases}
-e_1 & \text{si } j=1 \\
(\lambda_{01}-1)e_1+\lambda_{0,2}e_j & \text{si } j\geq 2
\end{cases}
\qquad \sigma_0 s_1
\]

\[
\sigma_0 e_j = \sigma_0 s_j-\sigma_0 s_0 = \sigma_0 s_j - s_1, \qquad \det(\sigma_0|_V) = -(\lambda_{0,2})^n
\]

\[
\sigma_0^2 s_j =
\begin{cases}
s_j & \text{si } j\leq 1 \\
\struck{\ill{}}\ \sigma_0(s_0+\lambda_{01}e_1+\lambda_{0,2}e_j)
\end{cases}
\]

\[
\struck{= s_0+e_1+}\qquad
s_0+e_1\ \uncertain{-}\ \lambda_{01}e_1+\lambda_{02}\bigl[(\lambda_{01}-1)e_1+\lambda_{02}e_j\bigr]
= s_0+(1-\lambda_{01})(1-\lambda_{02})e_1+\lambda_{02}^2e_j
\]

\[
\begin{cases}
(1-\lambda_{01})(1-\lambda_{02})=0 \\
(\lambda_{02})^2=1
\end{cases}
\]

\[
\boxed{\ \sigma_i s_j =
\begin{cases}
s_j & \text{si } j<i \\
s_{i+1} & \text{si } j=i \\
s_i & \text{si } j=i+1 \\
s_0+\lambda_i(e_i+e_{i+1})+e_j & \text{si } j\geq i+2
\end{cases}
\quad \text{on pose } e_0=0\ }
\]

\[
\boxed{\ \begin{aligned}
\sigma_0 e_j &=
\begin{cases}
-e_1 & j=1 \\
(\lambda_0-1)e_1+e_j & \text{si } j\geq 2
\end{cases} \\
\sigma_i e_j &=
\begin{cases}
e_j & \text{si } j<i \\
e_{i+1} & \text{si } j=i \\
e_i & \text{si } j=i+1 \\
\lambda_i(e_i+e_{i+1})+e_j & \text{autre (si } j\geq i+2)
\end{cases}
\quad (i\geq 1)
\end{aligned}\ }
\]

\[
\boxed{\ \sigma_i s_j = s_j \quad\text{si } j\leq i-1\ }
\qquad
\boxed{\ \begin{cases} \sigma_i s_i = s_{i+1} \\ \sigma_i s_{i+1} = s_i \end{cases}\ }
\]

\[
\sigma_i^2(s_j) = s_j \quad\text{si } j\leq i+1
\]

\[
\sigma_i^2 s_{i+2} = s_{i+2} \overset{?}{\Longrightarrow} \sigma_i^2 s_j = s_j \quad\text{si } j\geq i+3
\]

\[
\begin{aligned}
\sigma_i(\sigma_i s_j) &= \sigma_i(s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j) \\
&= \underbrace{\sigma_i(s_0+\dots+\lambda_{i,i-1}e_{i-1})}_{s_0+\lambda_{i1}e_1+\dots+\lambda_{i,i-1}e_{i-1}} + (\lambda_{i,i}e_{i+1}+\lambda_{i,i+1}e_i) \\
&\quad + \lambda_{i,i+2}\underbrace{\sigma_i e_j}_{(\lambda_{i1}e_1+\dots+\lambda_{i,i+1}e_{i+1})+\lambda_{i,i+2}e_j} \qquad (i\geq 1) \\
&= s_0+(\lambda_{i1}+\lambda_{i,i+2}\lambda_{i1})e_1+(\lambda_{i2}+\lambda_{i,i+2}\lambda_{i2})e_2+\dots \\
&\quad +(\lambda_{i,i-1}+\lambda_{i,i+2}\lambda_{i,i-1})e_{i-1} \\
&\quad +(\lambda_{i,i+1}+\lambda_{i,i+2}\lambda_{i,i})e_i \\
&\quad +(\lambda_{i,i}+\lambda_{i,i+2}\lambda_{i,i+1})e_{i+1} \\
&\quad +(\lambda_{i,i+2})^2e_j
\end{aligned}
\]

\[
\sigma_i^2 s_j = s_j \iff
\begin{cases}
\lambda_{i1}(1+\lambda_{i,i+2}) = \lambda_{i2}(1+\lambda_{i,i+2}) = \dots = \lambda_{i,i-1}(1+\lambda_{i,i+2}) = 0 \\
\lambda_{i,i+1}+\lambda_{i,i+2}\lambda_{ii} = \lambda_{i,i}+\lambda_{i,i+2}\lambda_{i,i+1} = 0 \\
(\lambda_{i,i+2})^2 = 1
\end{cases}
\]

\[
\sigma_i s_j =
\begin{cases}
s_j & \text{si } j<i \\
s_{i+1} & \text{si } j=i \\
s_i & \text{si } j=i+1 \\
s_0+\lambda_{i,1}e_1+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j & \text{si } i+2\leq j\leq n+1 \\
 & (\text{donc } 0\leq i\leq n-1)
\end{cases}
\]

\[
\sigma_i e_j =
\begin{cases}
e_j & \text{si } j<i \\
e_{i+1} & \text{si } j=i \\
e_i & \text{si } j=i+1 \\
\lambda_{i,1}e_1+\dots+\lambda_{i,i+1}e_{i+1}+\lambda_{i,i+2}e_j & \text{si } i+2\leq j\leq n+1 \\
 & (\text{donc } 0\leq i\leq n-1)
\end{cases}
\qquad 1\leq i\leq n
\]

\[
\sigma_0 e_j =
\begin{cases}
-e_1 & \text{si } j=1 \\
(\lambda_{0,1}-1)e_1+\lambda_{0,2}e_j & \text{si } 2\leq j\leq n+1
\end{cases}
\]

\[
\sigma_0^2=\mathrm{id} \iff
\begin{cases}
(\lambda_{01}-1)(\lambda_{02}-1)=0 \\
\lambda_{02}^2=1
\end{cases}
\]

\[
\sigma_i^2=\mathrm{id} \ (1\leq i\leq n-1) \iff
\begin{cases}
\lambda_{i1}(1+\lambda_{i,i+2}) = \lambda_{i2}(1+\lambda_{i,i+2}) = \dots = \lambda_{i,i-1}(1+\lambda_{i,i+2}) = 0 \\
\lambda_{i,i+1}+\lambda_{i,i+2}\lambda_{ii} = \lambda_{i,i}+\lambda_{i,i+2}\lambda_{i,i+1} = 0 \\
(\lambda_{i,i+2})^2 = 1
\end{cases}
\]

\[
\sigma_0 =
\begin{pmatrix}
-1 & \lambda_{0,1}-1 & \lambda_{0,1}-1 & \cdots & \lambda_{01}-1 & 1 \\
0 & \lambda_{0,2} & 0 & & 0 & 0 \\
0 & 0 & \lambda_{02} & & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & & \lambda_{02} & 0 \\
 & & & & 0 & 1
\end{pmatrix}
\]

\[
\sigma_0(x_1,\dots,x_{n+1}) = \struck{\ill{}}\ \bigl(1-x_1+(\lambda_{0,1}-1)(x_2+\dots+x_{n+1}),\ \lambda_{0,2}x_2,\ \dots,\ \lambda_{0,2}x_{n+1}\bigr)
\]

\[
H_0 :
\begin{cases}
(\lambda_{02}-1)x_i = 0 & \text{si } 2\leq i\leq n+1 \\
2x_1+(1-\lambda_{01})(x_2+\dots+x_{n+1}) = 1
\end{cases}
\]

\[
\begin{cases}
x_i = 0 & 2\leq i\leq n+1 \\
2x_1\ \struck{\ill{}} = 1
\end{cases}
\]

\[
\begin{cases}
\nu x_i = 0 & 2\leq i\leq n+1 \\
2x_1+(1-\lambda_{01})(x_2+\dots+x_{n+1}) = 1
\end{cases}
\]

\[
\lambda_{02} = 1
\]

\[
\boxed{\ \lambda_{01} = 1+\alpha_0\ }
\]

\[
\begin{cases}
\sigma_0 =
\begin{pmatrix}
-1 & \alpha_0 & \alpha_0 & \cdots & \alpha_0 & 1 \\
0 & 1 & 0 & \cdots & 0 & 0 \\
0 & 0 & 1 & \cdots & 0 & 0 \\
\cdots & & & & & \\
0 & 0 & 0 & \cdots & 1 & 0 \\
0 & 0 & 0 & \cdots & 0 & 1
\end{pmatrix} \\[2ex]
\sigma_0(x_1,\dots,x_{n+1}) = \bigl(1-x_1+\alpha_0(x_2+\dots+x_{n+1}),\ x_2,\ \dots,\ x_{n+1}\bigr) \\[1ex]
H_0 : 2x_1\ \uncertain{-}\ \alpha_0(x_2+\dots+x_{n+1}) = 1
\end{cases}
\]

\[
\begin{aligned}
\sigma_i(x_1,\dots,x_{n+1}) = \bigl(&x_1+\lambda_{i1}(x_{i+2}+\dots+x_{n+1}),\ x_2+\lambda_{i2}(x_{i+2}+\dots+x_{n+1}),\ \dots, \\
&x_{i-1}+\lambda_{i,i-1}(x_{i+2}+\dots+x_{n+1}),\ x_{i+1}+\lambda_{i,i}(x_{i+2}+\dots+x_{n+1}), \\
&x_i+\lambda_{i,i+1}(x_{i+2}+\dots+x_{n+1}),\ \lambda_{i,i+2}x_{i+2},\ \dots,\ \lambda_{i,i+2}x_{n+1}\bigr)
\end{aligned}
\]

\[
\struck{\sigma_i(x_1,\dots,x_{n+1}) = (x_1,\dots,x_{i-1},x_{i+1},x_i,\lambda_{i,i+2}x_{i+2},\lambda_{i,i+2}x_{i+3},\dots,\lambda_{i,i+2}x_{n+1})}
\]

\[
\struck{H_i : \begin{cases} x_i = x_{i+1} \\ (\lambda_{i,i+2}-1)x_{i+2} = \dots = (\lambda_{i,i+2}-1)x_{n+1} = 0 \end{cases}}
\]

\[
(H_i)\quad
\begin{cases}
\lambda_{i1}(x_{i+2}+\dots+x_{n+1}) = \lambda_{i2}(x_{i+2}+\dots+x_{n+1}) = \dots \\
\qquad = \lambda_{i,i-1}(x_{i+2}+\dots+x_{n+1}) = 0 \\
\lambda_{i,i}(x_{i+2}+\dots+x_{n+1}) = x_i-x_{i+1} \\
\lambda_{i,i+1}(x_{i+2}+\dots+x_{n+1}) = x_{i+1}-x_i \\
(\lambda_{i,i+2}-1)x_{i+2} = \dots = (\lambda_{i,i+2}-1)x_{n+1} = 0
\end{cases}
\]

\[
\begin{cases}
x_{i+2} = \dots = x_{n+1} = 0 \\
\struck{\ill{} = \lambda_{i,i+1}\Sigma_i(x) = 0 \quad \Sigma_i(x)=x_{i+2}+\dots+x_{n+1}} \\
\struck{\ill{}\ x_i = x_{i+1}} \\
\struck{\ill{}}
\end{cases}
\]

\[
\begin{cases}
\lambda_{i1}\Sigma_i(x) = \dots = \lambda_{i,i-1}\Sigma_i(x) = 0 \\
\lambda_{i,i}\Sigma_i(x) = x_i-x_{i+1} \\
\lambda_{i,i+1}\Sigma_i(x) = x_{i+1}-x_i
\end{cases}
\qquad \text{où } \Sigma_i(x) = x_{i+2}+\dots+x_{n+1}
\]

\[
\begin{cases}
\Sigma_i(x) = 0 \\
x_i-x_{i+1} = 0
\end{cases}
\]

\[
\lambda_{i1} = \dots = \lambda_{i,i-1} = 0
\]

\[
\lambda_{ii}\Sigma_i(x) = -\lambda_{i,i+1}\Sigma_i(x) = x_i-x_{i+1} .
\]

\[
\begin{cases}
\begin{aligned}
\sigma_i(x_1,\dots,x_{n+1}) = \bigl(&x_1,\dots,x_{i-1},\ x_{i+1}-(1+\alpha_i)(x_{i+2}+\dots+x_{n+1}), \\
&x_i+(1+\alpha_i)(x_{i+2}+\dots+x_{n+1}), \\
&x_{i+2},\dots,x_{n+1}\bigr)
\end{aligned} \\[2ex]
H_i :\ x_i-x_{i+1}+(1+\alpha_i)(x_{i+2}+\dots+x_{n+2}) = 0
\end{cases}
\]

\[
\lambda_{i,i+2} = 1+\nu \qquad \nu \text{ nilpotent}
\]

\[
\begin{cases}
\nu = 0 \\
\lambda_{i,1} = \dots = \lambda_{i,i-1} = 0 \\
\lambda_{i,i+1}+\lambda_{i,i+1}\ \struck{\ill{}} = \lambda_{i,i}+\lambda_{i,i+1}\ \struck{\ill{}} = 0 \\
\struck{\nu^2=\ill{}}
\end{cases}
\]

\[
\begin{cases}
\underbrace{(2+\nu)}_{\text{nilpot.}}\lambda_{i,1} = (2+\nu)\lambda_{i,2} = \dots = (2+\nu)\lambda_{i,i-1} = 0 \\
\lambda_{ii}+\lambda_{i,i+1} = -\nu\lambda_{ii} = -\nu\lambda_{i,i+1} \\
\struck{\ill{}}\ \nu^2+2\nu = 0 \quad \text{i.e.}\quad \nu(2+\nu) = 0
\end{cases}
\]

\[
\Longrightarrow\quad \lambda_{i,i+1} = -(1+\nu)\lambda_{ii} \qquad \struck{-\nu\lambda_{ii}=}\quad \nu\underbrace{\bigl[\lambda_{i,i+1}-\lambda_{ii}\bigr]}_{-(2+\nu)\lambda_{ii}} = 0
\]

\[
\text{II}\qquad
\left.\begin{cases}
(2+\nu)\lambda_{i,1} = \dots = (2+\nu)\lambda_{i,i-1} = 0 \\
\lambda_{i,i+1} = -(1+\nu)\lambda_{i,i} \\
\nu(2+\nu) = 0
\end{cases}\right]
\]

\[
\lambda_{ii}\Sigma_i(x) = x_i-x_{i+1} .
\]

\[
\begin{aligned}
\sigma_0\sigma_1(x_1,x_2,x_3,\dots,x_{n+1}) &= \sigma_0\bigl(x_2-(1+\alpha_1)\Sigma_1,\ x_1+(1+\alpha_1)\Sigma_1,\ x_3,\dots\bigr) \\
&= \bigl(1-x_2+(1+\alpha_1)\Sigma_1+\alpha_0(x_1+(1+\alpha_1)\Sigma_1+\Sigma_1), \\
&\qquad x_1+(1+\alpha_1)\Sigma_1,\ x_3,\dots,x_{n+1}\bigr) \\
&= \bigl(1+\alpha_0x_1-x_2+(1+\alpha_1+2\alpha_0+\alpha_0\alpha_1)\Sigma_1,\ x_1+(1+\alpha_1)\Sigma_1, \\
&\qquad x_3,\dots,x_{n+1}\bigr)
\end{aligned}
\]

\[
(\sigma_0\sigma_1)^p(x_1,x_2,\dots,x_{n+1}) = \bigl(F_p(\alpha_0,\alpha_1;x_1,\dots,x_{n+1}),\ G_p(\alpha_0,\alpha_1;x_1,\dots,x_{n+1}),\ x_3,\dots,x_{n+1}\bigr)
\]

\[
(i\geq 1)\quad
\begin{cases}
\sigma_i|f_{i-1} = \mathrm{id} \\
\sigma_i|(f_{i+1}/f_{i-1}) = \text{échange des coordonnées } x_i,x_{i+1} \\
\sigma_i|E/f_{i+1} = \mathrm{id}
\end{cases}
\]

\[
\begin{cases}
\sigma_0|f_1 = \text{échange de } s_0,s_1 \\
\sigma_0|E/f_1 = \mathrm{id}
\end{cases}
\]

\[
\struck{\ill{}}\ (i\geq 1)\quad
\begin{cases}
\sigma_i\sigma_{i+1}|f_{i-1} = \mathrm{id} \\
\sigma_i\sigma_{i+1}|f_{i+2}/f_{i-1} \overset{?}{=} \\
\sigma_i\sigma_{i+1}|E/f_{i+2} = \mathrm{id}
\end{cases}
\]

\[
\begin{cases}
\sigma_0\sigma_1|f_2 \overset{?}{=} \\
\sigma_0\sigma_1|E/f_2 = \mathrm{id}
\end{cases}
\]

\[
\underbrace{\sigma_0\sigma_1}_{\pi}(x_1, x_2, x_3, \dots, x_n) = \bigl(1 + \alpha_0 x_1 - x_2 + \underset{\substack{\| \\ 1+\alpha_1+2\alpha_0+\alpha_0\alpha_1}}{a}\,\Sigma_1,\ x_1 + \underset{\substack{\| \\ 1+\alpha_1}}{b}\,\Sigma_1,\ x_3, \dots\bigr)
\]

\[
\begin{aligned}
\pi^2(x_1, \dots, x_n) = \bigl(&1 + \alpha_0(1 + \alpha_0 x_1 - x_2 + a\Sigma_1) - x_1 - b\Sigma_1 + a\Sigma_1,\\
&1 + \alpha_0 x_1 - x_2 + a\Sigma_1 + b\Sigma_1,\ x_3, \dots\bigr)\\
= \bigl(&(1+\alpha_0) + (\alpha_0^2 - 1)x_1 - \alpha_0 x_2 + (a\alpha_0 + a - b)\Sigma_1,\\
&1 + \alpha_0 x_1 - x_2 + (a+b)\Sigma_1,\ x_3, \dots\bigr)
\end{aligned}
\]

\[
\begin{aligned}
\pi^3(x_1, \dots, x_n) = \ &1 + \alpha_0(1+\alpha_0) + \alpha_0(\alpha_0^2 - 1)x_1 - \alpha_0^2 x_2 + \alpha_0(a\alpha_0 + a - b)\Sigma_1\\
&-1\ \struck{\ill{}}\quad - \alpha_0 x_1 \quad + x_2 \quad - (a+b)\Sigma_1 + a\Sigma_1\\
&\uncertain{(1+\alpha_0) + (\alpha_0^2-1)x_1 - \alpha_0 x_2 + [a(\alpha_0 + 1)]\Sigma_1,\ x_3 \ldots}\\
= \ &\struck{(1+}\ \alpha_0(1+\alpha_0) + \alpha_0(\alpha_0^2 - 2)x_1 + (1 - \alpha_0^2)x_2 + \struck{(a\alpha_0^2 + a\alpha_0 \ldots}\\
&(\alpha_0+1)(a\alpha_0 - b)\ \struck{\ill{}}\ \Sigma_1,\ \ldots
\end{aligned}
\]

\[
\begin{cases}
\alpha_0(1 + \alpha_0) = 0 \qquad \alpha_0(\alpha_0^2 - 2) = \struck{1}\ \uncertain{-1}? \qquad \alpha_0^2 = 1 \quad \big|\ \alpha_0 = -1\\
\struck{\ill{}}
\end{cases}
\]

\[
\begin{aligned}
\sigma_0(x_1, \dots, x_{n+1}) &= (1 - x_1 + \alpha_0\Sigma_0,\ x_2, \dots, x_{n+1})\\
\sigma_i(x_1, \dots, x_{n+1}) &= (x_1, \dots, x_{i-1},\ x_{i+1} - (1+\alpha_i)\Sigma_i,\\
&\qquad x_i + (1+\alpha_i)\Sigma_i,\ x_{i+2}, \dots, x_{n+1}) \quad (1\le i\le n-1)\\
\sigma_n(x_1, \dots, x_{n+1}) &= (x_1, \dots, x_{n-1},\ x_{n+1},\ x_n)
\end{aligned}
\]

\[
\struck{f(x_1, \dots, x_{n+1}) = \sum_{1\le i\le n} a_{ii}\, x_i^2 + \sum_{1\le i<j\le n+1} a_{ij}\, x_i x_j + \sum_{1\le i\le n+1} c_i x_i}
\]

\[
\begin{array}{ll}
H_0 : & 2x_1 \struck{=} - \alpha_0(x_2 + \cdots + x_{n+1}) = z\\
D_0 : & x_2 = x_3 = \cdots = x_{n+1} = z = 0 \qquad (\text{pt à l'infini de la } \struck{\ill{}}\ \mathrm{dr}(s_0 s_1))\\[4pt]
H_i : & x_i - x_{i+1} + (1+\alpha_i)(\struck{x_{i+2} =}\ x_{i+2} + \cdots + x_{n+1}) = 0\\
D_i : & x_1 = \cdots = x_{i-1} = x_{i+2} = \cdots = x_{n+1} = z = 0
\end{array}
\]

\[
\begin{aligned}
\overline{\sigma}_0(\underbrace{x_1, \dots, x_{n+1}, \overset{x_0}{\overset{\|}{z}}}_{\overline{x}}) &= (x_0 - x_1 + \alpha_0(x_2 + \cdots + x_{n+1}),\ x_2, \dots, x_{n+1},\ x_0)\\
&= \overline{x} + \struck{\ill{}}\ [e'_0 - 2e'_1 + \alpha_0(e'_2 + \cdots + e'_{n+1})] \otimes e_1\\
\overrightarrow{\sigma_i}(\overline{x}) &= \overline{x} + [e'_i - e'_{i+1} + (1+\alpha_i)(e'_{i+2} + \cdots + e'_{n+1})] \otimes (-e_i + e_{i+1}) \quad (1\le i\le n)\\
\overrightarrow{\sigma_n}(\overline{x}) &= \overline{x} + (e'_n - e'_{n+1}) \otimes (-e_n + e_{n+1})
\end{aligned}
\]

\[
\begin{array}{lll}
H_0 : & \overset{\substack{z\ \add{\text{variable d'homogénéité}}\\ \|}}{\struck{x_0}} - 2x_1 + \alpha_0(x_2 + \cdots + x_{n+1}) = 0 &\\
\delta_0 : & \text{pt à l'infini dans la direction de } e_1 = s_1 - s_0 &\\[4pt]
H_i : & x_i - x_{i+1} + (1+\alpha_i)(x_{i+2} + \cdots + x_{n+1}) = 0 & (1\le i\le n)\\
\delta_i : & \text{pt à l'infini dans la direction de } e_{i+1} - e_i = s_{i+1} - s_i &
\end{array}
\]

\[
\sigma_0 s_i = -(1+\alpha_i)s_0 + (1+\alpha_i)s_1 + s_i \qquad i\ge 2
\]

\[
\sigma_0 s_{i+1} = \sigma_0\sigma_i s_i = \sigma_i\sigma_0 s_i = -(1+\alpha_i)s_0 + (1+\alpha_i)s_1 + s_{i+1}
\]

\[
\begin{aligned}
\sigma_i s_j &= s_j \quad 0\le j\le i-1\\
\sigma_i s_i &= s_{i+1}\\
\sigma_i s_{i+1} &= s_i\\
\sigma_i s_{i+2} &= (\underbrace{0, \dots, 0}_{i-1},\ -(1+\alpha_i),\ (1+\alpha_i),\ 1,\ 0, 0, \dots) = s_{i+2} + (1+\alpha_i)(s_{i+1} - s_i)\\
&= -(1+\alpha_i)s_i + (1+\alpha_i)s_{i+1} + s_{i+2}
\end{aligned}
\]

\[
\boxed{\sigma_i s_j = \begin{cases}
s_j & \text{si } 0\le j\le i-1\\
s_{i+1} & \text{si } j = i\\
s_i & \text{si } j = i+1\\
-(1+\alpha_i)s_i + (1+\alpha_i)s_{i+1} + s_j & \text{si } i+2\le j\le n+1
\end{cases}}
\]

\[
\struck{\sigma_i s_j - s_j =} \quad \sigma_i - \mathrm{id} = \ell_i \otimes (s_{i+1} - s_i), \qquad \ell_i = s'_i - s'_{i+1} + (1+\alpha_i)(s'_{i+2} + \cdots + s'_{n+1})
\]

\[
\sigma_i s_j - s_j = \begin{cases}
0 & \text{si } 0\le j\le i-1\\
s_{i+1} - s_i & \text{si } j = i\\
-(s_{i+1} - s_i) & \text{si } j = i+1\\
(1+\alpha_i)(s_{i+1} - s_i) & \text{si } j\ge i+2
\end{cases}
\]

\[
(1)\quad \begin{cases}
\sigma_0,\dots,\sigma_n \in \mathrm{Aff}(E), & E \text{ affine de rang } n+1\\
\sigma_j\sigma_i = \sigma_i\sigma_j & \text{si } 0\le i,\ \struck{i+1}\ i+2\le j\le n
\end{cases}
\]

\[
(2)\quad E_0 = \{s_0\} \subset E_1 \subset \cdots \subset E_n \quad \text{drapeau de } E
\]

\[
(3)\quad \begin{cases}
\sigma_i E_j = E_j & \text{si } j\ne i\\
\sigma_i E_i \ne E_i
\end{cases}
\qquad
(4)\quad \begin{cases}
s_0\\
s_1 = \sigma_0 s_0\\
\cdots\\
s_i = \sigma_{i-1} s_{i-1} & (1\le i\le n+1)\\
\cdots\\
s_{n+1} = \sigma_n s_n
\end{cases}
\]

\[
\begin{aligned}
(5)&\quad E_i = \struck{\ill{}}\ \mathrm{Env}(s_0,\dots,s_i)\\
(6)&\quad \sigma_k s_i = s_i \quad \text{si } 0\le i< k\le \struck{n+1}\ n
\end{aligned}
\]

\[
\begin{cases}
s_i \notin E_{i-1} \ (= \mathrm{Env}(s_0,\dots,s_{i-1}) \text{ par hyp.\ réc.})\\
E_i = \mathrm{Env}(E_{i-1}, s_i)\\
\sigma_k s_i = s_i \quad i<k\le n
\end{cases}
\]

\[
\sigma_{i-1}E_{i-1} = \mathrm{Env}(\underbrace{\sigma_{i-1}s_0}_{=s_0},\dots,\underbrace{\sigma_{i-1}s_{i-2}}_{=s_{i-2}},\underbrace{\sigma_{i-1}s_{i-1}}_{\in E_{i-1}}) \subset E_{i-1}
\]

\[
\sigma_k s_i = \sigma_k\sigma_{i-1}s_{i-1} = \sigma_{i-1}(\sigma_k s_{i-1}) \overset{\text{réc}}{=} \sigma_{i-1}(s_{i-1}) = s_i.
\]

\[
(7)\quad e_i = s_i - s_0 \qquad 1\le i\le n+1
\]

\[
(8)\quad \underset{\textstyle \sigma_i^2 s_i}{\sigma_i s_{i+1}} = \struck{u_{i+1}}\ u_i, \qquad \sigma_i s_{i+2} = v_i
\]

\[
(9)\quad \begin{cases}
\sigma_i s_j = s_j & \text{si } 0\le j< i\\
\sigma_i s_i = s_{i+1}\\
\sigma_i s_{i+1} = u_i \in E_{i+1} & E_{i+1} = \mathrm{Env}(E_{i-1}, s_{i+1}, u_i)\\
\sigma_i s_{i+2} = v_i \in E_{i+2} & E_{i+2} = \mathrm{Env}(E_{i+1}, v_i) \quad (0\le i\le n-1)\\
\sigma_i s_j = \sigma_{j-1}\cdots\sigma_{i+2}(v_i) & \text{si } \uncertain{i+2< j\le n+1}
\end{cases}
\]

\[
\sigma_i s_j = \sigma_i\sigma_{j-1}\cdots\sigma_{i+2}\,s_{i+2} = \sigma_{j-1}\cdots\sigma_{i+2}\,\underbrace{\sigma_i s_{i+2}}_{v_i}
\]

\[
(\ill)\quad \begin{cases}
u_i = (u_{i,1},\dots,u_{i,i+1}) = s_0 + \sum_{1\le k\le i+1} u_{i,k}\,e_k\\
v_i = (v_{i,1},\dots,v_{i,i+2}) = s_0 + \sum_{1\le k\le i+2} v_{i,k}\,e_k
\end{cases}
\]

\[
\begin{aligned}
\sigma_i s_j &= \sigma_{j-1}\cdots\sigma_{i+2}\,(s_0 + v_{i,1}e_1 + \cdots + v_{i,i+1}e_{i+1} + v_{i,i+2}e_{i+2})\\
&= s_0 + \sum_{1\le k\le i+1} v_{i,k}\,e_k + v_{i,i+2}\,e_j
\end{aligned}
\]

\[
(10)\quad \begin{cases}
\sigma_i s_j = (v_{i,1},\dots,v_{i,i+1},\ 0,\dots,0,\ \overset{j}{v_{i,i+2}},\ 0,\dots,0)\\
\qquad = s_0 + \sum_{1\le k\le i+1} v_{i,k}\,e_k + v_{i,i+2}\,e_j
\end{cases}
\quad \text{si } \struck{\ill{}}\ i+2\le j\le n+1
\]

\[
(11)\quad
\begin{cases}
\sigma_i e_j = e_j & \text{si } 1\le j< i\\
\sigma_i e_i = e_{i+1} \quad \struck{(\text{si } i\ge 1)} & \\
\sigma_i e_{i+1} = \struck{\ldots}
  \begin{cases} u_0 - s_1 = (u_0-s_0) - e_1 = (u_{01}-1)\,e_1 & \text{si } i=0\\
  u_i - s_0 = \sum_{1\le k\le i+1} u_{ik}\, e_k & \text{si } i\ge 1\end{cases}\\
\struck{\sigma_i e_{i+2} = \ldots}\\
\Bigl[\sigma_0 e_{i+2} = \Bigr]
  \begin{cases} v_0 - s_1 = (v_0-s_0) - e_1 = (v_{01}-1)\,e_1 + v_{0,2}\, e_2 & \text{si } i=0\\
  v_i - s_0 = \sum_{1\le k\le i+2} v_{i,k}\, e_k & \text{si } i\ge 1\end{cases}\\
\sigma_i e_j = \begin{cases} (v_{01}-1)\,e_1 + v_{0j}\, e_j & \text{si } i=0\\
  \sum_{1\le k\le i+1} v_{ik}\, e_k + v_{i,i+2}\, e_j & \text{si } i\ge 1\end{cases}\\
\qquad \text{si } i+2\le j\le n+1
\end{cases}
\]

\[
(12)\quad \sigma_0 =
\begin{pmatrix}
u_{01}-1 & v_{01}-1 & v_{01}-1 & v_{01}-1 & \cdots & v_{01}-1 & 1\\
0 & v_{02} & 0 & 0 & \cdots & 0 & 0\\
0 & 0 & v_{02} & 0 & \cdots & 0 & 0\\
0 & 0 & 0 & v_{02} & \cdots & 0 & 0\\
\vdots & & & & \ddots & & \vdots\\
0 & 0 & 0 & 0 & \cdots & v_{02} & 0\\
0 & 0 & 0 & 0 & \cdots & 0 & 1
\end{pmatrix}
\]

\[
(13)\quad \sigma_0(x_1,\dots,x_{n+1}) = \bigl(1 + (u_{01}-1)\,x_1 + (v_{01}-1)\,\Sigma_0,\ v_{02}\,x_2,\ \dots,\ v_{02}\,x_{n+1}\bigr),
\]

\[
\Sigma_0 = x_2 + \cdots + x_{n+1}
\]

\[
(14)\quad H_0 : \begin{cases}
1 + (u_{01}-2)\,x_1 + (v_{01}-1)\,\Sigma_0 = 0\\
(v_{02}-1)\,x_2 = \cdots = (v_{02}-1)\,x_{n+1} = 0
\end{cases}
\]

\[
(15)\quad \sigma_0(x_1,\dots,x_{n+1}) = \bigl(1 + (\lambda_0-1)\,x_1 + (\mu_0-1)\,\Sigma_0,\ x_2,\ \dots,\ x_{n+1}\bigr),
\qquad \Sigma_0 = x_2+\cdots+x_{n+1}
\]

\[
(16)\quad \sigma_i =
\left(\begin{array}{ccc|c|ccccc|c}
1 & & 0 & 0 & u_{i1} & v_{i1} & v_{i1} & \cdots & v_{i1} & 0\\
 & \ddots & & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots\\
0 & & 1 & 0 & u_{i,i-1} & v_{i,i-1} & v_{i,i-1} & \cdots & v_{i,i-1} & 0\\
 & & & 0 & u_{i,i} & v_{i,i} & v_{i,i} & \cdots & v_{i,i} & 0\\
 & 0 & & 1 & u_{i,i+1} & v_{i,i+1} & v_{i,i+1} & \cdots & v_{i,i+1} & 0\\
 & & & 0 & 0 & v_{i,i+2} & 0 & \cdots & 0 & 0\\
 & & & 0 & 0 & 0 & v_{i,i+2} & & 0 & 0\\
 & & & \vdots & \vdots & & & \ddots & & \vdots\\
 & & & 0 & 0 & 0 & 0 & & v_{i,i+2} & 0\\
\hline
0 & \cdots & 0 & 0 & 0 & 0 & 0 & \cdots & 0 & 1
\end{array}\right)
\]

\[
(17)\quad \begin{aligned}
\sigma_i(x_1,\dots,x_{n+1}) = \bigl(&x_1 + u_{i1}\,x_{i+1} + v_{i1}\,\Sigma_i,\ x_2 + u_{i2}\,x_{i+1} + v_{i2}\,\Sigma_i,\ \dots\\
&\dots,\ x_{i-1} + u_{i,i-1}\,x_{i+1} + v_{i,i-1}\,\Sigma_i ;\\
&u_{i,i}\,x_{i+1} + v_{i,i}\,\Sigma_i,\ x_i + u_{i,i+1}\,x_{i+1} + v_{i,i+1}\,\Sigma_i ;\\
&v_{i,i+2}\,x_{i+2},\ \dots,\ v_{i,i+2}\,x_{n+1}\bigr),
\end{aligned}
\]

\[
\Sigma_i = x_{i+2} + \cdots + x_{n+1}
\]

\[
(18)\quad H_i = \begin{cases}
u_{i1}\,x_{i+1} + v_{i1}\,\Sigma_i = \uncertain{u_{i2}\,x_{i+1} + v_{i2}\,\Sigma_i} = \cdots = u_{i,i-1}\,x_{i+1} + v_{i,i-1}\,\Sigma_i = 0\\
\add{-x_i+}\ u_{ii}\,x_{i+1} + v_{ii}\,\Sigma_i = 0,\quad x_i + (u_{i,i+1}-1)\,x_{i+1} + v_{i,i+1}\,\Sigma_i = 0\\
(v_{i,i+2}-1)\,x_{i+2} = \cdots = (v_{i,i+2}-1)\,x_{n+1} = 0
\end{cases}
\]

\[
(19)\quad \begin{cases}
u_{ii} + u_{i,i+1} = 1, & v_{i,i} + v_{i,i+1} = 0\\
v_{i,i+2} = 1\\
u_{i1} = \cdots = u_{i,i-1} = 0, & v_{i1} = \cdots = v_{i,i-1} = 0
\end{cases}
\]

\[
(20)\quad \begin{aligned}
\sigma_i(x_1,\dots,x_{n+1}) = \bigl(&x_1,\dots,x_{i-1},\ \lambda_i\,x_{i+1} + \mu_i\,\Sigma_i,\\
&x_i + (1-\lambda_i)\,x_{i+1} - \mu_i\,\Sigma_i,\ x_{i+2},\dots,x_{n+1}\bigr),
\end{aligned}
\]

\[
\Sigma_i = x_{i+2}+\cdots+x_{n+1}
\]

\[
\sigma_i^2(x_1,\dots,x_{n+1}) = (x_1,\dots,x_{i-1};\ X_i,\ X_{i+1},\ x_{i+2},\dots,x_{n+1})
\]

\[
\begin{aligned}
X_i &= \lambda_i\bigl[x_i + (1-\lambda_i)\,x_{i+1} - \mu_i\,\Sigma_i\bigr] + \mu_i\,\Sigma_i = \lambda_i\,x_i + \lambda_i(1-\lambda_i)\,x_{i+1}\\
X_{i+1} &= \bigl[\lambda_i\,x_{i+1} + \mu_i\,\Sigma_i\bigr] + (1-\lambda_i)\bigl[x_i + (1-\lambda_i)\,x_{i+1} - \mu_i\,\Sigma_i\bigr] - \mu_i\,\Sigma_i\\
&= (1-\lambda_i)\,x_i + x_{i+1} + (1-\lambda_i)\,\mu_i\,\Sigma_i
\end{aligned}
\]

\[
\sigma_0^2(x_1,\dots,x_{n+1}) = (X_1,\ x_2,\ \dots,\ x_{n+1})
\]

\[
\begin{aligned}
X_1 &= 1 + (\lambda_0-1)\bigl[1 + (\lambda_0-1)\,x_1 + (\mu_0-1)\,\Sigma_0\bigr] + (\mu_0-1)\,\Sigma_0\\
&= \lambda_0 + (\lambda_0-1)^2\,x_1 + \lambda_0(\mu_0-1)\,\Sigma_0
\end{aligned}
\]

\[
\begin{cases}
\sigma_0(x,y) = \bigl(1 + (\lambda_0-1)\,x + (\mu_0-1)\,y,\ y\bigr) = \bigl(1 + (\lambda_0-1)\,x + \alpha_0\,y,\ y\bigr)\\
\qquad = (1 + \eta_0\,x + \alpha_0\,y,\ y)\\
\sigma_1(x,y) = \bigl(\lambda_1\,y,\ x + (1-\lambda_1)\,y\bigr)
\end{cases}
\]

\[
\begin{pmatrix}a & c\\ 0 & b\end{pmatrix}\begin{pmatrix}a' & c'\\ 0 & b'\end{pmatrix}
= \begin{pmatrix}aa' & ac' + b'c\\ 0 & bb'\end{pmatrix}
\overset{\text{si } b'=1}{=} \begin{pmatrix}aa' & c + ac'\\ 0 & b\end{pmatrix}
\]

\[
\begin{pmatrix}a & c\\ 0 & b\end{pmatrix}\begin{pmatrix}\eta_0 & \alpha_0\\ 0 & 1\end{pmatrix}
= \begin{pmatrix}\eta_0\,a & c + \alpha_0\,a\\ 0 & b\end{pmatrix}
\]

\[
\begin{pmatrix}\eta_0 & \alpha_0\\ 0 & 1\end{pmatrix}^2 = \begin{pmatrix}\eta_0^2 & \alpha_0 + \uncertain{\alpha_0\eta_0}\\ 0 & 1\end{pmatrix}
\qquad
\begin{pmatrix}\eta_0 & \alpha_0\\ 0 & 1\end{pmatrix}^3 = \begin{pmatrix}\eta_0^3 & \alpha_0(1 + \eta_0 + \eta_0^2)\\ 0 & 1\end{pmatrix}
\]

\[
\begin{pmatrix}\eta_0 & \alpha_0\\ 0 & 1\end{pmatrix}^n = \begin{pmatrix}\eta_0^n & \alpha_0(1 + \eta_0 + \cdots + \eta_0^{n-1})\\ 0 & 1\end{pmatrix}
\]

\[
\sigma_{0v}^n = 1 \iff
\begin{cases}
\eta_0^n = 1 & \text{i.e. } (\eta_0 - 1)(1 + \eta_0 + \cdots + \eta_0^{n-1}) = 0\\
\struck{\alpha_0 \ldots} & \alpha_0(1 + \eta_0 + \cdots + \eta_0^{n-1}) = 0
\end{cases}
\]

\[
\begin{aligned}
\sigma_0(x,0) &= (1 + \eta_0\,x,\ 0)\\
\sigma_0^2(x,0) &= 1 + \eta_0 + \eta_0^2\,x\\
\sigma_0^3(x,0) &= 1 + \eta_0 + \eta_0^2 + \eta_0^3\,x\\
&\cdots\\
\sigma_0^n(x,0) &= 1 + \eta_0 + \cdots + \eta_0^{n-1} + \eta_0^n\,x =\\
&\qquad \begin{cases}
(\text{cas a}) & \eta_0^n\,x = x\\
(\text{cas b}) & \text{translation par } (n,0) \text{ (qui est } \ill \text{ par l'} \ill\text{)}
\end{cases}
\end{aligned}
\]

\[
\sigma_0^{\varepsilon} = \mathrm{id} \iff 1 + \eta_0 + \cdots + \eta_0^{\varepsilon-1} = 0
\]

\[
\eta_0 = 1\
\begin{cases}
\text{a)}\ \alpha_0 = \uncertain{1} & \sigma_0 = \mathrm{id}\quad \emph{cas impropre}\\[2pt]
\text{b)}\ \alpha_0 \ne 1 & \begin{cases}
p = 0 & \sigma_{0v} \text{ d'ordre infini, a fortiori } \sigma_0 \text{ d'ordre inf.}\\
p > 0 & \sigma_{0v} \text{ d'ordre } p,\ \sigma_0^p\ \struck{= \text{translation}}\ \ldots
\end{cases}
\end{cases}
\]

\[
\begin{aligned}
\sigma_1^2(x,y) &= \bigl(\lambda_1\,x + \lambda_1(1-\lambda_1)\,y,\ (1-\lambda_1)\,x + (\lambda_1 + (1-\lambda_1)^2)\,y\bigr)\\
&= \bigl(\lambda_1\,x + \lambda_1(1-\lambda_1)\,y,\ (1-\lambda_1)\,x + (\lambda_1^2 - \lambda_1 + 1)\,y\bigr)
\end{aligned}
\]

\[
\sigma_1 = \begin{pmatrix}0 & \lambda_1\\ 1 & 1-\lambda_1\end{pmatrix}
\]

\[
\det(t\,\mathrm{id} - \sigma_1) = t^2 - (1-\lambda_1)\,t - \lambda_1 = \struck{t^2 - (1+\delta)\,t + \delta}
\]

\[
\begin{cases}
\sigma_0(x_1,\dots,x_{n+1}) = (1 + \eta_0\,x_1 + \alpha_0\,\Sigma_0,\ x_2,\ \dots,\ x_{n+1})\\
\cdots\\
\sigma_i(x_1,\dots,x_{n+1}) = \bigl(x_1,\dots,x_{i-1},\ -\eta_i\,x_{i+1} + (1+\alpha_i)\,\Sigma_i, & \\
\qquad x_i + (\eta_i + 1)\,x_{i+1} - (1+\alpha_i)\,\Sigma_i,\ x_{i+2},\dots,x_{n+1}\bigr) & (1\le i\le n-1)\\
\cdots\\
\sigma_n(x_1,\dots,x_{n+1}) = \bigl(x_1,\ \dots,\ x_{n-1},\ -\eta_n\,x_{n+1},\ x_n + (1+\eta_n)\,x_{n+1}\bigr)
\end{cases}
\]

\[
\Gamma_n = \{\sigma_0, \dots, \sigma_n \mid \sigma_0^2 = \cdots = \sigma_n^2 = 1,\ \sigma_i\sigma_j = \sigma_j\sigma_i \text{ si } j\ge i+2\}
\]

\[
D_i = E/(\sigma_0, \dots, \widehat{\sigma_i}, \dots, \sigma_n) = E/\Gamma_n^{-}(i)\times\Gamma_n^{+}(i)
\]

\[
E^{-}(i) = E/\Gamma_n^{-}(i) \longrightarrow D_i
\]

\[
E^{-}(i)/(\sigma_0, \dots, \widehat{\sigma_j}, \dots, \sigma_{i-1}) \simeq E/(\sigma_0, \dots, \widehat{\sigma_j}, \dots, \widehat{\sigma_i}, \dots, \sigma_n)
\]

\[
E^{+}(i) = E/\Gamma_n^{+}(i) \longrightarrow D_i \ \overset{\text{\struck{?}}}{=}\ D_{i,i+1,\dots,n}
\]

\[
f_i < f_j
\]

\[
E(i,j) = E/(\sigma_0,\dots,\sigma_{i-1};\ \sigma_{j+1},\dots,\sigma_n) = E/G^{-}(i)\times G^{+}(j) = D_{i,i+1,\dots,j}
\]

\[
E(i,j) \longrightarrow D_{i,j} .
\]

\[
\begin{cases}
p_0 = \text{ordre de } \sigma_0\sigma_1\\
p_1 = \text{ordre de } \sigma_1\sigma_2\\
\quad\dots\\
p_{n-1} = \text{ordre de } \sigma_{n-1}\sigma_n
\end{cases}
\]

\[
\bigl(E/G^{+}_{n}(2)\bigr)_{f_2} \quad\text{ou}\quad E^{-}(2)_{f_2}
\]

\[
(\sigma_0\sigma_1)^{p}x = g_x x \qquad g_x\in G^{+}_{n}(2)
\]

\[
\mathfrak{G}_n = \bigl\{\sigma_0,\dots,\sigma_n \bigm| \sigma_0^2=\dots=\sigma_n^2=1,\ \sigma_i\sigma_j=\sigma_j\sigma_i \text{ si } j\geq i+2\bigr\}\ )
\]

\[
(1)\qquad R \xrightarrow{\ \varphi_R\ } \mathrm{Drap\,max}(E) = \bigl\{(E_0\subset E_1\subset\dots\subset E_i\subset\dots\subset E_n\subset E)\bigr\}
\]

\[
(2)\qquad
\begin{cases}
\pi_j(\varphi_R(\sigma_i r)) = \pi_j\varphi_R(r) \quad\text{si}\\
0\leq i,j\leq n,\ i\neq j
\end{cases}
\]

\[
(3)\qquad D_j(\Pi) \xrightarrow{\ \varphi_j\ } \mathrm{Grass\,aff}_j(E) \qquad (0\leq j\leq n)
\]

\[
\begin{cases}
\varphi_R(r) = (E_0\subset E_1\subset\dots\subset E_n\subset E)\\
\sigma_i E_j = E_j \quad\text{si } i\neq j \qquad \dots
\end{cases}
\]

\[
(4)\qquad D_{\{j_0,j_1,\dots,j_p\}}(\Pi) \xrightarrow{\ \varphi_{\{j_0,\dots,j_p\}}\ } \mathrm{Drap\,aff}_{\{j_0,j_1,\dots,j_p\}}(\Pi)
\]

\[
R_{ij} = R/(\sigma_0,\dots,\sigma_{i-1},\sigma_{j+1},\dots,\sigma_n)
\]

\[
\begin{matrix}
\sigma_{i+1}, & \dots, & \sigma_{j-1}\\
\shortparallel & & \shortparallel\\
\sigma'_0, & \dots, & \sigma'_{j-i-2=\nu}
\end{matrix}
\]

\[
= \text{image réciproque de } \{f_i\} \text{ par } D_{0,1,\dots,i}(\Pi)\to D_i(\Pi).
\]

\[
\begin{cases}
\varphi_S : \mathrm{S}\to E\\
\varphi_A : \mathrm{A}\to \mathrm{Dr}(E)\\
\varphi_F : \mathrm{F}\to \mathrm{Pl}(E)
\end{cases}
\]

\[
\left.\begin{aligned}
a_0&=\mathrm{dr}(s_0,s_1)\\
f_0&=\mathrm{pl}(s_0,s_1,s_2)
\end{aligned}\right\}\Longrightarrow \varphi_A,\varphi_F \text{ connus quand on connaît } \varphi_S
\]

\[
\left.\begin{aligned}
a_0&=f_0\cap f_1\\
s_0&=f_0\cap f_1\cap f_2
\end{aligned}\right\}\Longrightarrow \varphi_S,\varphi_A \text{ connus quand on connaît } \varphi_F
\]

\[
\left.\begin{aligned}
s_0&=a_0\cap a_1\\
f_0&=\mathrm{pl}(a_0,a_1)
\end{aligned}\right\}\Longrightarrow \varphi_S,\varphi_F \text{ connus quand on connaît } \varphi_A
\]

\[
\begin{cases}
g\circ\varphi_S = \varphi_S\circ\mathrm{g}\\
g\circ\varphi_A = \varphi_A\circ\mathrm{g}\\
g\circ\varphi_F = \varphi_F\circ\mathrm{g}
\end{cases}
\]

\[
\begin{cases}
\sigma_0(s_0)\overset{\mathrm{df}}{=}s_1, & \sigma_0(a_0)=a_0,\quad \sigma_0(f_0)=f_0\\
\sigma_1(s_0)=s_0 & \sigma_1(a_0)\overset{\mathrm{df}}{=}a_1,\quad \sigma_1(a_0)=a_0\\
\sigma_2(s_0)=s_0 & \sigma_2(a_0)=a_0,\quad \sigma_2(f_0)\overset{\mathrm{df}}{=}f_1
\end{cases}
\]

\[
\varphi_R : \mathrm{R}\longrightarrow \mathrm{Drap}_{012}(E)
\]

\[
\begin{matrix}
 & \sigma_0(a_0)=a_0 & \sigma_0(f_0)=f_0\\
\sigma_1(s_0)=s_0 & & \sigma_1(f_0)=f_0\\
\sigma_2(s_0)=s_0 & \sigma_2(a_0)=a_0 &
\end{matrix}
\]

\[
1\to\Gamma^0\to\Gamma\to\mathrm{Aut}(G)
\]

\[
\begin{matrix}
K_1\simeq G/H_1 \\
\downarrow \\
K_0=G/H_0
\end{matrix}
\qquad \sigma\in N(H)/H \qquad \sigma^2=1
\]

\[
\begin{matrix}
1\to\Gamma^0\to\Gamma\to\mathrm{Aut}(\mathcal{C})\\
*\\
\Gamma\to G \qquad 1\to\Gamma^1\to\Gamma^0\to N(H)/H_0\\
*\\
1\to\Gamma^2\to\Gamma^1\hookrightarrow\mathrm{Aut}_G\,\pi_e\\
*\\
\pi_e,\ \mathrm{Hom}(e,e')\qquad \pi_e\\
\shortparallel\\
T_{e,e'} \qquad N_{F(\pi_e)}H_0
\end{matrix}
\]

\[
1 \to R_n \to \mathrm{Sl}(2,\mathbb{Z})/\pm 1 \to \mathrm{Sl}(2,\mathbb{Z}_2) \to 0
\]

\[
\begin{cases}
\rho_s = \sigma_0\sigma_1 \\
\rho_f = \sigma_1\sigma_2 \\
\rho_s\rho_f = \sigma_0\sigma_2 = \sigma
\end{cases}
\qquad \rho_s = \sigma\rho_f^{-1},
\qquad \boxed{\rho_f^3 = 1,\ \ \sigma^2 = 1}
\]

\[
\frac{az+b}{cz+d} = z, \qquad cz^2 + (d-a)z - b = 0, \qquad z = x+iy \ (y \neq 0)
\]

\[
\begin{cases}
c(x^2-y^2) + (d-a)x - b = 0 \\
\bigl[2cx + (d-a)\bigr]y = 0 \iff 2cx + (d-a) = 0
\end{cases}
\Longleftrightarrow
\begin{cases}
d-a = -2cx \\
c(x^2+y^2) + b = 0
\end{cases}
\]

\[
\begin{cases}
d = a - c\,\mathrm{Tr}(z) \\
b = -c\,N(z)
\end{cases}
\]

\[
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
= \begin{pmatrix} a & -cN(z) \\ c & a - c\,\mathrm{Tr}\,z \end{pmatrix}
\]

\[
\det = a^2 - ac\,\mathrm{Tr}\,z + c^2N(z) = (a-cz)(a-c\bar z) = N(a-cz) = 1
\]

\[
G_z = G_i = \left\{ \begin{pmatrix} a & -c \\ c & a \end{pmatrix} \,\middle|\, a^2+c^2 = 1 \right\}
\]

\[
\boxed{\sigma = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}}
\qquad \sigma^4 = \mathrm{id}, \qquad \boxed{\sigma^2 = -1}
\]

\[
G_z = G_\zeta = \left\{ \begin{pmatrix} a & -c \\ c & a-c \end{pmatrix} \,\middle|\, a^2 - ac + c^2 = 1 \right\},
\qquad N(a - c\zeta) = 1
\]

\[
\boxed{\rho_f = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}}
\qquad \rho^6 = \mathrm{id}, \qquad \boxed{\rho^3 = -1}
\]

\[
\boxed{\rho'_s = \sigma\rho_f = \sigma_2\rho_f\sigma_2^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}}
\qquad \rho'^{\,n}_s = \begin{pmatrix} 1 & -n \\ 0 & 1 \end{pmatrix}
\]

\[
\Gamma^0_{\infty,3} \simeq \mathrm{Sl}(2,\mathbb{Z})/\pm 1
\]

\[
\Gamma^0_{n,3} \xrightarrow{\ \text{épi}\ } \mathrm{Sl}(2,\mathbb{Z}_n)/\pm 1 \quad (n \geq 2) \quad (\text{iso \emph{ssi} } n \leq 5)
\]

\[
\begin{array}{c|c|c|l}
n & \text{ordre } \Gamma^0_{n,3} & \text{ordre } \mathrm{Sl}(2,\mathbb{Z}_n)/\pm 1 & \\ \hline
2 & (\mathfrak{S}_3)\ 6 & 6 & \text{cas 3-diédral} \\
3 & (\mathfrak{A}_4)\ 12 & 12 & \text{tétraèdre} \\
4 & (\mathfrak{S}_4)\ 24 & 24 & \text{cube orienté} \\
5 & (\mathfrak{A}_5)\ 60 & 60 & \text{icos. orienté} \\
6 & (\mathbb{Z}_6\cdot\mathbb{Z}^2)\ \infty & 72 & \text{pavage par triangles orienté du plan}
\end{array}
\]

\[
\mathrm{card}\,\mathrm{Sl}(2,\mathbb{F}_q) = (q+1)q(q-1) = q(q^2-1)
\]

\[
\mathrm{card}\,\mathrm{Sl}(2,\mathbb{Z}/p^\nu\mathbb{Z}) = \bigl(\mathrm{card}\,\mathrm{Sl}(2,\mathbb{F}_p)\bigr)\,p^{3(\nu-1)}
\]

\[
\begin{array}{lll}
\sigma_0 s_0 = s_1 & \sigma_0 a_0 = a_0 & \sigma_0 f_0 = f_0 \\
\sigma_1 s_0 = s_0 & \sigma_1 a_0 = \{s_0, s_2\} & \sigma_1 f_0 = f_0 \\
\sigma_2 s_0 = s_0 & \sigma_2 a_0 = a_0 & \sigma_2 f_0 = \{s_0, s_1, s_3, \ldots\}
\end{array}
\qquad
\begin{array}{l}
a_0 = \{s_0, s_1\} \\
f_0 = \{s_0, s_1, s_2, \ldots\}
\end{array}
\]

\[
\sigma_0 s_0 = s_1, \qquad \sigma_1 s_1 = s_2, \qquad \sigma_2 s_2 = s_3
\]

\[
\begin{cases}
e_1 = s_1 - s_0 \\
e_2 = s_2 - s_0 \\
e_3 = s_3 - s_0
\end{cases}
\]

\[
\sigma_1 \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_2 \\ s_2 \mapsto s_1 \\ s_3 \mapsto s''_3 \end{cases}
\qquad
\sigma_2 \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_1 \\ s_2 \mapsto s_3 \\ s_3 \mapsto s_2 \end{cases}
\qquad
\sigma_0 \begin{cases} s_0 \mapsto s_1 \\ s_1 \mapsto s_0 \\ s_2 \mapsto s'_2 \\ s_3 \mapsto s'_3 \end{cases}
\]

\[
\boxed{\mathrm{Pl}(s_0, s_1, s_2) = \mathrm{Pl}(s_0, s_1, s'_2)}
\qquad
\boxed{\mathrm{Pl}(s_0, s_1, s_3) = \mathrm{Pl}(s_0, s_1, s'_3)}
\]

\[
\begin{array}{l|ll}
s_0 \mapsto & s_1 & s_1 \\
s_1 \mapsto & s_0 & s_0 \\
s_2 \mapsto & s'_3 & \sigma_2 s'_2 \\
s_3 \mapsto & s'_2 & \sigma_2 s'_3
\end{array}
\]

\[
\boxed{\sigma_0^2 = 1} \quad \boxed{\sigma_1^2 = 1} \quad \sigma_2^2 = 1 \text{ automatique}
\]

\[
\boxed{\begin{array}{l} \sigma_0 s'_2 = s_2 \\ \sigma_0 s'_3 = s_3 \end{array}}
\qquad
\boxed{\sigma_1 s''_3 = s_3}
\qquad
\boxed{\sigma_2 s'_2 = s'_3}, \quad \sigma_2 s'_3 = s'_2
\]

\[
s'_2 = s_0 + \lambda e_1 + \mu e_2 = (\lambda, \mu, 0),
\qquad
s'_3 = s_0 + \lambda e_1 + \mu e_3 = (\lambda, 0, \mu)
\]

\[
\Bigl[\ \underbrace{\mu = 1}_{?},\ \lambda = 1 + \beta \ldots\ \Bigr]
\]

\[
s''_3 = s_3 + \xi e_1 + \eta e_2 + \zeta e_3 = s_0 + \xi e_1 + \eta e_2 + (1+\zeta)e_3 = (\xi, \eta, 1-\zeta)
\]

\[
\Bigl[\ \underbrace{\xi = -\eta}_{?},\ \eta = 1 + \alpha,\ \underbrace{\zeta = 0}_{?}\ \Bigr]
\]

\[
\begin{cases}
\sigma_0 s_0 = s_1 = (1,0,0) \\
\sigma_0 e_1 = -e_1 \\
\sigma_0 e_2 = (\lambda-1)e_1 + \mu e_2 \\
\sigma_0 e_3 = (\lambda-1)e_1 + \mu e_3
\end{cases}
\qquad
\sigma_0 = \begin{pmatrix}
-1 & \lambda-1 & \lambda-1 & 1 \\
0 & \mu & 0 & 0 \\
0 & 0 & \mu & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

\[
\boxed{\sigma_0(x,y,z) = \bigl(1 - x + (\lambda-1)y + (\lambda-1)z,\ \mu y,\ \mu z\bigr)}
\]

\[
\sigma_0(s'_2) = \sigma_0(\lambda,\mu,0) = \bigl(1-\lambda+(\lambda-1)\mu,\ \mu^2,\ 0\bigr) \overset{?}{=} s_2 = (0,1,0)
\]

\[
\boxed{\mu^2 = 1 \qquad (\lambda-1)(\mu-1) = 0}
\]

\[
\begin{cases}
\sigma_1 s_0 = s_0 = (0,0,0) \\
\sigma_1 e_1 = e_2 \\
\sigma_1 e_2 = e_1 \\
\sigma_1 e_3 = \xi e_1 + \eta e_2 + (1+\zeta)e_3
\end{cases}
\qquad
\sigma_1 = \begin{pmatrix}
0 & 1 & \xi & 0 \\
1 & 0 & \eta & 0 \\
0 & 0 & 1+\zeta & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\]

\[
\boxed{\sigma_1(x,y,z) = \bigl(y + \xi z,\ x + \eta z,\ (1+\zeta)z\bigr)}
\]

\[
\sigma_1(s''_3) = \sigma_1(\xi, \eta, 1-\zeta) = \Bigl(\eta + \xi(1-\zeta),\ \xi + \eta(1-\zeta),\ \underbrace{(1+\zeta)(1-\zeta)}_{1-\zeta^2}\Bigr)
\overset{?}{=} s_3 = (0,0,1)
\]

\[
\zeta^2 = 0, \qquad \eta + \xi - \xi\zeta = \eta + \xi - \eta\zeta = 0
\]

\[
\boxed{\zeta^2 = 0 \qquad \eta + \xi = \xi\zeta = \eta\zeta}
\]

\[
\det\sigma_{0V} = -\mu^2 = -1, \qquad
\det\sigma_{1V} = -(1+\zeta), \qquad
\det\sigma_{2V} = -1
\]

\[
\boxed{(\det\sigma_{1V} = -1) \iff \zeta = 0} \ \Longrightarrow\ \xi = -\eta
\]

\[
\sigma_1(x,y,z) = (x,y,z) \iff -x + y + \xi z = 0,\ -x + y - \eta z = 0,\ \struck{(1+\zeta)}\,\zeta z = 0
\]

\[
\Updownarrow
\]

\[
\xi + \eta \struck{+1+\zeta = 0} = \zeta \qquad (= \xi\zeta = \eta\zeta)
\]

\[
\begin{cases}
\lambda = 1 + \beta \\
\eta = 1 + \alpha
\end{cases}
\]

\[
\sigma_0 = \begin{pmatrix}
-1 & \beta & \beta & 1 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\qquad
\sigma_0(x,y,z) = \bigl(1 - x + \beta(y+z),\ y,\ z\bigr)
\]

\[
\sigma_1 = \begin{pmatrix}
0 & 1 & -(1+\alpha) & 0 \\
1 & 0 & 1+\alpha & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\qquad
\sigma_1(x,y,z) = \bigl(y - (1+\alpha)z,\ x + (1+\alpha)z,\ z\bigr)
\]

\[
\sigma_2 = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\qquad
\sigma_2(x,y,z) = (x, z, y)
\]

\[
\rho_s = \sigma_1\sigma_2 \qquad \rho_f = \sigma_0\sigma_1
\]

\[
\rho_s^p = \mathrm{id} \iff F_p(\alpha) = 0 \qquad (\rho_s^d \neq \mathrm{id} \text{ si } 0 < d < p)
\]

\[
\rho_f^q = \mathrm{id} \iff F_q(\beta) = 0 \qquad (\rho_f^d \neq \mathrm{id} \text{ si } 0 < d < q)
\]

\[
ax^2 + by^2 + cz^2 + uyz + vzx + wxy + px + qy + rz + d
\]

\[
f(x,y,z) = a\bigl[x^2 + y^2 + z^2 \struck{\ill{}} - \beta x(y+z) - x - y - z\bigr] + uyz
\]

\[
\struck{u = -a(\beta + 2(1+\alpha) + {}}
\]

\[
u = -a\bigl(2 + 2(\alpha+\beta) + \alpha\beta\bigr)
\]

\[
f(x,y,z) = x^2 + y^2 + z^2 - \underbrace{\bigl(2 + 2(\alpha+\beta) + \alpha\beta\bigr)}_{\gamma = (\alpha+2)(\beta+2) - 2} yz - \beta zx - \beta xy - (x+y+z)
\]

\[
\begin{cases}
\text{Matrice de } f_0 \ (\text{quadratique}) = \begin{pmatrix} 2 & -\beta & -\beta \\ -\beta & 2 & -\gamma \\ -\beta & -\gamma & 2 \end{pmatrix} \\[4ex]
\delta'\ (\text{discrim.\ divisé}) = 4 - \beta^2(\gamma+2) - \gamma^2 = -(\alpha+2)(\beta+2)^2(\alpha-\beta) \\[1ex]
\gamma = 2(1+\alpha+\beta) + \alpha\beta
\end{cases}
\]

\[
\text{Matrice de } f = \begin{pmatrix}
2 & -\beta & -\beta & -1 \\
-\beta & 2 & -\gamma & -1 \\
-\beta & -\gamma & 2 & -1 \\
-1 & -1 & -1 & 0
\end{pmatrix}
\]

\[
\delta(f) = (\gamma+2)(\gamma - 4\beta - 6) = (\alpha+2)(\alpha-2)(\beta+2)^2
\]

\[
\sigma = \Bigl(\frac{\alpha}{2(\alpha+\beta)},\ \frac{-1}{2(\alpha+\beta)},\ \frac{-1}{2(\alpha+\beta)}\Bigr)
\]

\[
\mathbf{a}(x,y,z) = \Bigl(\frac{\alpha}{\alpha+\beta} - x,\ \frac{-1}{\alpha+\beta} - y,\ \frac{-1}{\alpha+\beta} - z\Bigr)
\]

\[
\struck{f(\sigma) = \frac{-\alpha^2 - \alpha\beta + 2(\alpha+\beta)}{4(\alpha+\beta)^2}}
\qquad
\boxed{f(\sigma) = \frac{2-\alpha}{4(\alpha+\beta)}}
\]

\[
\boxed{F(X,Y,Z) = (-X+Y+Z)^2 - 4(\alpha+2)YZ + \underbrace{\frac{2-\alpha}{4(\alpha+2)}}_{\lambda > 0}}
\]

\[
-2 < \alpha < 2, \qquad 0 < 4(\alpha+2) < 16
\]

\[
4YZ = (Y+Z)^2 - (Y-Z)^2
\qquad
\begin{cases}
-X + Y + Z = U\sqrt{\lambda} \\
\sqrt{\alpha+2}\,(Y+Z) = W\sqrt{\lambda} \\
\sqrt{\alpha+2}\,(Y-Z) = V\sqrt{\lambda}
\end{cases}
\]

\[
F_1(U,V,W) = U^2 \struck{+ \ldots}\ (V^2 - W^2) + \ldots\ 1
\]

\[
U^2 + V^2 - W^2 + \struck{\lambda} = 0, \qquad U^2 = V^2 + W^2 + \struck{\lambda}
\]

\[
\struck{W^2 - \rho UV + \sigma = 0} \qquad \struck{\rho UV = W^2 + \sigma}
\]

\[
\struck{W^2 - \rho YZ + \sigma = 0} \qquad \struck{\rho YZ = W^2 + \sigma}
\]

\[
W^2 = U^2 + V^2 + 1
\]

\[
\beta = 2 \qquad \alpha \neq -2
\]

\[
\gamma = 2(3+\alpha) + 2\alpha = 2(2\alpha+3) \qquad \gamma + 2 = 2(2\alpha+4) = 4(\alpha+2)
\]

\[
\delta' = 4 - 4(\gamma+2) - \gamma^2 = -(\gamma^2 + 4\gamma + 4) = -(\gamma+2)^2 = -16(\alpha+2)^2
\]

\[
f_0(x,y,z) = x^2 + y^2 + z^2 - 2(2\alpha+3)yz - 2zx - 2xy = (-x+y+z)^2 - 2(2\alpha+4)yz
\]

\[
f_0(x,y,z) = (-x+y+z)^2 - 4(\alpha+2)yz
\]

\[
\sigma_0 : \begin{cases} s_0 \leftrightarrow s_1 \\ s_2 \mapsto s_2 \\ s_3 \mapsto s_3 \end{cases}
\qquad
\begin{cases} e_1 \mapsto -e_1 \\ e_2 \mapsto -e_1 + e_2 \\ e_3 \mapsto -e_1 + e_3 \end{cases}
\qquad
\sigma_0 = \begin{pmatrix} -1 & -1 & -1 & 1 \\ 0 & +1 & 0 & 0 \\ 0 & 0 & +1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\]

\[
\boxed{\sigma_0(x,y,z) = \bigl(1 - (x+y+z),\ y,\ z\bigr)}
\]

\[
\sigma_1 : \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_2 \\ s_2 \mapsto s_1 \\ s_3 \mapsto s'_3 = s_0 + (1+\alpha)(e_2 - e_1) + e_3 \end{cases}
\]

\[
\begin{cases} e_1 \mapsto e_2 \\ e_2 \mapsto e_1 \\ e_3 \mapsto (1+\alpha)(e_2 - e_1) + e_3 = \alpha' e_1 - \alpha' e_2 + e_3 \end{cases}
\]

\[
\sigma_1 = \begin{pmatrix} 0 & 1 & \alpha' & 0 \\ 1 & 0 & -\alpha' & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\qquad
\boxed{\sigma_1(x,y,z) = (y + \alpha' z,\ x - \alpha' z,\ z)}
\]

\[
\sigma_2 : \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_1 \\ s_2 \mapsto s_3 \\ s_3 \mapsto s_2 \end{cases}
\qquad
\begin{cases} e_1 \mapsto e_1 \\ e_2 \mapsto e_3 \\ e_3 \mapsto e_2 \end{cases}
\qquad
\sigma_2 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
\qquad
\boxed{\sigma_2(x,y,z) = (x, z, y)}
\]

\[
\begin{cases}
H_0 : 2x + y + z - 1 = 0 \\
H_1 : x - y - \alpha' z = 0 \\
H_2 : y - z = 0
\end{cases}
\qquad
\begin{cases}
\Delta_0 : y = z = 0 & \text{i.e. } \Delta_0 = \mathcal{O}_S\cdot e_1 \\
\Delta_1 : z = 0,\ x + y = 0 & \text{i.e. } \Delta_1 = \mathcal{O}_S(e_1 - e_2) \\
\Delta_2 : x = 0,\ y + z = 0 & \text{i.e. } \Delta_2 = \mathcal{O}_S(e_2 - e_3)
\end{cases}
\]

\[
\begin{cases}
D_0 = H_1 \cap H_2 : y = z = \alpha' x \\ \qquad D_0 = \mathcal{O}_S(1, \alpha', \alpha') = \mathcal{O}_S(-\alpha, 1, 1) \\
D_1 = H_2 \cap H_0 : y = z = \tfrac12 - x \ (\text{car.} \neq 2) \\ \qquad \mathrm{hom}\,D_1 : (y = z = -x) = \mathcal{O}_S(1, -1, -1) \\
D_2 = H_0 \cap H_1 : y = (1-3\alpha)x + \alpha,\ z = \ldots(1-\alpha)x + (1-\alpha) \\ \qquad \mathrm{hom}(D_2) = \mathcal{O}_S\bigl(1, 1-3\alpha, \ldots(1-\alpha)\bigr)
\end{cases}
\]

\[
\begin{array}{ll}
s_0 = (0,0,0) & \mathbf{a}s_0 = (\alpha', 1-\alpha', 1-\alpha') \\
s_1 = (1,0,0) & \mathbf{a}s_1 = (\alpha'-1, 1-\alpha', 1-\alpha') \\
s_2 = (0,1,0) & \mathbf{a}s_2 = (\alpha', -\alpha', 1-\alpha') \\
s_3 = (0,0,1) & \mathbf{a}s_3 = (\alpha', 1-\alpha', -\alpha') \\
s'_3 = (\alpha', -\alpha', 1) & \mathbf{a}s'_3 = (0, 1, -\alpha') \\
s''_3 = (0, -\alpha', 1) & \mathbf{a}s''_3 = (\alpha', 1-2\alpha', -\alpha')
\end{array}
\]

\[
\begin{array}{l}
\mathbf{a}(x,y,z) = \bigl(\alpha' - x,\ (1-\alpha') - y,\ (1-\alpha') - z\bigr) \\[1ex]
c = \tfrac12(\alpha', 1-\alpha', 1-\alpha') \quad (\text{centre}) \\[1ex]
u = 2c = (\alpha', 1-\alpha', 1-\alpha', 2)
\end{array}
\]

\[
q_0(x,y,z) = x^2 + y^2 + z^2 - \alpha yz + x(y+z) - (x+y+z)
\]

\[
\Bigl[\ = (\text{en car.} \neq 2)\ \alpha\bigl[(x - \alpha y)^2 + \alpha'(x-\alpha y)(x - \alpha z) + (x - \alpha z)^2\bigr] - (x+y+z)\ \Bigr]
\]

\[
\begin{cases}
q(e) = 1 \quad (\text{c'est la normalisation de } q\,!) \\
q(f) = \alpha + 2 \\
q(f)/q(e) = \alpha + 2 = (\alpha+1)^2 = \alpha'^2
\end{cases}
\]

\[
\struck{\ldots} = \frac{q(f)}{q(e)} = \frac{4y^2}{4x^2} = \Bigl(\frac{y}{x}\Bigr)^2 = \alpha'^2 \quad \text{donc} \quad \frac{y}{x} = \pm\alpha'
\]

\[
\delta'(q_0) = 1 \qquad \text{Matrice } \varphi_{q_0} = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & -\alpha \\ 1 & -\alpha & 2 \end{pmatrix}
\]

\[
\struck{\ldots} \qquad \boxed{\Delta' = \alpha'\Delta, \quad \Delta = \alpha\Delta'}
\]

\[
\Delta = \struck{\ldots}\,\mathcal{A}', \quad \Delta' = \struck{\ldots}\,\mathcal{A}
\qquad
\begin{cases} \mathcal{A} = \alpha'\mathcal{A}' \\ \mathcal{A}' = \alpha\mathcal{A} \end{cases}
\]

\[
\Phi' = \lambda\Phi, \quad \Phi = \lambda'\Phi' \qquad (\lambda\lambda' = \mathrm{Norm}(\lambda) = 1) \qquad \lambda = 2\alpha - 1
\]

\[
\boxed{\Psi' = \mu\Psi, \ \ \Psi = \mu'\Psi'} \qquad \struck{\mu\mu' = \mathrm{Norm}(\mu) = 1} \quad \gamma = -\lambda
\]

\[
\boxed{\Psi = \mu\Phi, \ \ \Psi' = \mu'\Phi'} \qquad \mu = 1 + 2\alpha = -(1+2\alpha') \quad (\mu\mu' = -5),\ \mu^2 = 5 \text{ i.e. } \mu = \sqrt5
\]

\[
\Sigma = \rho S = \rho' S \qquad \rho' = -\rho \quad \text{i.e. } \mathrm{Tr}\,\rho = 0 \quad \struck{\ldots} \qquad \rho = \mu = 1 + 2\alpha\ldots
\]

\[
\begin{array}{rl}
6 & \text{directions de sommets} \\
15 & \text{directions d'arêtes} \\
10 & \text{directions de plans} \\ \hline
31 & = 5^2 + 5 + 1 \text{ nb de pts du plan projectif sur } \mathbb{F}_5
\end{array}
\]

\[
1 + 12 + 60 + 40 = 113 \qquad 125 = 1 + \underbrace{24}_{\substack{\text{pts sur la conique}\\ = 2\times 12\\ (\text{deux icos.})}} + 60 + 40
\]

\[
y = z \qquad x = (1+\alpha')y = -\alpha y \qquad y = \alpha' x
\]

\[
(-\alpha y, y, y) \text{ ou } (x, \alpha' x, \alpha' x), \qquad (-\alpha, 1, 1) \qquad (1, \alpha', \alpha')
\]

\[
-\alpha = \alpha'(1-\alpha') = \alpha' - \alpha'^2 \qquad \alpha' = -\alpha(1-\alpha') = -\alpha - 1 \ \text{ok}
\]

\[
\mathcal{O}_S(-\alpha, 1, 1)
\qquad
\begin{array}{l} y + z + 1 = 0 \\ x + y + \alpha z = 0 \\ y = z \end{array}
\]

\[
\begin{cases}
\bar s_0 = (-\alpha, 1, 1) \\
\bar s_1 = \sigma_0 \bar s_0 = (\alpha - 1, 1, 1) \\
\text{d'où } \bar s_1 - \bar s_0 = (2\alpha - 1, 0, 0), \quad \alpha' = 1 - \alpha'
\end{cases}
\qquad \alpha + \alpha' = -1
\]

\[
f(0) = f(\sigma_1\sigma_2) = 0
\qquad
f(\sigma_0^2) = f(\sigma_0) + \sigma_0 f(\sigma_0) \ldots
\qquad e + \sigma_0 e = 0, \quad \boxed{\sigma_0 e = -e}
\]

\[
\begin{cases}
\Gamma \to SO(q) \\
+ \text{ 1-cocycle } \Gamma \xrightarrow{\ f\ } E
\end{cases}
\qquad
f(gg') = f(g) + g f(g')
\]

\[
f(\sigma_1) = f(\sigma_2) = 0 \qquad f(\sigma_0) = \struck{\ldots}
\]

\[
\sigma_0\sigma_2\sigma_0\sigma_2 \qquad f(\sigma_0\sigma_2) = \struck{\ldots}\, e \qquad e + (\sigma_0\sigma_2)e = \ldots \qquad \struck{\sigma_0\sigma_2 e = \ldots} \qquad \sigma_0(\sigma_2 e - e) \ldots
\]

\[
\sigma_0\sigma_1 = u \qquad f(u) = e,\ f(u^2) = e + ue,\ \ldots
\qquad
\frac{e_1 \wedge e_2 \wedge e_3}{q}
\]

\[
f(u^3) = f(u \cdot u^2) = e + u(e + ue) = e + ue + u^2 e
\]

\[
s_0 \mapsto s_1, \qquad s_2 - s_0 = e_2 - e_3 = (0, 1, -1)
\qquad
\boxed{q_0(e) = 1}
\]

\[
1 + 1 + \alpha = 2 + \alpha = \struck{\ldots}\ 1 - \alpha' = \boxed{\alpha'^2}
\]

\[
\boxed{\begin{cases} \sigma_0 e = -e \\ \sigma_2 e = e \end{cases}}
\iff (1 + \sigma_0)e = 0
\iff (1 + \sigma_0\sigma_2)e = 0
\qquad
\boxed{(1 + u + u^2)e = 0}\ \struck{\ldots}
\]

\[ R = (f_0, f_1, f_2) \qquad
\begin{cases}
\sigma_0 f_0 \neq f_0, & \sigma_0 f_1 = f_1, \quad \sigma_0 f_2 = f_2 \\
\sigma_1 f_0 = f_0, & \sigma_1 f_1 \neq f_1, \quad \sigma_1 f_2 = f_2 \\
\sigma_2 f_0 = f_0, & \sigma_2 f_1 = f_1
\end{cases} \]

\[ \sigma_0 : \begin{cases} s_0 \mapsto s_1 \\ s_1 \mapsto s_0 \\ s_2 \mapsto s_2 \\ s_3 \mapsto s_3 \end{cases}
\qquad
\sigma_1 : \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_2 \\ s_2 \mapsto s_1 \\ s_3 \mapsto s'_3 \overset{?}{=} \struck{s''_3} \end{cases}
\qquad
\sigma_2 : \begin{cases} s_0 \mapsto s_0 \\ s_1 \mapsto s_1 \\ s_2 \mapsto s_3 \\ s_3 \mapsto s_2 \end{cases} \]

\[ \begin{cases} e_1 = s_1 - s_0 \\ e_2 = s_2 - s_0 \\ e_3 = s_3 - s_0 \end{cases} \]

\[ s'_3 = s_0 + \lambda e_1 + \mu e_2 + \nu e_3 \quad \bigl(= s_0 + \lambda e_1 - \nu\lambda e_2 + \nu e_3\bigr) \]

\[ \sigma_0 : \begin{cases} e_1 \mapsto -e_1 \\ e_2 \mapsto -e_1 + e_2 \\ e_3 \mapsto -e_1 + e_3 \end{cases}
\qquad \struck{\uncertain{\text{matrice}}} \quad
\sigma_0 = \begin{pmatrix} -1 & -1 & -1 & \ill{} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

\[ \sigma_1 : \begin{cases} e_1 \mapsto \struck{e_0}\ e_2 \\ e_2 \mapsto e_1 \\ e_3 \mapsto \lambda e_1 + \mu e_2 + \nu e_3 \quad (= \lambda e_1 - \lambda e_2 + e_3) \end{cases}
\qquad \struck{\uncertain{\text{matrice}}} \quad
\sigma_1 = \begin{pmatrix} 0 & 1 & \lambda & 0 \\ 1 & 0 & \mu & 0 \\ 0 & 0 & \nu & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

\[ \sigma_2 : \begin{cases} e_1 \mapsto e_1 \\ e_2 \mapsto e_3 \\ e_3 \mapsto e_2 \end{cases}
\qquad
\sigma_2 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

\[ \sigma_1^2 = \mathrm{id} \]

\[ \sigma_1^2(s_3) = s_3 \quad \text{i.e.} \quad \sigma_1(s'_3) = s_3 \quad \text{or} \]

\begin{align*}
\sigma_1(s'_3) &= \sigma_1 s_0 + \struck{\sigma_1 e_3 = s_0 +} \lambda\sigma_1 e_1 + \mu\sigma_1 e_2 + \nu\sigma_1 e_3 \\
&= s_0 + \lambda e_2 + \mu e_1 + \nu(\lambda e_1 + \mu e_2 + \nu e_3) \\
&= s_0 + (\mu + \nu\lambda) e_1 + (\lambda + \nu\mu) e_2 + \nu^2 e_3 .
\end{align*}

\[ \begin{cases} \nu^2 = 1 \quad \struck{\mu + \nu\lambda} & (1^\circ) \\ \mu = -\nu\lambda & (2^\circ) \\ \lambda = -\nu\mu \end{cases} \]

\[ \begin{cases} \nu = 1, & \mu = -\lambda \\ \nu = -1, & \mu = \lambda \end{cases} \]

\[ \det \sigma_1 = \struck{\det} -\nu \]

\[ s'_3 - s_3 = \struck{\lambda e_1 + \ldots} (s'_3 - s_0) - (s_3 - s_0) = \lambda(e_2 - e_1) = -\lambda(s_2 - s_1) \]

\[ (1) \qquad -\lambda = (1 + \alpha), \qquad \lambda = -(1+\alpha) = \alpha' \]

\[ (2) \qquad s'_3 - s_3 = (1 + \alpha)(s_2 - s_1) \]

\[ (3) \qquad \sigma_1 = \begin{pmatrix} 0 & 1 & -(1+\alpha) & 0 \\ 1 & 0 & 1+\alpha & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}
= \begin{pmatrix} 0 & 1 & \uncertain{-\alpha'} & 0 \\ 1 & 0 & -\alpha' & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

\[ (4) \qquad \begin{cases} u = \sigma_0\sigma_1 & \quad s_0 \mapsto s_1 \mapsto s_2 \mapsto s_1\uncertain{\text{ ?}},\quad s_3 \mapsto s''_3 \ \text{(à calculer)} \\ v = \sigma_1\sigma_2 & \quad s_0 \mapsto s_0,\quad s_3 \mapsto s_1 \mapsto s_2 \mapsto s'_3 \end{cases} \]

\begin{align*}
s''_3 = u s_0 + u e_3 &= s_{\uncertain{1}} + \sigma_0(\sigma_1(e_3)) = s_0 + e_1 + \sigma_0\bigl[(1+\alpha)(e_2 - e_1) + e_3\bigr] \\
&= s_0 + e_1 + (1+\alpha)\, e_2 \struck{\ill{}} + e_3 - e_1
\end{align*}

\[ (5) \qquad s''_3 = s_0 + (1+\alpha)\, e_2 \struck{\ill{}} + e_3 = s_3 + (1+\alpha)\, e_2 \]

\[ (6) \qquad s''_3 - s_3 = (1+\alpha)(s_2 - s_0) \]

\[ \struck{u s'_3 = u s_0 + u e_3 = s_1 + \sigma_0\sigma_1\bigl((1+\alpha)(e_2 - e_1) + e_3\bigr)} = s_1 + (1+\alpha)\, u(s_2 - s_1) \]

\[ u(s'_3) = u(s_3) + (1+\alpha)\underbrace{u(s_2 - s_1)}_{s_0 - s_2 = -e_2} = s''_3 - (1+\alpha)\, e_2 \underset{(5)}{=} s_3 \]

\[ \struck{(7) \quad u(s_3) \longmapsto s'} \]

\[ \struck{(7)}\ \ u(s''_3) = u(s_3) + (1+\alpha)\underbrace{u(s_2 - s_3)}_{s_0 - s_1 = -e_1} = s''_3 - (1+\alpha)\, e_1 \]

\[ (7) \qquad u : s_3 \longmapsto s''_3 \longmapsto s'_3 \longmapsto s_3 \]

\[ (8) \qquad u^3 = \mathrm{id} \]

\[ (9) \qquad v^5 = \mathrm{id} \iff \boxed{\alpha^2 + \alpha - 1 = 0} \]

\[ (10) \qquad G = \Bigl\{ \sigma_0, \sigma_1, \sigma_2 \Bigm| \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = 1,\ \sigma_0\sigma_2 = \sigma_2\sigma_{\uncertain{1}},\ (\sigma_0\sigma_1)^3 = 1,\ (\sigma_1\sigma_2)^5 = 1 \Bigr\} \]

\[ = \mathrm{Aut}(\text{icosaèdre comb. épinglé} \ldots) \]

\[ (11) \qquad \begin{cases}
s_0 = (0, 0, 0) \\
s_1 = (1, 0, 0) \\
s_2 = (0, 1, 0) \\
s_3 = (0, 0, 1) \\
s'_3 = \bigl(-(1+\alpha), 1+\alpha, 1\bigr) = (\alpha', -\alpha', 1) \\
s''_3 = (0, 1+\alpha, 1) = (0, -\alpha', 1)
\end{cases} \]

\[ \alpha + \alpha' = -1 \qquad \begin{cases} \alpha = -(\alpha' + 1) \\ \alpha' = -(\alpha + 1) \end{cases} \]

\[ \alpha\alpha' = -1 \qquad \begin{cases} \alpha = -1/\alpha' \\ \alpha' = -1/\alpha \end{cases} \]

\begin{gather*}
(1 - \alpha)(1 - \alpha') = 1 \\
(1 + \alpha)\alpha = (1 + \alpha')\alpha' = 1 \qquad (1 + \alpha)(1 + \alpha') = -1 \\
(\alpha + 2)(\alpha' + 2) = 1 \\
(2\alpha - 1)(2\alpha' - 1) = -1
\end{gather*}

\[ (12) \quad \begin{cases} \sigma_0(x, y, z) = \bigl(1 - (x + y + z), y, z\bigr) \\ \text{pts fixes définis par } 2x + y + z - 1 = 0 \quad \struck{\ill{}} \quad (\text{si } 2 \text{ inv.}) \end{cases} \]

\[ (13) \quad \begin{cases} \sigma_1(x, y, z) = \bigl(y - (1+\alpha)z,\ x + (1+\alpha)z,\ z\bigr) = (y + \alpha' z,\ x - \alpha' z,\ z) \\ \text{pts fixes définis par } x + (1+\alpha)z = y \quad (\text{vis. en } x, y, z) \\ \phantom{\text{pts fixes définis par }} \struck{x + \alpha' z = y} \quad x - y - \alpha' z = 0 \end{cases} \]

\[ (14) \quad \begin{cases} \sigma_2(x, y, z) = (x, z, y) \\ \text{pts fixes définis par } y = z \end{cases} \]

\[ x = \tfrac{1}{2}\alpha', \quad y = \alpha' x = \struck{\ill{}}\ \tfrac{1 - \alpha'}{2}, \quad z = y \qquad \struck{x = \tfrac{\ill{}}{2},\ y = z = -\tfrac{\ill{}}{2\alpha}} \]

\[ (15) \qquad \struck{o = \tfrac{1}{2}(\ill{}) = -\tfrac{1}{2}\Bigl(-1, \tfrac{1}{\alpha}, \tfrac{1}{\alpha}\Bigr)} \]

\[ o = \tfrac{1}{2}(\alpha', 1 - \alpha', 1 - \alpha') = \tfrac{1}{2}\bigl(-(1+\alpha), 2 + \alpha, 2 + \alpha\bigr) \]

\[ (16) \qquad \boxed{\alpha(\alpha + 1) = 1} \quad \text{ou} \]

\[ (17) \qquad \underline{a}(x, y, z) = \bigl(\alpha' - x,\ (1 - \alpha') - y,\ (1 - \alpha') - z\bigr) \]

\[ \mathbb{F}_4 \overset{\text{déf}}{=} \mathbb{F}_2[T]/(T^2 + T + 1) = \text{corps des racines primitives } 3^{\text{ièmes}} \text{ de l'unité sur } \mathbb{F}_2 \]

\[ (18) \qquad \begin{cases}
\underline{a}\, s_0 = (\alpha', 1 - \alpha', 1 - \alpha') \\
\underline{a}\, s_1 = (\alpha' - 1, 1 - \alpha', 1 - \alpha') \\
\underline{a}\, s_2 = (\alpha', -\alpha', 1 - \alpha') \\
\underline{a}\, s_3 = (\alpha', 1 - \alpha', -\alpha') \\
\underline{a}\, s'_3 = (0, 1 \ill{}, -\alpha') \\
\underline{a}\, s''_3 = (\alpha', 1 \ill{}, -\alpha')
\end{cases} \]

\[ \struck{(18)} \quad \begin{cases}
\underline{a}\, s_0 = \bigl(1, -\frac{1}{\alpha}, -\frac{1}{\alpha}\bigr) = (1, -1-\alpha, -1-\alpha) = (1, \alpha', \alpha') \\
\underline{a}\, s_1 = \bigl(0, -\frac{1}{\alpha}, -\frac{1}{\alpha}\bigr) = (0, -1-\alpha, -1-\alpha) = (0, \alpha', \alpha') \\
\underline{a}\, s_2 = \bigl(1, -\frac{1}{\alpha} - 1, -\frac{1}{\alpha}\bigr) = (1, -2-\alpha, -1-\alpha) = (1, \alpha' - 1, \alpha') \\
\underline{a}\, s_3 = \bigl(1, -\frac{1}{\alpha}, -\frac{1}{\alpha} - 1\bigr) = (1, \uncertain{-1-\alpha}, -2-\alpha) = (1, \alpha', \alpha' - 1) \\
\underline{a}\, s'_3 = \bigl(2 + \alpha, -2(1+\alpha), -(2+\alpha)\bigr) = (1 - \alpha', -2\alpha', \alpha' - 1) \\
\underline{a}\, s''_3 = \bigl(1, -2(1+\alpha), -(2+\alpha)\bigr) = (1, 2\alpha', \alpha' - 1)
\end{cases} \]

\[ f(x, y, z) = ax^2 + by^2 + cz^2 + uyz + vzx + wxy + \lambda x + \mu y + \nu z + d \]

\[ f(x, y, z) = ax^2 + b(y^2 + z^2) + uyz + vx(y + z) + \lambda x + \mu(y + z) \]

\begin{align*}
f \circ \sigma_0(x, y, z) &= a\bigl(1 + x^2 + y^2 + z^2 + 2yz + 2zx + 2xy - 2(x + y + z)\bigr) \\
&\quad + b(y^2 + z^2) + uyz + v\bigl(1 - (x + y + z)\bigr)(y + z) \\
&\quad + \lambda\bigl(1 - (x + y + z)\bigr) + \mu(y + z) \\
&= ax^2 + (b + a - v)(y^2 + z^2) + (u + 2a - 2v)yz \\
&\quad + (2a - v)x(y + z) + (-2a - \lambda)x \\
&\quad + (-2a - \lambda + v + \struck{\mu})(y + z) + (a + \struck{\ill{}} + \lambda)
\end{align*}

\begin{gather*}
\boxed{v = a} \\
2v = 2a \quad \text{ok} \\
2a - v = v \quad \text{ok} \\
-2a - \lambda = \lambda \quad \text{i.e.} \quad \boxed{2(\lambda + a) = 0} \\
\mu = -2a - \lambda + v + \mu \quad \text{i.e.\ (connu)} \quad \boxed{\lambda = -a}
\end{gather*}

\[ f(x, y, z) = ax^2 + b(y^2 + z^2) + uyz + ax(y + z) \add{- ax} \struck{+ \mu(y + z)} \qquad \boxed{\uncertain{2a = 0}} \]

\[ \struck{= ax(x + y + z) + b(y^2 + z^2) + uyz + \mu(y + z)} \]

\[ \struck{= a\bigl[x^2 + x(y + z) - x\bigr]} \]

\[ f(x, y, z) = ax(x + y + z - 1) + b(y^2 + z^2) + uyz + \mu(y + z) \]

\[ \begin{array}{lcll}
f(s_0) = 0 & \Longrightarrow & d = 0 & \\
f(s_1) = 0 & \Longrightarrow & a + \lambda = 0 & \text{i.e.}\ \lambda = -a \\
f(s_2) = 0 & \Longrightarrow & b + \mu = 0 & \phantom{\text{i.e.}}\ \mu = -b \\
f(s_3) = 0 & \Longrightarrow & c + \nu = 0 & \phantom{\text{i.e.}}\ \nu = -c
\end{array} \]

\[ f(x, y, z) = a(x^2 - x) + b(y^2 - y) + c(z^2 - z) + uyz + vzx + wxy \]

\[ f(s''_3) = b(1+\alpha)\alpha + u(1+\alpha) = 0 \qquad \underline{u = -\alpha b} \]

\begin{align*}
f(\underline{a}\,s_1) &= \uncertain{\alpha}(\alpha+1)(\alpha+2)(b + c) - b(\alpha + 2) = 0 \\
\text{\uncertain{d'où}}\quad & (\alpha+1)(b + c) - b = 0 \quad \text{i.e.} \quad \alpha b + (\alpha+1)c = 0 \\
\text{i.e.}\quad & b = -\Bigl(1 + \frac{1}{\alpha}\Bigr)c = -(2 + \alpha)c
\end{align*}

\begin{align*}
f\sigma_1(x, y, z) &= a(y + \alpha' z)(x + y + z - 1) + b\bigl(x^2 + (\alpha'^2 + 1)z^2 - 2\alpha' xz\bigr) \\
&\quad + u(x - \alpha' z)z + \mu\bigl(x + (1 - \alpha')z\bigr) \\
&= bx^2 + ay^2 + \bigl(a\alpha' + b(1 + \alpha'^2) \add{- u\alpha'}\bigr)z^2 \\
&\quad + (\uncertain{-a\alpha})yz + (a\alpha' - 2b\alpha' + u)zx + axy \\
&\quad + \mu x + (-a)y + \bigl(-a\alpha' + \mu(1 - \alpha')\bigr)z
\end{align*}

\begin{gather*}
a\alpha' + b(1 + \alpha'^2) - u\alpha' = b \\
\text{i.e.} \quad a(\underbrace{\alpha'^2 + \alpha'}_{1} + \struck{\ill{}}\,\uncertain{1}) = u\alpha' \quad \text{i.e.} \quad \boxed{u = -\alpha a} \\
-a\alpha = u \quad \text{ok} \\
a\alpha' - 2b\alpha' + u = a \quad \text{i.e.} \quad a(\uncertain{-}\alpha' - 2\alpha' \struck{\ill{}}\,\underbrace{-\alpha}_{\alpha'} - 1) = 0 \quad \text{ok} \\
a = a \quad \text{ok} \\
\boxed{\mu = -a} \\
-a = \mu \quad \text{ok} \\
-a\alpha' + \mu(\struck{1} - \alpha') = \struck{\mu} \quad \text{i.e.} \quad \struck{a(-\alpha' - a\alpha')}\ -\alpha'(a + \mu) = 0 \quad \text{ok}
\end{gather*}

\[ f(x, y, z) = a\, q(x, y, z) \]

\[ (19) \qquad \boxed{\begin{aligned} q(x, y, z) &= x(x + y + z - 1) + (y^2 + z^2) - \alpha yz - (y + z) \\ &= x^2 + y^2 + z^2 - \alpha yz + zx + xy - (x + y + z) \end{aligned}} \]

\begin{align*}
&= bx^2 + ay^2 + \bigl[a(\alpha + 2) + b(\alpha + 3) + u(1 + \alpha)\bigr]z^2 \\
&\quad + \bigl(-2a(1 + \alpha)\bigr)yz + \bigl[2b(1 + \alpha) + u\bigr]zx \\
&\quad + \mu x + (-a)y + \bigl[a(1 + \alpha) + \mu(2 + \alpha)\bigr]z
\end{align*}

\[ \begin{cases}
a = b \\
a(2\alpha + 5) + u(1 + \alpha) = \underset{a}{b} \\
-2a(1 + \alpha) = u & \text{déjà vu} \\
2b(1 + \alpha) + u = 0 & \text{déjà vu } (b = a) \\
\mu = -a \\
-a = \mu & \text{déjà vu} \\
a(1 + \alpha) + \mu(2 + \alpha) = \mu \\
\quad \text{i.e.}\ (\mu + a)(1 + \alpha) = 0
\end{cases} \]

\[ \text{i.e.} \qquad
\begin{cases}
b = a \\
u = -2a\,\dfrac{\alpha + 2}{1 + \alpha} = -2a\alpha(\alpha + 2) = -2a(\alpha + 1) \\
\mu = -a
\end{cases} \]

\[ f(x, y, z) = a\, q(x, y, z) \quad \text{où} \]

\[ \boxed{q(x, y, z) = x^2 + y^2 + z^2 \struck{\ill{}} - 2(1 + \alpha)yz - (x + y + z)} \]

\[ f(x, y, z) = ax^2 + by^2 + cz^2 + uyz + vzx + wxy + \lambda x + \mu y + \nu z + d \]

\[ \begin{array}{lll|l}
f(s_0) = 0 & \boxed{d = 0} & & f(x, y, z) = \\
f(s_1) = 0 & a + \lambda = 0 & \boxed{\lambda = -a} & a(x^2 - x) + b(y^2 - y) + c(z^2 - z) \\
f(s_2) = 0 & b + \mu = 0 & \boxed{\mu = -b} & \quad + uyz + vzx + wxy \\
f(s_3) = 0 & c + \nu = 0 & \boxed{\nu = -c} &
\end{array} \]

\[ f(s''_3) = 0 \qquad b(\alpha'^2 + \alpha') \struck{\ill{}} + u\alpha' = 0 \qquad \boxed{u = \struck{\ill{}} -\alpha b} \]

\[ u = -(\alpha' + 1)b = -\alpha b \]

\[ \struck{f = a(x^2 - x) + b\bigl(y^2 - \ill{}\bigr) + c(z^2 - z)} \]

\[ f(x, y, z) = a(x^2 - x) + by(y - \alpha z - 1) + c(z^2 - z) + \struck{\ill{}} \add{x(vz + wy)} \]

\[ f(\struck{\ill{}}\,s'_3) = 0 \qquad b\bigl(\underbrace{1 - \underbrace{\alpha(-\alpha')}_{1} - 1}_{-1}\bigr) + c(\underbrace{\alpha'^2 + \alpha'}_{1}) = 0 \qquad \boxed{b = c} \]

\[ f(x, y, z) = a(x^2 - x) + b\bigl(\struck{y(y - \alpha z - 1) + z(z - 1)}\ \add{y^2 + z^2 - \alpha yz - (y + z)}\bigr) + x(vz + wy) \]

\begin{align*}
f(s'_3) = 0 \qquad & a(\alpha'^2 - \alpha') + b\bigl(-\alpha'(\underbrace{-\alpha' - \alpha - 1}_{0}) + 0\bigr) \\
& \struck{+ \alpha'(}\ v\alpha' - \alpha'^2 w = 0 \\
& a(\alpha' - 1) + v - \alpha' w = 0 \qquad \boxed{v = \alpha' w - (\alpha' - 1)a}
\end{align*}

\begin{align*}
f(\underline{a}\,s''_3) = 0 \qquad & a(\alpha'^2 - \alpha') + b\Bigl(\underbrace{\struck{1} + \underbrace{\alpha\alpha'}_{-1} \struck{- 1} + \underbrace{(-\alpha')(-\alpha' - 1)}_{\alpha'^2 + \alpha' = 1}}_{0}\Bigr) \\
& + (-\alpha'^2)v + \alpha' w = 0 \\
& (\alpha' - 1)a \struck{\ill{}} - \alpha' v + w = 0 \qquad \text{ou encore, \uncertain{tout} \uncertain{court}} \quad \boxed{w = \alpha' v - (\alpha' - 1)a}
\end{align*}

\[ \struck{(\alpha' - 1)a - 2\alpha b + (1 - \alpha'^2)w + (\alpha'^2 - \alpha')a = 0} \]

\[ \struck{(\alpha'^2 - 1)a - 2\alpha b + \underbrace{(1 - \alpha'^2)}_{\alpha'}w = 0} \]

\[ h : X \times I \longrightarrow Y \]

\[ h \mid F' \times I = (f \mid F') \circ \mathrm{pr}_1 \]

\[ h \mid F \times I = \struck{\ill{}}\bigl[(b, t) \longmapsto c\bigl(t\varphi(b)\bigr)\bigr] \]

\[ f(x) = c(0) \]

\[ f_t(x) = x \quad \text{si}\ x \in \complement\overset{\circ}{F} \]

\[ \struck{g_t(x) = x} \]

\[ F \times I \longrightarrow F \]

\[ \begin{cases} g_t(x) = x & x \in B = F - \overset{\circ}{F} \\ g_{\struck{t}}(s) = s \\ g_1(s) \end{cases} \]

\[
w = \alpha'\bigl(\alpha' w - (\alpha'-1)a\bigr) - (\alpha'-1)a
\]

\[
\underbrace{(1-\alpha'^2)}_{\alpha'} w = \underbrace{(\alpha'-1)\underbrace{(-\alpha'-1)}_{\alpha}}_{} a
\qquad \text{i.e. } \alpha' w = \alpha' a \qquad \text{i.e. } \emph{$w = a$}
\]

\[
\struck{1-\alpha}\quad \alpha\alpha' - \alpha = -1-\alpha = \alpha'
\]

\[
\boxed{v = w = a}
\]

\[
f(x,y,z) = a x(x+y+z-1) + b(y^2+z^2-y-z-\alpha yz)
\]

\[
f(a s_2) = 0 \quad\text{i.e.}\quad a\alpha'(-\alpha') + b\Bigl[\struck{\ill{}}\,\add{\alpha'^2} + (1-\alpha')(-\alpha') - \alpha(-\alpha')(1-\alpha')\Bigr] = 0
\]

\[
-\alpha'^2 a + \alpha'^2 b = 0 \qquad \boxed{b = a}
\]

\[
f(x,y,z) = a\, q(x,y,z)
\]

\[
q_0(x,y,z) = x^2 + y^2 + z^2 - \alpha yz + zx + xy \tag{19}
\]

\[
\struck{\ill{}}\;
\begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & -\alpha \\ 1 & -\alpha & 2 \end{pmatrix} \tag{20}
\]

\[
\delta = 8 - 2\alpha - 2 - 2\alpha^2 - 2 = 4 - 2\underbrace{(\alpha^2+\alpha)}_{1} = 2
\]

\[
\delta = 2 \quad\text{donc}\quad \emph{$\delta' = 1$} \tag{21}
\]

\[
q(x,y,z,t) = x^2 + y^2 + z^2 - \alpha yz + zx + xy - t(x+y+z)
\]

\[
\struck{\ill{}}\;
\begin{pmatrix} 2 & 1 & 1 & -1 \\ 1 & 2 & -\alpha & -1 \\ -\alpha & -1 & 2 & -1 \\ -1 & -1 & -1 & 0 \end{pmatrix} \tag{22}
\]

\[
W^+_T \simeq \mu_{n\,T} \qquad V_T \simeq D \oplus D^{-1}
\]

\[
\mu^{*}_{n\,T\times_S\Omega} \ni \zeta \qquad \sigma_T\sigma_\omega \qquad \sigma_T\zeta = \zeta^{-1}, \quad \sigma_\omega\zeta = \zeta^{-1}, \quad \sigma_T\sigma_\omega\zeta = \zeta
\]

\[
\mu^{*}_{n\,T\wedge\Omega}
\]

\[
\begin{array}{c} T \\ | \\ S \end{array} \quad \begin{array}{c} A \\ | \\ \mathcal{O}_S \end{array} \quad 1
\qquad
\zeta + \zeta^{-1} = c, \qquad \zeta^2 - c\zeta + 1 = 0, \qquad \zeta = \frac{c \pm \sqrt{c^2-4}}{2} \qquad c^2 = 4,\ c = \pm 2
\]

\[
\Delta_1 = \det\begin{pmatrix} 1 & 1 & -1 \\ 2 & -\alpha & -1 \\ -\alpha & 2 & -1 \end{pmatrix} = \alpha + \alpha - 4 - (-\alpha^2) \struck{-(-1) - (-2)}
= 2\alpha + \alpha^2 = \alpha + 1 = -\alpha'
\]

\[
\Delta_2 = \det\begin{pmatrix} 2 & 1 & -1 \\ 1 & -\alpha & -1 \\ 1 & 2 & -1 \end{pmatrix} = 2\alpha - 2 \struck{-1} - \alpha \struck{-(-1)} - (-4) = \alpha + 2
\]

\[
\Delta_3 = \det\begin{pmatrix} 2 & 1 & -1 \\ 1 & \struck{2} & -1 \\ 1 & -\alpha & -1 \end{pmatrix} = -4 + \alpha \struck{-1} - (-2) - 2\alpha \struck{-(-1)} = -2 - \alpha
\]

\[
\det(\ ) = \Delta_1 - \Delta_2 + \Delta_3 = \underbrace{-\alpha' - \alpha}_{1} - 2 - \alpha = -1 - \alpha = \alpha'
\]

\[
\mathbb{V} \simeq \struck{\ill{}} \add{\operatorname{Sym}^2(F) \otimes (\det F)^{-1}} \simeq \operatorname{Sym}^2 \check F \otimes (\det \check F)^{-1} \quad \struck{\simeq \ill{} \otimes (\det F)^{\ill{}}}
\]

\[
\Omega = \check{\mathbb{V}} = \Gamma^2(\check F) \otimes (\det F)^{\ill{}} = \operatorname{Bilsym}(F, F; \det F^{\ill{}})
\]

\[
\struck{\operatorname{Sym}^2(F)} \xrightarrow{\ \delta\ } (\struck{\det F})^{\otimes 2}\ \mathcal{O}_S
\]

\[
\mathbb{V} = \operatorname{Sym}^2(F)(\det F)^{-1} \xrightarrow{\ Q\ } \mathcal{O}_S
\]

\[
\struck{\det\Omega \simeq \det\Gamma^2\check F\,(\det F)^{3} \simeq \ill{}\,(\det F)^{-3} \det F^{3} \simeq \mathcal{O}_S}
\]

\[
\Gamma' = \bigl\{\sigma_0, \sigma_1, \sigma_2 \bigm| \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = (\sigma_0\sigma_1)^3 = (\sigma_1\sigma_2)^5 = (\sigma_0\sigma_2)^2 = 1\bigr\}
\]

\[
\begin{cases}
H_0 : & 2x + y + z - 1 = 0 \\
H_1 : & -x + y + \alpha' z = 0 \\
H_2 : & \struck{\ill{}}\ y - z = 0
\end{cases}
\]

\[
\text{soit}\ \begin{cases} \nu^2 = 1 \\ \mu = -\nu\lambda \end{cases} \quad \Bigl|\ \text{exprimant } \sigma_1^2 = 1
\]

\[
\alpha^2 + \alpha - 1 = 0
\]

\[
\struck{\lambda + \mu + \nu = 1 \quad\text{i.e.}\quad \lambda - \nu\lambda + \nu = 1 \quad\text{i.e.}} \quad \struck{\ill{}}\quad \struck{(\lambda - 1)(1 - \nu) = 0 \quad ?}
\]

\[
s_1 = \sigma_0 s_0, \quad s_2 = \sigma_1 s_1, \quad s_3 = \sigma_2 s_2
\]

\[
\sigma_2(P) = \operatorname{Pl}\bigl(\sigma_2(s_0) = s_0,\ \sigma_2(s_1) = s_1,\ \sigma_2(s_2) = s_3\bigr) = P
\]

\[
\begin{aligned}
S &\xrightarrow{\ \rho_S\ } \text{ens.\ des sections de } E \text{ sur } S \\
A &\xrightarrow{\ \rho_A\ } \text{ens.\ des \add{sous-}fibrés en droites affines de } E \\
F &\xrightarrow{\ \rho_F\ } \text{ens.\ des sous-fibrés en plans affines}
\end{aligned}
\]

\[
D_0 = H_1 \cap H_2 : \quad y = z,\ x = \alpha' y \quad\text{i.e.}\quad y = z = \alpha x
\]

\[
D_1 = H_2 \cap H_0 \quad
\begin{cases}
\struck{\ill{}} : & D_1 = \varnothing,\ (\text{i.e. } H_0 /\!/ H_2) \\
2 \text{ inv.} & \struck{\ill{}}\ y = z = \tfrac{1}{2} - x
\end{cases}
\]

\[
D_2 = H_0 \cap H_1 \quad
\begin{cases}
y = \alpha + (1 - 3\alpha)x \\
z = (1 - \alpha) - 3(1 - \alpha)x
\end{cases}
\]

\[
\sigma_1 s_0 = \sigma_2 s_0 = s_0 \quad\text{i.e.}\quad s_0 \in E^{\sigma_1} \cap E^{\sigma_2}
\]

\[
s_0,\ s_1 = \sigma_0 s_0,\ s_2 = \sigma_1 s_1 = \sigma_1\sigma_0 s_0,\ s_3 = \sigma_2 s_2 = \sigma_2\sigma_1\sigma_0 s_0
\]

\[
\Gamma = \bigl\{\sigma_0, \sigma_1, \sigma_2 \bigm| \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = (\sigma_0\sigma_2)^2 = 1\bigr\} \tag{1}
\]

\[
\begin{cases} \sigma_0^2 = \sigma_1^2 = \sigma_2^2 = 1 \\ \sigma_0\sigma_2 = \sigma_2\sigma_0 \end{cases} \tag{2}
\]

\[
\begin{cases}
\Gamma_H = \text{ss-groupe engendré par les } \sigma_i \text{ avec } i \notin H \\
\mathfrak{F}_H = \Gamma/\Gamma_H \quad (\text{espace homogène})
\end{cases} \tag{3}
\]

\[
H \subset K \Rightarrow \Gamma_K \subset \Gamma_H \Rightarrow \mathfrak{F}_K \xrightarrow{\ \varphi_{HK}\ } \mathfrak{F}_H \tag{4}
\]

\[
\mathfrak{F}_{**} = \coprod_{H} \mathfrak{F}_H \tag{5}
\]

\[
x \in \mathfrak{F}_H,\ y \in \mathfrak{F}_K \qquad x \preccurlyeq y \iff H \subset K,\ x = \varphi_{HK}\, y \tag{6}
\]

\[
\mathfrak{F}_{\{0\}} \overset{\text{df}}{=} \mathfrak{F}_0, \quad \mathfrak{F}_{\{1\}} \overset{\text{df}}{=} \mathfrak{F}_1, \quad \mathfrak{F}_{\{2\}} \overset{\text{df}}{=} \mathfrak{F}_2 \tag{7}
\]

\[
\mathfrak{F}_* = \coprod_{i \in \{0,1,2\}} \mathfrak{F}_i \xrightarrow{\ \dim\ } [0, 2] \qquad \dim(\mathfrak{F}_i) \in \{i\} \tag{8}
\]

\[
x \leqslant y \iff
\begin{cases}
\text{a) } \dim(x) \leqslant \dim(y) \\
\text{b) } \exists\, z \in \mathfrak{F}_*, \text{ avec } \struck{x, y \preccurlyeq z}\ x, y \preccurlyeq z
\end{cases} \tag{9}
\]

\[
\mathfrak{F}_{**}(X/S), \quad \mathfrak{F}_*(X/S) \ \ldots
\]

\[
\begin{cases}
S = S_P \subset \mathfrak{F}_0(X/S) = \Gamma(X/S) & \text{ens.\ des sommets} \\
A = A_P \subset \mathfrak{F}_1(X/S) & \ldots \text{arêtes} \\
F = F_P \subset \mathfrak{F}_2(X/S) & \ldots \text{faces}
\end{cases} \tag{10}
\]

\[
\boxed{11}\quad
\begin{cases}
\text{(a) } \forall a \in A, \struck{\text{il y a } \ill{} \text{ exactement}} \text{ l'ens.\ des } s \in S \text{ incidents à } a \\
\quad \text{est de cardinal 2 exactement —} \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \text{ non vide} \\
\text{(b) } \forall s \in S,\ f \in F, \text{ l'ens.\ des } a \text{ incidents à } s \text{ et } f \text{ est \add{loc.} de cardinal égal à } 2, \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \neq \varnothing \\
\text{(c) } \forall a \in A, \text{ l'ens.\ des } f \in F \text{ incidents \struck{\ill{}} est de cardinal } = 2, \\
\quad \text{et ceci reste vrai après tt chgt de base } S' \to S,\ S' \neq \varnothing
\end{cases}
\]

\[
\boxed{\Psi_* \overset{\text{df}}{=} \Psi_{*P} = S \amalg A \amalg F} \tag{12}
\]

\[
\boxed{\Psi_H := \Psi_{H P}} \tag{12 bis}
\]

\[
\operatorname{Rep}(P) = \Psi_{\{0,1,2\}\,P} \tag{13}
\]

\[
\begin{cases}
\sigma_0(s, a, f) = (s', a, f) & s' \neq s \\
\sigma_1(s, a, f) = (s, a', f) & a' \neq a \\
\sigma_2(s, a, f) = (s, a, f') & f' \neq f
\end{cases} \tag{14}
\]

\[
\operatorname{Rep}(P) \longrightarrow \Psi_H(P) \tag{15}
\]

\[
\operatorname{Rep}(P)/\Gamma_H \longrightarrow \Psi_H(P) \tag{16}
\]

\[
\boxed{\Gamma/N = \Gamma_P} \tag{21}
\]

\[
\boxed{\operatorname{Aut}(X, P) \simeq \operatorname{Aut}_{\mathrm{comb}}(P)} \tag{22}
\]

\[
\struck{\ill{}}\quad \Gamma_P \xrightarrow{\ \sim\ } G = \operatorname{Aut}(X, P) \tag{23}
\]

\[
\struck{r}\ \gamma \in \Gamma \mapsto \varphi(\gamma) \text{ est l'unique automorphisme de } (X, P) \text{ tel que } \varphi(\gamma)(r) = \gamma.r \tag{24}
\]

\[
\Gamma \longrightarrow \operatorname{Rep}(P) \qquad \gamma \mapsto \gamma.r \tag{25}
\]

\[
N\backslash\mathfrak{F}_H \struck{\ill{}} = \Gamma_P/\Gamma_H = \Gamma/N.\Gamma_H \longrightarrow \Psi_H(P) \tag{26}
\]

\[
\Gamma \xrightarrow{\ \varphi\ } \operatorname{Aut}(X) \tag{27}
\]

\[
r \struck{=}\ (s_0, a_0, f_0) \in \operatorname{Rep}(X) \tag{28}
\]

\[
\struck{\sigma_0(a_0)}\quad
\begin{cases}
\sigma_0 a_0 = a_0, & \sigma_0 f_0 = f_0 \\
\sigma_1 s_0 = s_0, & \sigma_1 f_0 = f_0 \\
\sigma_2 a_0 = a_0, & \sigma_2(f_0) = f_0
\end{cases} \tag{29}
\]

\[
N = \operatorname{Ker} \varphi \subset \Gamma \tag{30}
\]

\[
s'_0 = \sigma_0(s_0), \quad \struck{\ill{}}\ a'_0 = \sigma_1(a_0), \quad f'_0 = \sigma_2(f_0) \tag{31}
\]

\[
\gamma s_0 \prec \gamma' a_0 \prec \gamma'' \struck{f_0 \Rightarrow}\, f_0 \Longrightarrow \exists\, \gamma''' \in \Gamma, \text{ tel que } \gamma''' s_0 = \gamma s_0,\ \gamma''' a_0 = \gamma' a_0,\ \gamma''' f_0 = \gamma'' f_0
\]

\[
\begin{array}{c} B \\ | \\ I \end{array}
\qquad
\begin{array}{c} k^B \\ \big\downarrow{\scriptstyle t = \operatorname{Tr}} \\ k^I \end{array}
\xleftarrow{\ \varphi = (\varphi_b)_{b \in B}\ } E = t^{-1}(\{1\})
\qquad
H_b = \varphi_b^{-1}(\{1\})
\]

\[
\begin{array}{c} B' \subset B \\ \big\downarrow \\ I \end{array}
\qquad
\struck{\ill{}}\ F_{B'} = \bigcap_{b \in B'} H_b
\]

\[
\mathfrak{F} = \{F_{B'}\}_{B' \in \Gamma_p(B/I)} \cup \{\varnothing\}
\]

\[
0 \to W \longrightarrow V \longrightarrow k \to 0 \qquad V = k^B/N, \qquad n\ (\text{sous } W)
\]

\[
  G_\tau \simeq \mathrm{Aut}_{\mathrm{aff}}(X_\tau) \simeq \mathrm{Aut}_{\mathrm{ord}}(\mathcal{F}_\tau)
\]

\[
\begin{array}{lll}
  (s_i)_{0 \leq i \leq n-1} & s_{n-1} = 2 & s_n = 1 \\
  (s'_i)_{0 \leq i \leq n-1} & s'_0 = 2 & s'_{-1} = 1 \\
  (\delta_i)_{0 \leq i \leq n-2} & & \\
  (f_i)_{0 \leq i \leq n-1} & f_{-1} = f_n = 1 &
\end{array}
\]

\[
  N = \prod_{1}^{n-1} s_i = \prod_{1}^{n-1} s'_i = \operatorname{card} G
\]

\[
  \begin{cases}
    s'_i f_{i+1} = f_i s_{i+1} \quad (0 \leq i \leq n-1) \\
    s
  \end{cases}
  \qquad
  \frac{f_i}{s'_i} = \frac{f_{i+1}}{s_{i+1}} = f
\]

\[
  x_{i-1} \;\bullet\; \bullet\; x_{i+2} \qquad \sigma_i \sigma_{i+1}
\]

\[
  \sigma_0\sigma_1,\ \sigma_1\sigma_2,\ \dots,\ \sigma_{n-2}\sigma_{n-1}
  \qquad G \ni \sigma_0, \dots, \sigma_{n-1}
\]

\[
\begin{array}{cccccccc}
  x_0 & x_1 & & x_{i-1} & x_i & & & x_{n-1} \\
  F_0 & F_1 & - & F_{i-1} & F_i & F_{i+1} & - & F_{n-1}
\end{array}
\]