\[
A_x = E(x) \cap \lbrace S^{*} - \sigma_T(x) \rbrace = E(x) \cap (E(v)^{*} \cup
E(w)^{*}) = E(x) - \lbrace u, \sigma(x) \rbrace
\]

\[
\omega_v(x) = E(x) \cap E(v)^{*}, \qquad \omega_v(y) = E(y) \cap E(v)^{*}
\]

\[
\delta_v(x, y) \in \mathbb{F}_2(P_v)
\]

\[
\left\lbrace
\begin{aligned}
\operatorname{supp} \delta_v(x, y) &= \lbrace \beta \in P_v \mid \omega_v(x) \cap
\beta \neq \omega_v(y) \cap \beta \rbrace \\
P_v - \operatorname{supp} \delta_v(x, y) &= \lbrace \beta \in P_v \mid
\omega_v(x) \cap \beta = \omega_v(y) \cap \beta \rbrace
\end{aligned}
\right.
\]

\[
c_{w,u}\bigl(\lbrace \pi_u\lbrace x, y \rbrace, \complement_{P_u} \pi_u\lbrace
x, y \rbrace \rbrace\bigr) = \lbrace \operatorname{supp} \delta_v(x, y),
\operatorname{supp}(\delta_v(x, y) + \mathbf{1}_{P_v}) \rbrace .
\]

\[
R_{u,v}(\alpha, \beta) \overset{\text{déf}}{\iff} \text{tt él.\ de } \alpha
\text{ est lié à tt él.\ de } \beta .
\]

\[
\tilde{\varphi}_{vu} : \mathfrak{P}'_2(E_u^{*}) \to \mathfrak{P}'_2(E_v^{*})
\]

\[
\tilde{\varphi}_{vu}(\sigma \tilde{\alpha}) = \sigma
\tilde{\varphi}_{vu}(\tilde{\alpha}), \qquad \pi_v \tilde{\varphi}_{vu}(\lbrace x,
\sigma y \rbrace) = \complement_{P_v} \pi_v \tilde{\varphi}_{vu}(\lbrace x, y
\rbrace)
\]

\[
\tilde{\varphi}_{wv} \tilde{\varphi}_{vu}(\tilde{\alpha}) = \sigma\bigl(
\tilde{\varphi}_{wu}(\tilde{\alpha}) \bigr) \quad \text{i.e.} \quad
\tilde{\varphi}_{wv} \tilde{\varphi}_{vu} = \sigma \tilde{\varphi}_{wu}
\]

\[
\left.
\begin{array}{ll}
x, y \in E(u)^{*} & x, y \text{ liés à } s, t \\
s, t \in E(v)^{*} & s, t \text{ liés à } \xi, \eta \\
\lbrace \xi, \eta \rbrace \in E(w)^{*} &
\end{array}
\right\rbrace \Longrightarrow x, y \text{ liés à } \sigma\xi, \sigma\eta
\]

\[
(x, s, \xi), \ (y, t, \xi') \quad (\xi, \xi' \in E(w)^{*} \text{ sont liés})
\]

\[
(x, t, \eta), \ (y, s, \eta') \quad (\eta, \eta' \in E(w)^{*} \text{ sont
liés})
\]

\[
\lbrace \pi_w(\lbrace \xi, \eta \rbrace), \complement_{P_w} \pi_w(\lbrace \xi,
\eta \rbrace) \rbrace = c_{wv}\bigl(\pi_v\lbrace s, t \rbrace, \complement
\pi_v(\lbrace s, t \rbrace)\bigr),
\]

\[
\lbrace \pi_w(\lbrace \xi, \eta \rbrace), \complement \pi_w(\lbrace \xi, \eta
\rbrace) \rbrace = c_{wu}\bigl(\pi_u(\lbrace x, y \rbrace), \complement
\pi_u(\lbrace x, y \rbrace)\bigr)
\]

\[
\lbrace p', q' \rbrace = \tilde{\varphi}_{wu}(\lbrace x, y \rbrace)
\]

\[
\tilde{\Gamma} \xrightarrow{\ \sim\ } \prod_{x \in E(u)^{*}}
\omega_v(x)
\]

\[
(\pi_u(x), \pi_v(t), \pi_w(\eta)) \in \operatorname{Im}(\Gamma \to P_u \times
P_v \times P_w),
\]

\[
\mathrm{I}_0 \left\lbrace
\begin{array}{lll}
\alpha_1, \alpha_2, \alpha_3 \ (\text{et } \alpha_4) & \text{liés à} &
\beta_4 \\
\beta_1, \beta_2, \beta_3 \ (\text{et } \beta_4) & \text{---} & \gamma_4 \\
\gamma_1, \gamma_2, \gamma_3 \ (\text{et } \gamma_4) & \text{---} & \alpha_4
\end{array}
\right.
\quad ; \text{ de } = \ldots
\]

\[
\alpha_i, \ \beta_i, \ \gamma_i, \qquad \alpha'_i = \sigma\alpha_i, \
\beta'_i = \sigma\beta_i, \ \gamma'_i = \sigma\gamma_i .
\]

\[
[0] \quad \alpha_i \text{ lié à } \alpha'_i \ / \ \beta_i \text{ à }
\beta'_i \ / \ \gamma_i \text{ à } \gamma'_i
\]

\[
\mathrm{I}' \left\lbrace
\begin{array}{llll}
\alpha'_i & \text{liés à } \beta'_4, & \text{non à } \beta_4 & \text{ni
entre eux} \\
\beta'_i & \text{---} \ \gamma'_4, & \text{non à } \gamma_4 & \text{ni entre
eux} \\
\gamma'_i & \text{---} \ \alpha'_4, & \text{non à } \alpha_4 & \text{ni entre
eux}
\end{array}
\right.
\qquad
\mathrm{I} \left\lbrace
\begin{array}{llll}
\alpha_i & \text{liés à } \beta_4, & \text{non à } \beta'_4 & \text{ni entre
eux} \\
\beta_i & \text{---} \ \gamma_4 & \text{---} \ \gamma'_4 & \text{---} \\
\gamma_i & \text{---} \ \alpha_4 & \text{---} \ \alpha'_4 & \text{---}
\end{array}
\right.
\]

\[
\mathrm{II} \left\lbrace
\begin{array}{lll}
\alpha_i \text{ liés à } \gamma'_4, & \text{non à } \gamma_4 \\
\beta_i \ \text{---} \ \alpha'_4, & \text{---} \ \alpha_4 \\
\gamma_i \ \text{---} \ \beta'_4, & \text{---} \ \beta_4
\end{array}
\right.
\qquad
\mathrm{II}' \left\lbrace
\begin{array}{lll}
\alpha'_i \ \text{---} \ \gamma_4, & \text{non à } \gamma'_4 \\
\beta'_i \ \text{---} \ \alpha_4, & \text{---} \ \alpha'_4 \\
\gamma'_i \ \text{---} \ \beta_4, & \text{---} \ \beta'_4
\end{array}
\right.
\]

\[
\mathrm{III} \left\lbrace
\begin{array}{lll}
\alpha_i \text{ lié à } \beta_j, \gamma_j, \beta'_i, \gamma'_i & (1
\leqslant i \neq j \leqslant 3) & \text{non à } \beta'_j, \gamma'_j, \beta_i,
\gamma_i \\
\beta_i \text{ lié à } \gamma_j, \alpha_j, \gamma'_i, \alpha'_i & (\text{---})
& \text{---} \ \gamma'_j, \alpha'_j, \gamma_i, \alpha_i \\
\gamma_i \ \text{---} \ \alpha_j, \beta_j, \alpha'_i, \beta'_i & (\text{---}) &
\text{---} \ \alpha'_j, \beta'_j, \alpha_i, \beta_i
\end{array}
\right.
\]

\[
\mathrm{III}' \left\lbrace
\begin{array}{lll}
\alpha'_i \text{ lié à } \beta'_j, \gamma'_j & \text{--} & \text{--} \\
\text{--} & \text{--} & \text{--} \\
\text{--} & \text{--} & \text{--}
\end{array}
\right.
\]

\[
\begin{array}{ll}
\alpha_4 \text{ lié à} & \beta'_1, \beta'_2, \beta'_3, \beta_4 \quad
(\text{lemme } 8) \\
\alpha_1 \text{ lié à} & ? \quad ? \quad ? \quad \beta_4
\end{array}
\]

\[
\alpha_1 \text{ lié à } \beta'_1, \beta_2, \beta_3, \beta_4 \qquad
\text{cqfd.}
\]

\[
\begin{array}{lll|l}
D_0 = \emptyset & & & 1 \\
\cap \ D_1 = \bullet\, u & \text{pt} & & 27 = 3^3 . \\
D_2 = u \,\text{---}\, v & \text{arête} & & 27 \cdot 10 = 2 \cdot 3^3 \cdot 5 = 270 \\
\cap \ D_3 = \text{triangle } u, v, w & \text{triangle} & & 27 \cdot 10 \cdot 1 = 2 \cdot 3^3 \cdot 5 = 270 \\
D_4 & \text{\struck{triangle} \add{balise}} & & 27 \cdot 10 \cdot 1 \cdot 8 = 2^4 \cdot 3^3 \cdot 5 = 2160 \\
\cap \ D_5 & \text{préprisme (triangulaire)} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 = 2^6 \cdot 3^3 \cdot 5 = 8640 \\
\cap \ D_6 & \text{prisme (triangulaire)} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 = 2^6 \cdot 3^3 \cdot 5 = 8640 \\
\cap \ D_7 & \text{bitriangle} & & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 \cdot 1 = 2^6 \cdot 3^3 \cdot 5 = 8640
\end{array}
\]

\[
\begin{array}{llll}
\operatorname{Conf}_{D_0} & 1 & \text{---} \operatorname{Aut} D_0 & e \\
\operatorname{Conf}_{D_1} & 27 & \text{---} \operatorname{Aut} D_1 & e \\
\operatorname{Conf}_{D_2} & 27 \cdot 10 : 2 = 135 & \text{---}
\operatorname{Aut} D_2 & \mathfrak{S}_2 \\
\operatorname{Conf}_{D_3} & 27 \cdot 10 \cdot 1 : 6 = 45 & \text{---}
\operatorname{Aut} D_3 & \mathfrak{S}_3 \\
\operatorname{Conf}_{D_4} & 27 \cdot 10 \cdot 1 \cdot 8 : 2 = 2^3 \cdot 3^3
\cdot 5 = 1080 & \text{---} \operatorname{Aut} D_4 & \mathfrak{S}_2 \\
\operatorname{Conf}_{D_5} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 : 2 = 2^5
\cdot 3^3 \cdot 5 = 4320 & \text{---} \operatorname{Aut} D_5 &
\mathfrak{S}_2 \\
\operatorname{Conf}_{D_6} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1 : 12
= 2^4 \cdot 3^2 \cdot 5 = 720 & \text{---} \operatorname{Aut} D_6 &
\mathfrak{S}_2 \times \mathfrak{S}_3 \\
\operatorname{Conf}_{D_7} & 27 \cdot 10 \cdot 1 \cdot 8 \cdot 4 \cdot 1
\cdot 1 : 72 = 2^3 \cdot 3 \cdot 5 = 120 & \text{---} \operatorname{Aut} D_7 &
\mathfrak{S}_2 . (\mathfrak{S}_3 \times \mathfrak{S}_3)
\end{array}
\]

\[
\operatorname{card} W = 2^7 \cdot 3^4 \cdot 5 = 51840
\]

\[
S = I \amalg (I \amalg I) \amalg I \times \Pi \times \omega(I)
= (I \times \lbrace 0, 1, 2 \rbrace) \amalg (I \times \Pi \times \omega(I)),
\]

\[
I \ni u \longmapsto \alpha(u), \ \alpha'(u)
\]

\[
\begin{array}{lll}
(i, \nu) \text{ et } (j, \mu) \ (\text{distincts}) & \text{sont liés ssi} &
i = j \text{ \emph{ou} } \nu = \mu \\
(i, \nu) \text{ et } (j, \pi, \rho) & \text{liés dans les cas} &
\left\lbrace
\begin{array}{ll}
i = j : & \nu = 0 \\
i \neq j & \left\lbrace \begin{array}{l} \nu = 1, \ j = \rho i \\ \nu =
2, \ i = \rho j \end{array} \right.
\end{array}
\right. \\
(j, \pi, \rho) \text{ et } (j', \pi', \rho') \ \text{distincts} & \text{liés
dans les cas} &
\left\lbrace
\begin{array}{ll}
j = j' & \pi = \pi', \ (\rho \neq \rho') \\
j \neq j' & \left\lbrace \begin{array}{ll} i \neq j, & \rho = \rho' \\ i =
j & \rho \neq \rho' \end{array} \right.
\end{array}
\right.
\end{array}
\]

\[
I \amalg (I \times \varepsilon) \amalg (I \times \Pi \times \varepsilon
\times \omega(I)/\mathbb{F}_2)
\]

\[
(I \times J) \amalg
\]

\[
1 \to \varepsilon \to E_0 \to G \to 1
\]

\[
p_j \alpha_i = \left\lbrace
\begin{array}{ll}
0 & \text{si } i = j \\
\mathrm{id}_G & \text{si } i \neq j
\end{array}
\right. ,
\]

\[
q : \dot{V} \to \varepsilon
\]

\[
\rho_1(g)\, \rho_2(g)\, \rho_3(g) = q(\dot{g})
\]

\[
p_1(\tilde{x}_2) = p_1(\tilde{x}_3), \quad p_2(\tilde{x}_3) =
p_2(\tilde{x}_1), \quad p_3(\tilde{x}_1) = p_3(\tilde{x}_2),
\]

\[
\nu = 2(\mu + 1), \qquad \operatorname{card} S = 3(2\mu + 1)
\]

\[
\operatorname{card} S - E(s) - \lbrace s \rbrace \ \ldots \
\operatorname{card} (S - E(s) - \lbrace s \rbrace) = 4\mu
\]

\[
4\mu \leqslant 2^{\mu + 1}, \qquad \mu \leqslant 2^{\mu - 1}
\]

\[
C \simeq \Bigl( \coprod_{i \in E} P \wedge_{\Psi} (\mathbb{F}_2, p_i) \Bigr)
\amalg P \amalg \lbrace u \rbrace .
\]

\[
\lbrace u \rbrace \cup \underbrace{\Bigl( \coprod_i P \wedge_{\Psi}
(\mathbb{F}_2, p_i) \Bigr)}_{E_p^{*}} \amalg P .
\]

\[
e'_i = \sum_{j \in Q_u - \lbrace i \rbrace} e_j \in \Psi
\]

\[
t - E(x) = t - E(y)
\]

\[
16 + 16 + 40 = 72
\]

\[
x \cdot y =
\begin{cases}
0 & \text{si } y \neq x, \sigma x \text{ i.e. } \pi(x) \neq \pi(y) \\
1 & \text{si } y = \sigma x \\
-1 & \text{si } y = x
\end{cases}
\]

\[
\sigma\tilde{q} = -\tilde{q}, \qquad (\tilde{q}, \tilde{q}) = -1, \qquad
(\tilde{q}, \tilde{q}') = 0 \ \text{si } \tilde{q}' \neq \tilde{q}, \sigma\tilde{q} \ \ldots
\]

\[
E(u) = \begin{pmatrix}
\xi_1 & \xi_2 & \xi_3 & \xi_4 & \xi_5 \\
\xi_{16} & \xi_{26} & \xi_{36} & \xi_{46} & \xi_{56}
\end{pmatrix},
\]

\[
\varepsilon_i = \underbrace{\tfrac{1}{2}(\xi'_6 - \eta)}_{\theta_0} + \xi_i ,
\]

\[
\operatorname{card} W = (27 \cdot 40) \times (2 \cdot 4!) = 27 \cdot 3^{3} \cdot 5
\]

\[
\Psi(Q_u) \xrightarrow{\ \sim\ } \mathbb{F}_2^{P_u}
\qquad (\subset \mathbb{F}_2^{Q_u},\ \text{pr.}),
\]

\[
C \simeq \lbrace u \rbrace \amalg
\overbrace{\bigl(P \times_{\mathbb{F}_2^{P_u}} (\mathbb{F}_2, \varepsilon)\bigr)}^{\lbrace v, w \rbrace}
\amalg
\overbrace{\Bigl(\coprod_{i \in P_u} \bigl(P \times_{\mathbb{F}_2^{P_u}} (\mathbb{F}_2, p_i)\bigr)\Bigr)}^{\widetilde{P}_u}
\amalg \overbrace{P}^{E}
\]

\[
e'_i = \sum_{j \in P_u - \lbrace i \rbrace} e_j
\quad \text{ou} \quad
e'_0 = \sum_{i \in I} e_i
\]

\[
1 \longrightarrow \bigl(\mathbb{F}_2^{P_u}\bigr)^{F} \longrightarrow W_{\alpha}
\longrightarrow \mathfrak{S}_{P_u} \longrightarrow 1
\]

\[
1 \longrightarrow \Psi_{P_u} \longrightarrow W_{\vec{\alpha}}
\overset{= W_t}{\longrightarrow} \mathfrak{S}_{P_u} \longrightarrow 1
\]

\[
W_{\vec{\alpha}} \simeq W_{D_4}
\]

\[
\frac{8 \cdot 7}{2} - 4 = 24 \ \text{racines}.
\]

\[
\widetilde{\Psi}(V, t) = \bigwedge_{i \in t} p_i^{*}(\widetilde{V}),
\]

\[
\widetilde{P}_i = \widetilde{P} / N'_i
\]

\[
C \simeq t \amalg \Bigl(\coprod_{i \in t} \widetilde{P}_i\Bigr)
\]

\[
1 \longrightarrow \widetilde{\Gamma} \longrightarrow W_t \longrightarrow
\underbrace{\mathfrak{S}_t \times \mathfrak{S}_{V^{*}}}_{\simeq\, \mathfrak{S}_3 \times \mathfrak{S}_3} \longrightarrow 1
\]

\[
B \simeq B(i) \quad \text{par } b \mapsto b \cap E(i),
\]

\[
\operatorname{card} B = \operatorname{card} B(i) = \frac{8 \cdot 6}{2} = 24 .
\]

\[
\binom{6}{2}\binom{4}{2}\frac{1}{3!} = 15
\]

\[
\operatorname{card} W = (2^{3} \cdot 3^{3} \cdot 5)(2^{4} \cdot 3) = 2^{7} \cdot 3^{4} \cdot 5 .
\]

\[
\sum_{i \in t} i \cdot r_b = 0
\]

\[
\begin{cases}
r_b(u) = 1 & \text{i.e. } u \in b \text{ i.e. } E(u) \cap b = \emptyset \\
r_b(v) = -1 & \text{i.e. } v \in b' \text{ i.e. } E(v) \cap b \text{ de card. } 5 \\
r_b(w) = 0 & \text{i.e. } w \notin b \cup b' \text{ i.e. } E(w) \cap b \text{ card } 2
\end{cases}
\]

\[
72 \cdot 6 \cdot 5 = 2160 = 27 \cdot 10 \cdot 8 \ \text{---}
\]

\[
72 \cdot 5! = \operatorname{card} W / 6 \quad \text{tels triples}
\]

\[
45 \cdot 32 \cdot 6
\]

\[
\underbrace{45}_{\text{triangles}} \cdot \underbrace{3}_{\text{sommets de } t} \cdot \underbrace{8}_{\in E(u)^{*}} = \operatorname{card} W_{E_6} : 48 \quad \text{OK.}
\]

\[
\underbrace{27}_{\text{sommets}} \cdot \underbrace{\binom{5}{2}}_{\text{partie à 2 él. dans } Q_u} = 270 = 2 \cdot 3^{3} \cdot 5
\]

\[
\simeq (\mathfrak{S}_2 \times \mathfrak{S}_3) \cdot \underbrace{\Psi(\mathbb{F}_2)}_{\simeq\, \mathbb{F}_2^{4}}
\quad \text{de cardinal } 2 \cdot 6 \cdot 16 = 2^{6} \cdot 3
\]

\[
2^{7} \cdot 3^{4} \cdot 5 = \operatorname{card} W_{E_6} \ \ldots
\]

\[
\Delta_0 \longrightarrow (\Delta_0 / \Pi \times \Delta_0 / \Pi')
\]

\[
\underset{\simeq\, \mathfrak{S}_2}{\mathfrak{S}_2} \cdot \prod_{i \in I} \mathfrak{S}_{T_i}
\]

\[
\operatorname{Aut}(C, \Delta) \longrightarrow
\underset{\mathfrak{S}_2 \cdot (\mathfrak{S}_3 \times \mathfrak{S}_3)}{\underset{\simeq}{\operatorname{Aut}(\Delta)}}
\times
\underset{\mathfrak{S}_3}{\underset{\simeq}{\operatorname{Aut}(V^{*})}}
\]

\[
\prod_{j \in I} T_j \simeq \Delta \longrightarrow T_i \quad (i \in I)
\]

\[
(C - \Delta) \longrightarrow \mathbb{F}' \times V^{*} .
\]

\[
C \simeq \Delta \amalg (\mathbb{F}' \times V^{*}) .
\]

\[
\begin{cases}
\text{ou bien} & \Delta(x) \cap \Delta(y) \neq \emptyset,\ v(x) = v(y) \quad (\text{donc card } \Delta(x) \cap \Delta(y) = 1) \\
\text{ou bien} & \Delta(x) \cap \Delta(y) = \emptyset,\ v(x) \neq v(y)
\end{cases}
\]

\[
\mathcal{J} = \bigwedge_{i \in I} \widetilde{I}_i ,
\]

\[
\varphi(\text{image de } \Delta(x) \text{ dans } \mathcal{J}) = v(x) ,
\]

\[
V^{*} \simeq \mathcal{J} \amalg \lbrace pt \rbrace ) .
\]

\[
b = (\alpha_1, \alpha'_2, \beta_1, \beta'_2, \gamma_1, \gamma'_2)
\]

\[
\alpha_2 - \alpha_1 = \alpha'_1 - \alpha'_2 = \beta_2 - \beta_1 = \beta'_1 - \beta'_2 = \gamma_2 - \gamma_1 = \gamma'_1 - \gamma'_2 ,
\]

\[
\begin{cases}
t \subset \Delta & \text{i.e. card}(t \cap \Delta) = 3 \\
\operatorname{card}(t \cap \Delta) = 1 & \\
t \cap \Delta = \emptyset & \text{i.e. card}(t \cap \Delta) = 0
\end{cases}
\qquad
\begin{array}{r}
6 \\ 3 \cdot 9 \\ 12 \\ \hline 45
\end{array}
\]

\[
(\ast)\quad
\begin{cases}
D_i \cap D_j = \emptyset & \text{si } i \neq j \\
D_i \cap D_{jk} \neq \emptyset & \text{ssi } i \in \{j, k\} \\
D'_i \cap D'_j = \emptyset & \text{si } i \neq j \\
D'_i \cap D_{jk} \neq \emptyset & \text{ssi } i \in \{j, k\} \\
D_i \cdot D'_j \neq \emptyset & \text{ssi } i \neq j \\
D_{ij} \cdot D_{kl} \neq \emptyset & \text{ssi } \{i,j\} \cap \{k,l\} \text{ de card.\ } 0 \text{ ou } 2
\end{cases}
\]

\[
(1) \qquad \eta, \ (\xi_i)_{1 \leq i \leq 6}
\]

\[
\begin{aligned}
&(2) & \xi_{ij} &= \eta - \xi_i - \xi_j \\
&(3) & \xi'_i &= 2\eta - \sum_{\alpha \neq i} \xi_\alpha = 2\eta - \xi + \xi_i
\end{aligned}
\]

\[
(3\,\text{bis}) \qquad \xi'_i - \xi_i = 2\eta - \xi
\]

\[
(4) \qquad \xi = \sum_{1}^{6} \xi_\alpha
\]

\[
(4') \qquad \xi' = \sum_{1}^{6} \xi'_\alpha = 12\eta - 5\xi
\]

\[
(5) \qquad 2\eta - \xi = -(2\eta' - \xi') \quad \text{i.e.} \quad
2(\eta + \eta') = \xi + \xi'
\]

\[
(5\,\text{bis}) \qquad \xi_i - \xi'_i = 2\eta' - \xi'
\]

\[
(6) \qquad \lambda = 3\eta - \xi
\]

\[
(7) \qquad \xi + \xi' = 12\eta - 4\xi = 4\lambda
\]

\[
(8) \qquad \eta + \eta' = 2\lambda
\]

\[
(8\,\text{bis}) \qquad \eta' = 5\eta - 2\xi
\]

\[
(10) \qquad \eta^2 = 1, \quad \xi_i^2 = -1, \quad \eta \cdot \xi_i = 0,
\quad \xi_i \xi_j = 0 \ \text{si } i \neq j
\]

\[
(11) \qquad \eta'^2 = 1, \quad \xi_i'^2 = -1, \quad \eta' \cdot \xi'_i = 0,
\quad \xi'_i \xi'_j = 0 \ \text{si } i \neq j .
\]

\[
\begin{aligned}
&(12) & \lambda^2 &= 3 \qquad (= \lambda\eta = \lambda\eta') \\
&(13) & \lambda\xi_i &= \lambda\xi'_i = \lambda\xi_{ij} = 1
\end{aligned}
\]

\[
(14) \qquad \xi_i \cdot \xi'_j =
\begin{cases} 0 & \text{si } i = j \\ 1 & \text{si } i \neq j \end{cases}
\]

\[
\begin{aligned}
&(15) & \xi_i \xi_{jk} = \xi'_i \xi_{jk} &=
\begin{cases} 0 & \text{si } i \notin \{j, k\} \\ 1 & \text{si } i \in \{j, k\} \end{cases} \\
&(16) & \xi_{ij} \xi_{kl} &=
\begin{cases} 0 & \text{si } \operatorname{card}(\{i,j\} \cap \{k,l\}) = 1 \\
1 & \text{si } \{i,j\} \cap \{k,l\} = \emptyset \end{cases}
\end{aligned}
\]

\[
\begin{aligned}
&(a)\ (\text{déf}) & \xi &= \sum_{1}^{6} \xi_i, \quad \xi' = \sum_{1}^{6} \xi'_i \\
&(b) & 2(\eta + \eta') &= \xi + \xi' \quad \text{i.e.} \quad
2\eta - \xi = -(2\eta' - \xi')
\end{aligned}
\]

\[
(c) \qquad \xi'_i - \xi_i = 2\eta - \xi \quad \text{i.e.} \quad
\xi_i - \xi'_i = 2\eta' - \xi'
\]

\[
(d) \quad
\begin{cases}
\xi'_i = 2\eta - \xi + \xi_i = 2\eta - \sum_{\alpha \neq i} \xi_\alpha \\
\xi' = 12\eta - 5\xi
\end{cases}
\]

\[
(d') \quad
\begin{cases}
\xi_i = 2\eta' - \xi' + \xi'_i = 2\eta' - \sum_{\alpha \neq i} \xi'_\alpha \\
\xi = 12\eta' - 5\xi'
\end{cases}
\]

\[
(e) \qquad \xi + \xi' = 4\lambda = 4\lambda' \qquad \text{avec} \quad
\lambda = 3\eta - \xi, \ \lambda' = 3\eta' - \xi'
\]

\[
(\ast) \qquad 3\eta - \xi = 3\eta' - \xi' \overset{\text{df}}{=} \lambda
\]

\[
\bigl(3\eta' = 3\eta - \xi + \xi' = 15\eta - 5\xi\bigr)
\]

\[
(f) \quad
\begin{cases}
\eta + \eta' = 2\lambda \\
\xi + \xi' = 4\lambda
\end{cases}
\qquad\qquad
(g, g') \quad
\begin{cases}
\eta' = 5\eta - 2\xi \\
\eta = 5\eta' - 2\xi'
\end{cases}
\]

\[
(h) \qquad \eta - \xi_i - \xi_j = \eta' - \xi'_i - \xi'_j
\quad \bigl(\overset{\text{df}}{=} \xi_{ij}\bigr)
\]

\[
(i) \qquad \eta^2 = 1, \quad \xi_i^2 = -1, \quad \eta\xi_i = 0, \quad
\xi_i\xi_j = 0 \ \text{si } i \neq j
\]

\[
(i') \qquad \eta'^2 = 1, \quad \xi_i'^2 = -1, \quad \eta'\xi'_i = 0, \quad
\xi'_i\xi'_j = 0 \ \text{si } i \neq j
\]

\[
\begin{aligned}
\xi_i &= \xi_i \\
\xi'_j &= 2\eta - \sum_{\alpha \neq j} \xi_\alpha \\
\xi_{ij} &= \eta - \xi_i - \xi_j \qquad (i \neq j) \bigr)
\end{aligned}
\]

\[
(j) \qquad \delta, \delta' \ (\in \Delta, \ \delta \neq \delta') \ \text{liés
ssi} \ \delta\delta' = 1 \quad \text{i.e.} \quad \delta\delta' \neq 0 .
\]

\[
\delta\delta' =
\begin{cases}
-1 & \text{si } \delta = \delta' \\
1 & \text{si } \delta \neq \delta', \ \delta \text{ et } \delta' \text{ liés} \\
0 & \text{si } \delta \neq \delta', \ \delta \text{ et } \delta' \text{ non liés}
\end{cases}
\]

\[
\widehat{E}_k(\Delta) = k^{[\Delta]} / N(\Delta),
\]

\[
\operatorname{Isom}(\Delta_0, \Delta) / \mathfrak{S}_6 \ \simeq \
\text{ens.\ des parties basiques de } \Delta .
\]

\[
\Delta \simeq \underbrace{I \amalg I \amalg \mathfrak{P}_2(I)}_{\substack{\text{avec la structure}\\ \text{de graphe déjà}\\ \text{explicitée}}}
\]

\[
\begin{cases}
\lambda = 3\eta(\alpha) - \xi(\alpha) & 1^\circ) \\
\eta(\alpha) + \underbrace{\eta(\alpha')}_{= \eta'(\alpha)} = 2\lambda & 2^\circ) \\
\xi(\alpha) + \underbrace{\xi(\alpha')}_{= \xi'(\alpha)} = 4\lambda & 3^\circ)
\end{cases}
\]

\[
\begin{cases}
\sum_{ij \in \mathfrak{P}_2} \xi_{ij}(\alpha) = 5\lambda & 4^\circ) \\
\sum_{s \in S} s = 9\lambda & 5^\circ)
\end{cases}
\]

\[
0 \to E \to \widehat{E} \to \mathbb{Z} \to 0
\]

\[
r^2 = \delta'^2 + \delta^2 = -2
\]

\[
\begin{cases}
\alpha_i \text{ lié à } \alpha'_j \text{ ssi } i \neq j, \text{ les } \alpha_i \text{ non liés mutuellement}, \\
\text{les } \alpha'_i \text{ itou, en prenant } r_\alpha = \alpha'_1 - \alpha_1 = \alpha'_2 - \alpha_2 = \cdots = \alpha'_6 - \alpha_6
\end{cases}
\]

\[
\begin{array}{lll}
1 & \begin{cases} r_0 = 2\eta - \sum \xi_i = \lambda - \eta \\ r_0 = \xi'_1 - \xi_1 = \xi'_2 - \xi_2 = \cdots = \xi'_6 - \xi_6 \end{cases}
& \text{déduite des } \begin{cases} (\xi_1, \dots, \xi_6) \\ (\xi'_1, \dots, \xi'_6) \end{cases} \\[3ex]
15 & \begin{cases} r_{ij} = \xi_i - \xi_j \\ 1 \leq i < j \leq 6 \end{cases}
& \text{déduites des } \begin{cases} (\xi_j, \xi'_j, (\xi_{ik})_{k \neq i,j}) \\ (\xi_i, \xi'_i, (\xi_{jk})_{k \neq i,j}) \end{cases} \\[3ex]
20 & \begin{cases} r_{ijk} = \eta - \xi_i - \xi_j - \xi_k \\ 1 \leq i < j < k \leq 6 \end{cases}
& \text{déduites des } \begin{cases} (\xi_i, \xi_j, \xi_k, \xi_{lm}, \xi_{mn}, \xi_{nl}) \\ (\xi_{jk}, \xi_{ki}, \xi_{ij}, \xi'_l, \xi'_m, \xi'_n) \end{cases}
\end{array}
\]

\[
r_\alpha \alpha_i = (\alpha'_i - \alpha_i)\alpha_i = 0 - (-1) = 1,
\]

\[
\begin{cases}
r_\alpha . \alpha_i = 1 \\
r_\alpha \alpha'_i = -1 \\
r_\alpha \delta = 0 \ \text{si } \delta \in \Delta_0 \text{ distinct des } \alpha_i, \alpha'_i
\end{cases}
\]

\[
\eta(\alpha) = \lambda - r(\alpha) = \lambda - r
\]

\[
\begin{aligned}
\eta(r_0) &= \eta(\xi_1, \dots, \xi_6) = \eta \\
\eta(-r_0) &= \eta(\xi'_1, \dots, \xi'_6) = \eta' = 5\eta - 2\xi = 5\eta - 2\textstyle\sum \xi_i \\
\eta(\xi_i - \xi_j) &= \eta(\xi_j, \xi'_i, (\xi_{ik})) = 3\eta - 2\xi_i - \textstyle\sum_{k \neq i,j} \xi_k \ (= \lambda - \xi_i + \xi_j = \lambda - r)
\end{aligned}
\]

\[
r = 2\eta(\alpha) - \underbrace{\sum \alpha_i}_{\xi(\alpha)}
\quad \text{i.e.} \quad
\underset{\substack{\Vert \\ \sum \alpha_i}}{\xi(\alpha)} = 2(\lambda - r) - r = 2\lambda - 3r
\]

\[
s_r : x \longmapsto x + (x.r)r \qquad \widehat{E} \to \widehat{E}
\]

\[
\boxed{s_r \alpha_i = \alpha'_i, \quad s_r \alpha'_i = \alpha_i}
\]

\[
\boxed{s_r \delta = \delta \quad \text{si } \delta \notin \{(\alpha_i), (\alpha'_i)\}}
\]

\[
s_{r_{ijk}}(r_0) = r_{l,m,n}
\]

\[
s_{r_{ijk}}(r_{il}) = r_{jkl}
\]

\[
\begin{aligned}
&r_1 = r_{12} = \xi_1 - \xi_2 \\
&r_2 = r_{23} = \xi_2 - \xi_3 \\
&\dots \\
&r_5 = r_{56} = \xi_5 - \xi_6 \\
&r_6 = r_{123} = \eta - \xi_1 - \xi_2 - \xi_3
\end{aligned}
\]

\[
r_6 = \eta - \xi_1 - \xi_2 - \xi_3
= (\eta - 3\xi_1) + (\xi_1 - \xi_2) + (\xi_1 - \xi_3), \ldots
\]

\[
\boxed{r_{ij} = r_i + r_{i+1} + \cdots + r_{j-1} \quad \text{si } i < j}
\]

\[
\begin{aligned}
r_{ijk} &= \eta - \xi_i - \xi_j - \xi_k = (\eta - \xi_1 - \xi_2 - \xi_3)
+ \underbrace{(\xi_1 - \xi_i)}_{r_{1i}} + \underbrace{(\xi_2 - \xi_j)}_{r_{2j}} + \underbrace{(\xi_3 - \xi_k)}_{r_{3k}} \\
&= r_{123} + r_{1i} + r_{2j} + r_{3k}
\end{aligned}
\]

\[
r_{ijk} = r_6 + (r_1 + \cdots + r_{i-1}) + (r_2 + \cdots + r_{j-1}) + (r_3 + \cdots + r_{k-1})
\]

\[
\boxed{r_{ijk} = r_6 + (r_1 + \cdots + r_{i-1}) + (r_2 + \cdots + r_{j-1}) + (r_3 + \cdots + r_{k-1})}
\quad \text{pour } i < j < k
\]

\[
r_0 = 2\eta - \sum \xi_i = (\eta - \xi_1 - \xi_2 - \xi_3) + (\eta - \xi_4 - \xi_5 - \xi_6)
\]

\[
= r_{123} + r_{456}
\]

\[
\boxed{r_0 = r_1 + 2r_2 + 3r_3 + 2r_4 + r_5 + 2r_6} \quad (\text{poids } 11)
\]

\[
W = \operatorname{Aut}(\Delta) \hookrightarrow \widetilde{W}
\]

\[
\Delta \xrightarrow{\ i\ } E
\]

\[
i(\delta) = 3\delta - \lambda \qquad (\in E \ \text{car} \ \delta.\lambda = 1, \ \lambda\lambda = 3)
\]

\[
i(\delta)\, i(\delta') = 9\delta\delta' - 6 + 3 = 9\delta\delta' - 3
\]

\[
\begin{cases}
i(\delta)^2 = -12 \\
i(\delta)\, i(\delta') = \begin{cases} 9 & \text{si } \delta, \delta' \text{ liés} \\ -3 & \text{si } \delta \neq \delta' \text{ non liés.} \end{cases}
\end{cases}
\]

\[
x . x = -12
\]

\[
x, x' \ \text{liés} \iff x . x' = 9
\]

\[
3R = \lbrace x' - x \mid x, x' \in \Delta' \rbrace .
\]

\[
-\Delta' = \lbrace -(3\delta - \lambda) = \lambda - 3\delta \mid \delta \in \Delta \rbrace
\]

\[
3\delta - \lambda = \lambda - 3\delta' \ \text{implique} \ 3(\delta + \delta') = 2\lambda = 2(3\eta - \xi)
\]

\[
\widetilde{W} \simeq W \times \underbrace{\mathbb{Z}/2\mathbb{Z}}_{\substack{\text{engendré par} \\ \operatorname{id}_E}}
\]

\[
\mathfrak{P}_2(I) \longrightarrow \Delta - (I \cup I')
\]

\[
\operatorname{Card} \underset{\substack{\Vert \\ \operatorname{Aut}(\Delta)}}{W} = 72 . 6! = 2^7 3^4 5
\]

\[
r_1+r_2+r_3,\qquad r_2+r_3+r_4,\qquad r_3+r_4+r_5 ;
\]

\[
\begin{array}{ll}
r_0=2\eta-\ldots & \\
r_1=\xi_1-\xi_2 & \\
r_2=\xi_2-\xi_3 & 3(r_1+r_2+r_3)=3(\xi_1-\xi_4)\\
r_3=\xi_3-\xi_4 & \\
r_4=\xi_4-\xi_5 & 2r_4=\ldots\xi_4-2\xi_5\\
r_5=\xi_5-\xi_6 & 2r_5=\ldots\xi_5-\xi_6\\
r_6=\eta-\xi_1-\xi_2-\xi_3 &
\end{array}
\qquad 3\xi_1-\xi_4-\xi_5-\xi_6
\]

\[
r_0=2\eta-\xi,\qquad r_{ij}=\xi_i-\xi_j\quad(i<j),
\]

\[
r_{\lbrace i,j,k\rbrace}=\xi_{jk}-\xi_i=\xi_{ki}-\xi_j=\xi_{ij}-\xi_k
\]

\[
=\xi'_\ell-\xi_{mn}=\xi'_m-\xi_{n\ell}=\xi'_n-\xi_{\ell m}=\eta-\xi_i-\xi_j-\xi_k
\]

\[
r_0\cdot r_{ij}=0,\qquad r_0\cdot r_{ijk}=-1,
\]

\[
r_{ij}\cdot r_{i'j'}=\begin{cases}0 & \lbrace i,j\rbrace\cap\lbrace i',j'\rbrace=\emptyset\\ \pm1 & \lbrace i,j\rbrace\cap\lbrace i',j'\rbrace\neq\emptyset\end{cases}
\]

\[
r_{ijk}\cdot r_{pqr}=1-\operatorname{card}(\lbrace i,j,k\rbrace\cap\lbrace p,q,r\rbrace)=\begin{cases}-1 & \operatorname{card}(\lbrace i,j,k\rbrace\cap\lbrace p,q,r\rbrace)=2\\ 0 & \ldots=1\\ 1 & \ldots=0\end{cases}
\]

\[
r_{ij}\cdot r_{pqr}=\begin{cases}0 & \lbrace i,j\rbrace\subset\lbrace p,q,r\rbrace\ \text{ou}\ \lbrace i,j\rbrace\cap\lbrace p,q,r\rbrace=\emptyset\\ \pm1 & \operatorname{card}(\lbrace i,j\rbrace\cap\lbrace p,q,r\rbrace)=1\end{cases}
\qquad
\begin{cases}+1 & i\in\lbrace p,q,r\rbrace\\ -1 & j\in\lbrace p,q,r\rbrace\end{cases}
\]

\[
\frac{-\langle r,r'\rangle}{\sqrt{\langle r,r\rangle\langle r',r'\rangle}}=\frac{\langle r,r'\rangle}{-2}
\]

\begin{align*}
\xi_1\xi_2\xi_3\ \ \xi_4,\xi_5\xi_6\quad &= r_0=2\eta-\xi=3(r_1+r_2+r_3)+2r_4+r_5+2r_6\\
\xi_1\xi_2\xi_3\ \ \xi_{56}\xi_{64}\xi_{45}\quad &= r_{123}=\eta-\xi_1-\xi_2-\xi_3=r_6\\
\xi_{23},\xi_{31},\xi_{12}\ \ \xi_4\xi_5\xi_6\quad &= r_{4,5,6}=\eta-\xi_4-\xi_5-\xi_6=r_6+3(r_1+r_2+r_3)+2r_4+r_5
\end{align*}

\[
\begin{array}{ll}
\operatorname{Pl}(L_1,C)=C & 27=3^3\\
\uparrow\ \operatorname{Pl}(L_2,C) & 27\cdot 16=2^4 3^3=432\\
\uparrow\ \operatorname{Pl}(L_3,C) & 27\cdot16\cdot10=2^5\cdot3^3\cdot5=4320\\
\uparrow\ \operatorname{Pl}(L_4,C)\xleftarrow{\ \approx\ }\operatorname{Pl}''(L_5,C) & 27\cdot16\cdot10\cdot6=2^6\cdot3^4\cdot5=25\,920\\
\uparrow\ \operatorname{Pl}'(L_5,C) & 27\cdot16\cdot10\cdot6\cdot2=2^7\cdot3^4\cdot5=51\,840\\
\uparrow\ \operatorname{Pl}(L_6,C) & 
\end{array}
\]

\[
\begin{array}{lll}
B_1(C) & 27:1=27=3^3 & \operatorname{aut}L_1=e\\
B_2(C) & 27\cdot16:2!=27\cdot8=2^3 3^3=216 & \operatorname{aut}L_2=\mathfrak{S}_2\\
B_3(C) & 27\cdot16\cdot10:3!=2^4\cdot3^2\cdot5=720=6! & \operatorname{aut}L_3=\mathfrak{S}_3\\
B_4(C) & 27\cdot16\cdot10\cdot6:4!=2^3\cdot3^3\cdot5=1080 & \operatorname{aut}L_4=\mathfrak{S}_4\\
\quad\simeq B''_4(C) & & \\
B'_5(C) & 27\cdot16\cdot10\cdot6\cdot2:5!=2^4\cdot3^3=432 & \operatorname{aut}L_5=\mathfrak{S}_5\\
B(C)=B_6(C) & 27\cdot16\cdot10\cdot6\cdot2\cdot1:6!=2^3 3^2=72 & \operatorname{aut}L_5\simeq\mathfrak{S}_6\\
\operatorname{Bib}(C) & 36 & \operatorname{aut}(L'_5)\simeq\mathfrak{S}_2\times\mathfrak{S}_6
\end{array}
\]

\[
\mathfrak{S}_\tau\times\mathfrak{S}_K\times\mathfrak{S}_{K'}\quad(\simeq\mathfrak{S}_2\times\mathfrak{S}_3\times\mathfrak{S}_3)
\]

\[
(\tau,K,K')\longmapsto(\tau,K',K).
\]

\[
\tau\times K'\subset\mathcal{B}=\struck{\ill{}}\ \tau\times(K\amalg K')\ \cdots
\]

\[
E(J)\simeq(b-J)=b\cap\mathcal{L}(J)\subset\mathcal{L}(J).
\]

\[
T\longrightarrow\tau\qquad(\operatorname{card}\tau=2).
\]

\[
\operatorname{card}\Bigl(\frac{D\times D-D}{2}\Bigr)-18=18
\]

\[
\Delta\xrightarrow{\ \sim\ }\prod_{i\in\sigma}\mathfrak{S}_i
\]

\[
\Delta\longmapsto\bar\Delta
\]

\[
c\cdot c'=\begin{cases}-1 & c=c'\\ 0 & c\neq c',\ c,c'\ \text{non liés}\\ 1 & c\neq c',\ c,c'\ \text{liés}\end{cases}
\]

\[
\dot e_1=e_2+e_3+e_4+e_5,\qquad \dot e_2=e_1+e_3+e_4+e_5,\qquad \dot e_1+\dot e_2=e_1+e_2 ;
\]

\[
3\cdot2^6\qquad 3\cdot2^3\qquad 3\cdot2=6\qquad 2
\]

\[
\underbrace{45\cdot3}\cdot\underbrace{32\cdot3}\ :\ \ldots=45\cdot8\cdot\ldots=2^3\,3^2\,5=360
\]

\[
U\mid P_u=\mathrm{id},\quad V\mid P_v=\mathrm{id},\quad W(P_w)=\mathrm{id},\qquad U(\alpha_4)=\alpha_4,\quad V(\beta_4)=\beta_4,\quad W(\gamma_4)=\gamma_4,
\]

\[
UVW\in\lbrace1,\sigma\rbrace\ \overset{?}{\Longrightarrow}\ U,V,W=1 ;
\]

\[
\underset{9}{I\times\lbrace0,1,2\rbrace}\ \amalg\ \underset{3}{I}\times\underset{3}{J}\times\underset{2}{\operatorname{circ}(I)}.
\]

\[
  \delta'_1-\delta_1=\delta'_2-\delta_2 \qquad
  \delta_1\neq\delta_2,\ \delta_1\neq\delta'_1
\]

\[
  \underbrace{\delta'_1+\delta_2}=\underbrace{\delta'_2+\delta_1}
\]

\[
  \xi_{ij}=\eta-\xi_i-\xi_j \qquad
  \sum\xi_{ij}=15\eta-5\xi=5\lambda
\]

\[
  \delta\longmapsto\lambda-\delta
\]

\[
  \xi_i\longmapsto\underbrace{\xi'_j,\ \xi_{ij}}\qquad
  \xi'_j-\xi_{ij}=2\eta-\xi+\xi_h
\]

\[
  \struck{=\eta-\xi_i-\xi_j}\qquad \struck{\ill{}}\ -\xi_i
\]

\[
  \xi'_i\longmapsto\underbrace{\xi_j,\ \xi_{ij}}\qquad
  \xi_h+\xi_{ij}=\eta-\xi_j
\]

\[
\begin{array}{c}
  A_3\ \text{ép.}\\
  \uparrow{\scriptstyle 4}\\
  A_4\ \text{ép.}\sim A_5\ \text{ép.}\sim\text{hex.\ ép.}\sim
  \text{prisme épinglé}\sim\text{carrous.\ ép.}\\
  \uparrow{\scriptstyle 3}\\
  \text{carrousel ép.}\sim\text{pent.\ ép.}\\
  \uparrow{\scriptstyle 2}\\
  (\text{\uncertain{repère}})
\end{array}
\]

\[
  a_1+a_4=\alpha_1,\qquad a_2+a_4=\alpha_2,\qquad a_3+a_4=\alpha_3
\]

\[
\begin{array}{ll}
  \text{\emph{prismes épinglés}}\ \ 27\cdot10\cdot8\cdot4\cdot1=[W]:6 &\\
  \text{prismes} & [W]:72\\[4pt]
  \text{A}_5\ \text{épinglés}\ \ 27\cdot10\cdot8\cdot4\cdot1=[W]:6 &\\
  \text{A}_5 & [W]:12\\[4pt]
  \text{A}_4\ \text{épinglé}\ \ 27\cdot10\cdot8^{\,4}=[W]:\uncertain{6} &\\
  \text{A}_4 & [W]:12\\[4pt]
  \text{A}_3\ \text{épinglés}\ \ 27\cdot10\cdot8=[W]:24 &\\
  \quad(\text{autres que les triangles épinglés}) &\\
  \text{A}_3\ \text{ép.} & [W]:48\\[4pt]
  \text{carrousels}\ \text{\add{simples}}\ \text{épinglés}\ \
  27\cdot10\cdot8\cdot4\cdot3=[W]:2 &\\
  \text{carrousels}\ \text{\add{simples}} & [W]:12
\end{array}
\]

\[
  \xi_1\quad\xi_{14}\quad\xi_4\quad\xi'_2 \qquad\qquad
  \xi_1\ \big|\ \xi'_2\quad\xi'_3\quad\xi_{31}\quad\xi_{12}
\]

\[
  \delta=\xi_1,\ \delta'=\xi_2,\ \delta''=\xi'_2 ;\qquad
  \delta'''=\xi_{14},\ \delta^{IV}=\xi_{25}
\]

\[
  \delta\delta'=0\quad\delta\delta''=1\quad
  \delta\delta'''=1\quad\delta\delta^{IV}=0
\]

\[
  \delta\delta'\ \struck{\bmod 2}\equiv
  \operatorname{card}(\delta\cap\delta')
  +\operatorname{card}\bigl(\varepsilon(\delta)\cap\varepsilon(\delta')\bigr)
\]

\[
\begin{array}{ll}
  \text{hex.\ pointés \add{orientés}}\ \
  27\cdot10\cdot8\cdot4\cdot1\cdot1=2^6\cdot3^3\cdot5=[W]:6 &\\
  \text{hexagones} & [W]:72\\[4pt]
  \text{pentagones pointés \add{orientés}}\ \
  27\cdot10\cdot8\cdot4\cdot3=2^6\,3^4\,5=[W]:2 &\\
  \text{pentagones} & [W]:20\\[4pt]
  \text{carr.\ pointés}\ \ 27\cdot10\cdot8\cdot4=[W]:6 &\\
  \text{carr.} & [W]:48
\end{array}
\]

\[
  \mathbb{F}_2^{A}\quad P\qquad x,y\qquad
  x-y=u\in\mathbb{F}_2^{A}\simeq\mathfrak{P}(A)
\]

\[
  x,y\ \text{liés}\iff
  \begin{cases}
    x\ \text{et}\ y\ \text{de même parité}\ (\text{i.e.\ card}\ u\
      \text{pair}) :\ x-y=\mathbb{1}^{A}\\
    \quad(\text{i.e.}\ E(x)\cap\tilde A\ \text{et}\ E(y)\cap\tilde A\
      \text{sont complém.})\\
    x\ \text{et}\ y\ \text{de parité contr.}\ (\text{i.e.\ card}\ u\
      \text{impair}) :\ \struck{\ill{}}\ \operatorname{card}u=3
  \end{cases}
\]

\[
  x,y\ \text{non liés}\iff
  \begin{cases}
    x\ \text{et}\ y\ \text{de même parité i.e.\ card}\ u\ \text{pair} :\
      x-y\neq\mathbb{1}\\
    \quad(\text{i.e.}\ \struck{\ill{}}\ E(x)\cap\tilde A\cap
      E(y)\cap\tilde A\neq\emptyset)\\
    x\ \text{et}\ y\ \text{de parité contr.} :\ \operatorname{card}u=1
  \end{cases}
\]

\[
  x,y\ \text{liés}\iff\operatorname{card}u=3\ \text{ou}\ 4 \qquad
  x,y\ \text{non liés}\iff\operatorname{card}u=0,1\ \text{ou}\ 2
\]

\[
\begin{array}{l}
  \text{triangles pointés orientés}\ \ 27\cdot10=270=[W]:
    \uncertain{2^3\cdot3^4}\\
  \text{triangles}:\ \dfrac{27\cdot10}{6}=45=[W]:\uncertain{2^7\cdot3^2}
\end{array}
\]

\[
\begin{array}{l}
  \text{bicarrousel épinglé}\\
  \text{\ill{} du bicarrousel}\\
  \quad\text{d'indice}\ 48=2^4\cdot3\\
  \text{nb des bicarrousels}\ \ 2^3\,3^3\,5=6^3\cdot5=1080\\
  \quad=\underbrace{72}_{\text{\uncertain{racines}}}\cdot
    \underbrace{15}_{\text{nb de \ill{} d'une \uncertain{racine}, 6}}
\end{array}
\]

\[
  \frac{8!}{2!\,\struck{\ill{}}\,2!}\cdot
  \binom{6}{2}\binom{4}{2}\,\frac{1}{3!}
  =\frac{15\cdot6}{6}=15
\]

\[
\begin{array}{lll}
  r'=r & \iff r\cdot r'=-2 & \iff b=b'\\
  r'=-r & \iff r\cdot r'=2 & \iff b'\ \text{associé à}\ b\\
  r'\ \text{orth.\ à}\ r & \iff rr'=0 & \iff \operatorname{card}(b\cap b')=1\\
  \widehat{r,r'}=60^\circ & \iff rr'=-1 & \iff
    \operatorname{card}(b\cap b')=3\ \ \struck{\text{\ill{}}}\\
  \widehat{r.r'}=120^\circ & \iff rr'=1 & \iff
    \operatorname{card}(b\cap b')=0\ \ \text{\add{et $b,b'$ non associés}}
\end{array}
\]

\[
\begin{array}{ll}
  1 & \text{racine égale à}\ r,\ \text{savoir}\ r\\
  1 & \text{racine opposée à}\ r,\ \text{savoir}\ -r\\
  30 & \text{racines orthogonales à}\ r\\
  20 & \struck{\text{\ill{}}}\ \text{faisant un angle de}\ 60^\circ\
    \text{avec}\ r\\
  20 & \text{-----}\ \ 120^\circ\ \ \text{-----}\ \ r
\end{array}
\]

\[
\begin{array}{lll}
  \text{couples de racines} & \text{égales} & 72\\
  & \text{opposées} & 72/2=36\\
  \text{couples de racines orthogonales} && 72\cdot15=1080\\
  \text{couples de racines} & 60^\circ & 72\cdot10=720\\
  & 120^\circ & 72\cdot10=720
\end{array}
\]

\[
  s_r(x)=x+(r,x)\,r \qquad s_r(x)-x=(r,x)\,r \qquad
  (r,x)\in\lbrace0,1,-1\rbrace
\]

\[
  (\struck{x-y})\ \ s_rx-x=\qquad \struck{x=wx}\qquad
  wx=x+\rho\quad \rho\ \text{\uncertain{vecteur}}
\]

\[
  s_rwx=s_rx+s_r\rho=x+\underbrace{(x,r)}_{\varepsilon}\,r+s_r\rho
\]

\[
  \struck{r_0-2r_1-r_2+r_6}\qquad
  r_1+2r_2+3r_3+2r_4+r_5+3r_6 \qquad \alpha_1
\]

\[
  3\xi_i-\lambda=3\xi_i-3\eta+\xi=-3\eta+\sum_{\alpha\neq i}\xi_\alpha+4\xi_i
  \qquad 3\uncertain{\eta_i}-\lambda
\]

\[
  \tilde\xi_i=3\xi_i-\lambda=
  \underbrace{3(\xi_i-\xi_1)}_{-3(r_1+r_2+\cdots+r_{i-1})}
  +\underbrace{(\uncertain{3\xi_1-\eta})}_{-r_6+2r_1+r_2}
  +\underbrace{\eta-\lambda}_{-r_0}
\]

\[
  -\tilde\xi_i=3(r_1+\cdots+r_{i-1})\ \struck{\ill{}}\ -r_2+r_6
  +(r_1+2r_2+3r_3)+2r_4+r_5+2r_6
\]

\[
\begin{aligned}
  -\tilde\xi_1&=-r_1+4r_2+3r_3+2r_4+r_5\ \struck{+r_6}\ +3r_6\\
  -\tilde\xi_2&=2r_1+4r_2+3r_3+2r_4+r_5+3r_6\\
  -\tilde\xi_3&=2r_1+4r_2+3r_3+2r_4+r_5+3r_6\\
  -\tilde\xi_4&=2r_1+4r_2+6r_3+2r_4+r_5+3r_6\\
  -\tilde\xi_5&=2r_1+4r_2+6r_3+5r_4+r_5+3r_6\\
  -\tilde\xi_6&=2r_1+\uncertain{9}r_2+6r_3+5r_4+4r_5+3r_6
\end{aligned}
\]

\[
  \struck{-\tilde\eta=\ill{}\,r_1+(r_2+r_3)+6r_4+3r_5+9r_6}
\]

\[
  \struck{\tilde\eta=3\eta-3\lambda=-3(\lambda-\eta)=-3r_0}
\]

\[
  -\tilde\eta=+3r_1+6r_2+9r_3+6r_4+3r_5+6r_6
\]

\[
\begin{array}{l}
  a+3b=-2\\
  3a+\uncertain{9}b=0\\
  \struck{2a=}\quad a=-\tfrac{6}{3}\\
  \uncertain{8}a=2\qquad a=\tfrac14\qquad b=
\end{array}
\qquad
\begin{array}{l}
  a+3b+\\
  a+3b+2,\ a+3b+3\\
  a+3b,\ a+3b+1\\
  \quad\parallel\\
  a+3b'-3,\ a+3b'-2
\end{array}
\]

\[
  2\quad -1\qquad a+3b=-1\qquad
  \left.\begin{array}{l}a+3b=-1\\ 3a+b=0\end{array}\right\rbrace\quad
  x=b\bigl(\xi-\tfrac{\uncertain{5}}{3}\bigr)+\xi_i
\]

\[
  x=a\cdot\eta+b\xi+\xi_i\qquad \struck{\ill{}\,6b}\qquad
  x=a\eta+b'\xi-\xi_i\qquad a=2b\qquad \struck{3a+6b}\quad
  \underline{3a+6b=0}
\]

\[
  x=a\eta+b\xi+\xi_i \qquad
  \begin{array}{l}3a+6b=0\\ a+2b=0\\ a=-2b\end{array}
  \qquad
  \begin{array}{ll}
    \alpha)\ a+3b=-1 & \text{i.e.}\ \boxed{b=-1,\ a=2}\quad
      2\eta-\xi+\xi_i=\boxed{\xi'_i}\\
    \beta)\ a+3b=0 & \text{i.e.}\ b=0,\ a=0\quad \boxed{\xi_i}
  \end{array}
\]

\[
  x=a\eta+b\xi-\xi_i\qquad \lambda_0=\eta-\frac{\xi}{3}
\]

\[
  \text{prenons}\ \ x'=2\lambda_0-x=\struck{\ill{}}\,2\eta-\tfrac23\xi
  -a\eta-b\xi+\xi_i
  =\underbrace{(2-a)}_{a'}\eta
  +\underbrace{\bigl(-\tfrac23-b\bigr)}_{b'}\xi+\xi_i
\]

\[
  x=a\eta+b\xi-\xi_i-\xi_j\qquad
  3a+6b=3\quad a+2b=1\quad a=1-2b\qquad
  a+3b=1\quad \boxed{b=0,\ a=1}\quad \boxed{\xi_{ij}}
\]

\[
  x=a\eta+b\xi+\xi_i+\xi_j
\]

\[
  \delta\sim-(\delta-\lambda_0)+\lambda_0=2\lambda_0-\delta
  =\tfrac23\lambda-\delta=\delta'
\]

\[
\begin{array}{l}
  r_1\longmapsto-r_5\\
  r_2\longmapsto-r_4\\
  r_3\longmapsto-r_3\\
  r_4\longmapsto-r_2\\
  r_5\longmapsto-r_1\\ \hline
  r_6\longmapsto r_{456}=r_6+3(\ill{})+2r_4+r_5
\end{array}
\]

\[
  V_{ijh}=\eta-\xi_i-\xi_j-\xi_h \qquad
  \eta-\xi_3-\xi_5-\xi_{\uncertain{6}}=\eta-\xi_1-\xi_2-\xi_3
  +(\xi_1-\xi_4)+(\xi_2-\xi_5)+\xi_{\ill{}}-\xi_{\ill{}}
\]

\[
  \xi'_i=\struck{\ill{}}\ r_0+\xi_i \qquad
  \tilde\xi'_i=\tilde r_0+\tilde\xi_i=3r_0+\tilde\xi_i
  \qquad \lbrace i\mapsto\ill{}\rbrace\qquad\lambda
\]

\[
  -\tilde\xi'_i=-\tilde\xi_i-3r_0=-\tilde\xi_i\
  \struck{-3(r_1+r_2+r_3)-6r_4-3r_5-6r_6}\ =-\tilde\xi_i-\tilde\eta
\]

\[
  \xi_{ij}=\eta-\xi_i-\xi_j
\]

\[
  \tilde\xi_{ij}=\tilde\eta-\tilde\xi_i-\tilde\xi_j
  =(-\tilde\xi_i)+(-\tilde\xi_j)-(-\tilde\eta)
\]

\[
\begin{array}{ll}
  1,4,-8,-5,-4,-1 & -8,-5,-4,-1,1,4\\
  2,5,-7,-4,-5,-2 & -7,-5,-4,-2,2,5\\
  3,6,-6,-3,-3,0 & -6,-3,0,3,6\\
  2,5,-4,-1,-2,1 & -4,-2,-1,1,2,5\\
  1,4,-2,1,-1,2 & -2,-1,1,2,4\\
  3,-3,-3 & -3,3
\end{array}
\qquad
\begin{array}{c}6\\6\\5\\6\\5\\2\end{array}
\]

\[
  -\tilde\delta=-w(-\tilde\delta)\qquad
  \tilde\delta=\struck{\ill{}}-w(\tilde\delta)=-\widetilde{w(\delta)}
  =-\tilde\delta' \qquad \widetilde{\delta\cdot\delta'}=0
\]

\[
  \ill{}\ \delta+\delta'=2\lambda=\qquad
  \tilde\delta=-w(\tilde\delta)=-\widetilde{w(\delta)}=-\tilde\delta'
  \qquad \tilde\delta\cdot\tilde\delta'=0
\]

\[
\begin{array}{c|cccccc}
  & r_1 & r_2 & r_3 & r_4 & r_5 & r_6\\ \hline
  -\tilde\xi_1 & -1 & 1 & 3 & 2 & 1 & 3\\
  -\tilde\xi_2 & 2 & 1 & 3 & 2 & 1 & 3\\
  -\tilde\xi_3 & 2 & 4 & 3 & 2 & 1 & 3\\
  -\tilde\xi_4 & 2 & 4 & 6 & 2 & 1 & 3\\
  -\tilde\xi_5 & 2 & 4 & 6 & 5 & 1 & 3\\
  -\tilde\xi_6 & 2 & 4 & 6 & 5 & 4 & 3\\ \hline
  -\tilde\xi'_1 & 2 & 7 & 12 & 8 & 4 & 9\\
  -\tilde\xi'_2 & 5 & 7 & 12 & 8 & 4 & 9\\
  -\tilde\xi'_3 & 5 & 10 & 12 & 8 & 4 & 9\\
  -\tilde\xi'_4 & 5 & 10 & 15 & 8 & 4 & 9\\
  -\tilde\xi'_5 & 5 & 10 & 15 & 11 & 7 & 9\\
  -\tilde\xi'_6 & 5 & 10 & 15 & 11 & 10 & 9\\ \hline
  & -2 & -4 & -3 & -2 & -2 & 0\\
  & 1 & -1 & 0 & 1 & 1 & 0
\end{array}
\]

\[
  -2,-1,1,2,5 \qquad -2,1,4,7,10
\]

\[
  \mathfrak{S}=\bigl\lbrace x\in E^{*}\ \big|\
  (x,r)\ \struck{\tfrac{2}{(r,r)}}\in\lbrace-1,0,1\rbrace\ \
  \forall r\in R\bigr\rbrace
\]

\[
  \mathfrak{S}_0=\bigl\lbrace x\in E^{*}\ \big|\ wx-x\in R\ \
  \forall w\in W\ \text{t.q.}\ wx\neq x\bigr\rbrace
  \qquad W=\text{groupe de Weyl}
\]

\[
  E=\mathbb{R}^{r+1},\qquad
  E=\Bigl\lbrace(x_1,\ldots,x_{r+1})\ \Big|\ \sum_{1}^{r+1}x_i=0\Bigr\rbrace
\]

\[
  R=\lbrace\alpha_{ij}=\xi_i-\xi_j\mid i\neq j,\ 1\leq i,j\leq r+1\rbrace
  \qquad \operatorname{card}R=r(r+1)
\]

\[
  \xi_1\ \ldots\ \xi_r\qquad \sum\xi_i=0
\]

\[
  \underbrace{\xi_1=\cdots=\xi_{r_1}}_{\alpha_1}<
  \underbrace{\xi_{r_1+1}=\cdots=\xi_{r_1+r_2}}_{\alpha_2}\ \cdots\
  \underbrace{\xi_{r_1+r_2+\cdots+r_{s-1}+1}=\cdots=
    \xi_{\underbrace{r_1+r_2+\cdots+r_s}_{r+1}}}_{\alpha_s}
\]

\[
  \begin{cases}
    r_1\alpha_1+r_2\alpha_2+\cdots+r_s\alpha_s=0\\
    r_1+r_2+\cdots+r_s=r+1
  \end{cases}
  \qquad
  \begin{array}{l}
    \alpha_1<\alpha_2<\cdots<\alpha_s\\
    r_1,r_2,\ldots,r_s\geq1
  \end{array}
\]

\[
  \alpha_i-\alpha_j\in\lbrace\pm1\rbrace\quad\forall i,j\ \neq
\]

\[
  \begin{cases}
    r_1\alpha_1+r_2\alpha_2=0\\
    r_1+r_2=r+1\\
    \alpha_2=\alpha_1+1
  \end{cases}
  \Longrightarrow
  \begin{array}{l}
    (r_1+r_2)\alpha_1+r_2=0\\
    \text{i.e.}\ (r+1)\alpha_1+r_2=0\\
    \text{i.e.}\ \alpha_1=\dfrac{-r_2}{r+1}
  \end{array}
\]

\[
  \struck{\text{Il y a}\ r+}\quad
  \alpha_1=\frac{-r_2}{r+1},\quad\alpha_2=\frac{r_1}{r+1}\qquad
  r_1+r_2=r+1
\]

\[
  \underbrace{-\frac{r_2}{r+1},\ldots,-\frac{r_2}{r+1}}_{r_1},\
  \underbrace{\frac{r_1}{r+1},\ldots,\frac{r_1}{r+1}}_{r_2}
  \qquad
  -\frac{r_1}{r+1}\ \ -\frac{r_2}{r+1}\qquad\pm1
\]

\[
  [1,r+1]=\underbrace{[1,r_1]}_{I_1}\cup
  \underbrace{[r_1+1,r+1]}_{I_2}
  \qquad \sigma I_1,\ \sigma I_2
\]

\[
  \xi-\sigma\xi\qquad
  \begin{array}{ll}
    0 & \text{sur}\ (I_1\cap\sigma I_1)\cup(I_2\cap\sigma I_2)\\
    \struck{-1} & \text{sur}\ I_1\cap\sigma I_2\\
    & \text{sur}\ I_2\cap\sigma I_1
  \end{array}
\]

\[
\begin{array}{ll}
  \operatorname{card}(I_1\cap\sigma I_2)=1 &
    \to\ \text{c'est possible (\uncertain{ssi}}\
    \operatorname{card}I_1\leq1\ \text{ou}\ \operatorname{card}I_2\leq1\
    \forall\sigma\ldots)\\
  \operatorname{card}(I_2\cap\sigma I_1)=1 & \text{--- \,--- }
\end{array}
\]

\[
  \underbrace{-\frac{1}{r+1},\ \frac{r}{r+1},\ldots,\frac{r}{r+1}}_{r}
  \qquad
  \underbrace{-\frac{r}{r+1},\ \struck{\ill{}},\ -\frac{r}{r+1}}_{r},\
  \frac{1}{r+1}
\]

\[
  R=\lbrace\pm\varepsilon_i\ (1\leq i\leq r),\ \pm\varepsilon_i\pm
  \varepsilon_j\ (1\leq i<j\leq r)\rbrace
  \qquad 2r+4\,\frac{r(r-1)}{2}=2r+2r(r-1)=2r^2
\]

\[
  \xi=(\xi_1,\xi_2,\ldots,\xi_r)\qquad
  \struck{\xi_i}\ \ 0\leq\xi_1\leq\xi_2\leq\cdots\leq\xi_r
\]

\[
  \underbrace{\xi_1=\cdots=\xi_{r_1}}_{\alpha_1}\
  \underbrace{\xi_{r_1+1}=\cdots=\xi_{r_1+r_2}}_{\alpha_2}\ \cdots\
  \underbrace{\xi_{r_1+\cdots+r_{s-1}+1}=\cdots=
    \xi_{r_1+\cdots+r_s=r}}_{\alpha_s}
\]

\[
  (\xi,\varepsilon_i)\in\struck{2}\lbrace-1,0,1\rbrace\ \ \text{d.c.}\ \
  \boxed{\xi_i\ \text{égale}\ 0\ \text{ou}\ 2}
\]

\[
  (\xi,\varepsilon_i\pm\varepsilon_j)=\xi_i\pm\xi_j\in\lbrace1,0,-1\rbrace
  \qquad \struck{\ill{}}\
  \begin{array}{l}\alpha_i+\alpha_j=1\\ \alpha_i-\alpha_j=-1\end{array}
  \qquad \struck{\ill{}}\ \ \text{\uncertain{si}}\ 1\leq i<j\leq s
\]

\[
  \xi=(2,2,\ldots,2)\qquad \mathfrak{S}=\emptyset
\]

\[
  R=\lbrace\pm2\varepsilon_i\ (1\leq i\leq r),\ \pm\varepsilon_i\pm
  \varepsilon_j\ (1\leq i<j\leq r)\rbrace
\]

\[
  \struck{\xi_i\ \ldots}\ \ (\xi_i)_{1\leq i\leq r}\qquad
  \xi_i\in\tfrac24\lbrace-1,0,+1\rbrace=\lbrace\ \rbrace
\]

\[
  0\leq\xi_1\leq\cdots\leq\xi_r\quad\text{i.e.}\quad
  \xi_1\in\lbrace0,\tfrac12\rbrace\qquad
  \pm\xi_i\pm\xi_j\in\lbrace-1,0,+1\rbrace
\]

\[
  \struck{\ill{}}\ \bigl(\tfrac12,\tfrac12,\ldots,\tfrac12\bigr)\qquad
  \struck{r\geq3}\ \ r\geq3
\]

\[
  \mathfrak{S}=\struck{\ill{}}\ \struck{\varepsilon_1,\varepsilon_2}\
  \ill{}\,\lbrace\pm\tfrac12\rbrace^{r}\qquad
  \mathfrak{S}_0=\emptyset
\]

\[
  \struck{\ill{}}\ \xi_1\ \ldots\ \xi_r\qquad
  \xi_1\leq\xi_2\leq\cdots\leq\xi_r\qquad
  \pm\xi_i\pm\xi_j\in\lbrace-1,0,1\rbrace
\]

\[
  2\alpha+1=0\quad\alpha=-\tfrac12\quad\alpha+1=\tfrac12\qquad
  2\alpha+1=1,\ \alpha=0\qquad 2\alpha+1=-1,\ \alpha=-1
\]

\[
  \left\lbrace
  \begin{array}{l}
    -\frac12,\ \underbrace{\frac12,\frac12,\ldots,\frac12}_{r-1}\\[4pt]
    (\underbrace{0,\ldots,0}_{r-1},1)\ \ \struck{\ill{}}\
      \boxed{r_2=1}\ \longrightarrow r\\[4pt]
    (-1,\underbrace{0,\ldots,0}_{r-1})\ \ \struck{\ill{}}\
      \text{\uncertain{conjugués}}\ (r_2=1)\\[4pt]
    (\frac12,\frac12,\ldots,\frac12)
  \end{array}\right.
\]

\[
  \bigl(\tfrac12\ \struck{\ill{}}\bigr)\qquad \struck{\ill{}}\qquad
  \prod(\ill{})=1\qquad \prod(\ill{})=-1\qquad
  \struck{\ill{}\,\lbrace\pm\varepsilon_i\rbrace}
\]

\[
  \mathfrak{S}'_0=\bigl\lbrace x\in E\ \big|\ w\in W,\
  w(\struck{\ill{}}\,\mathbb{R}x)\neq\mathbb{R}x\Longrightarrow
  wx-x\in R\bigr\rbrace
\]

\[
  x\in\mathfrak{S}'_0,\ r\in R\qquad
  s_rx-x=2\,\frac{(x,r)}{(r,r)}\,r
\]

\[
  s_r(\struck{\ill{}}\,\mathbb{R}x)=(\mathbb{R}x)\iff
  s_rx=x\ \text{ou}\ s_rx=-x\iff(x,r)=0\ \text{ou}\ x\in\mathbb{R}r
\]

\[
  (x,r)\neq0\ \text{et}\ x\notin\mathbb{R}r\Longrightarrow
  2\underbrace{\frac{(x,r)}{(r,r)}}_{\uncertain{n_{x,r}}}=\pm1
\]

\[
  \mathfrak{S}'_0\subset\mathfrak{S}'=\Bigl\lbrace x\in E\ \Big|\
  \forall r\in R\cap\complement\,\mathbb{R}x,\ \text{on a}\
  (x,r)\in\frac{(r,r)}{2}\lbrace-1,0,+1\rbrace\Bigr\rbrace
  \qquad
  \begin{array}{ccc}
    \mathfrak{S}'_0&\subset&\mathfrak{S}'\\
    \cup&&\cup\\
    \mathfrak{S}_0&\subset&\mathfrak{S}
  \end{array}
\]

\[
  \mathfrak{S}'=\mathfrak{S}\cup\Bigl\lbrace\lambda s\ \Big|\ s\in R,\
  \lambda\in\mathbb{R}^{*+},\ (\lambda s,r)\in\frac{(r,r)}{2}
  \lbrace-1,0,1\rbrace\ \ \forall r\in R-\lbrace s,-s\rbrace\Bigr\rbrace
\]

\[
  \text{i.e.}\ \ \forall r\in R-\lbrace s,-s\rbrace\ \struck{\ill{}}\
  (R\cap s^{\perp})\qquad
  \struck{\text{i.e.}\ \lambda\in\frac{(r,r)}{2(s,r)}}\qquad
  \lambda\in\frac{(r,r)}{2(s,r)}\lbrace-1,0,1\rbrace
\]

\[
  \text{i.e.}\ \ \lambda=\pm\frac{(r,r)}{2(s,r)}\qquad
  \text{i.e.}\ \ \frac1\lambda=\pm\frac{2(s,r)}{(r,r)}=\pm n(s,r)
\]

\[
  \mathfrak{S}'_0=
  \begin{cases}
    \mathfrak{S}_0 & \text{si toutes les racines ne sont pas de même
      longueur}\\
    \mathfrak{S}_0\cup R & \text{si toutes les racines de même long.}
  \end{cases}
\]

\[
  \text{\uncertain{rel.\ donnés}}\quad
  \struck{\underbrace{-\tfrac{r_2}{r+1},\ \ldots,\ -\tfrac{r_2}{r+1}}}
  \quad
  \left\lbrace
  \begin{array}{l}
    -\frac{r_2}{r+1}(e_1+\cdots+e_{r_1})+\frac{r_1}{r+1}
      (e_{r_1+1}+\cdots+e_{r_1+r_2})\\
    \quad 1\leq r_2\leq r-1\\
    e_1-e_2
  \end{array}\right.
\]

\[
  \mathfrak{S}'_0=
  \begin{cases}
    \mathfrak{S}_0 & \text{si}\ r\neq3\ \text{i.e.}\ r+1\neq4\\
    \struck{\ill{}}\ \mathfrak{S}_0\cup W.\bigl(-\tfrac12,-\tfrac12,
      \tfrac12,\tfrac12\bigr)
  \end{cases}
\]

\[
  G\qquad N_1\ N_2\ N_3
\]

\[
  N_1\cap N_2=\lbrace e\rbrace\qquad
  \forall x_1\in G,\ x_2\in G\quad \exists x\in G,\quad
  x\in x_1N_1\cap x_2N_2\quad
  \text{i.e.}\ \begin{cases}x_1^{-1}x\in N_1\\ x_2^{-1}x\in N_2\end{cases}
\]

\[
  N_1\times N_2\to G\qquad(x_1,x_2)\mapsto x_1x_2\qquad
  \struck{N_1N}\ N_1.N_2=G\iff N_2N_1=G
\]

\[
  x_1x_2=y_1y_2\qquad(y_1^{-1}x_1)=(y_2x_2^{-1})\in N_1\cap N_2=\lbrace
  e\rbrace\qquad x_1=y_1,\ x_2=y_2
\]

\[
  \struck{x_1\in G}\qquad a_1b_2=x_1\quad d_2c_1=x_2\qquad a_1\ \ a
\]

\[
  N_1\times N_2\simeq G\qquad
  \begin{array}{l}N_1\simeq G/N_2\\ N_2\simeq G/N_1\end{array}
\]

\[
  \struck{(x_1,x_2)\mapsto x_1x_2}\qquad
  \begin{array}{l}
    x=a_1a_2\ \longrightarrow\ a_1\in N_1\simeq G/N_2\qquad a_1=x_1\\
    \phantom{x}=b_2b_1\ \longrightarrow\ \struck{\ill{}}\ \
      N_2\simeq G/N_1
  \end{array}
\]

\[
  x=x_1a_2\qquad x=x_2b_1\qquad x_1a_2=x_2b_1\qquad
  x_2^{-1}x_1=b_1a_2^{-1}=b_1a'_2
\]

\[
  \begin{cases}
    N_2\cap N_3=N_3\cap N_1=N_1\cap N_2=\lbrace e\rbrace\\
    N_2N_3=N_3.N_1=N_1.N_2=G
  \end{cases}
  \qquad
  G\simeq N_2\rtimes N_1\simeq N_3\rtimes N_1
\]

\[
N_2 \times N_3 \hookleftarrow N_1
\qquad\qquad
G \times G = \Gamma \hookleftarrow G
\]

\[
G \times e = N_2 \qquad e \times G = N_3 \qquad \delta G = N_1
\]

\[
(g, h)(k, k)(g, h)^{-1} = (g k g^{-1}, h k h^{-1})
\]

\[
\begin{array}{ccc}
G & & \\
\big\uparrow{\scriptstyle \mathrm{pr}_2} & & \\
G \times G & \xrightarrow{\ \mathrm{pr}_3\ } & G
\end{array}
\]

\[
G \times G/\delta G \simeq G \qquad \delta(g, h) = g h^{-1} \qquad (g, h) \mapsto g h^{-1}
\]

\[
(g, h) \longmapsto (g, h, g h^{-1}) \qquad (g, h
\]

\[
\hookrightarrow (g, h^{-1}, h g^{-1})
\]

\[
\begin{array}{ccccc}
P & \times & Q & \to & R \\
e & & e & & e
\end{array}
\qquad
\begin{array}{l}
Q \simeq R \\
P \simeq R
\end{array}
\]

\[
\left\lbrace
\begin{array}{l}
y \underset{p(\gamma)}{\sim} z \\
z \underset{q(\gamma)}{\rightsquigarrow} x
\end{array}
\right.
\iff
\begin{array}{l}
\exists\,(p(\gamma), y, z) \in \Gamma \\
(x, q(\gamma), z) \in \Gamma
\end{array}
\]

\[
\overset{?}{\Downarrow}
\]

\[
x \underset{r(\gamma)}{\sim} y \qquad (x, y, r(\gamma)) \in \Gamma
\]

\[
\begin{array}{ccc}
& \widetilde{P}_u/\sigma & \\
& \big\uparrow & \\
& \widetilde{\Gamma}/\sigma & \\
\swarrow & & \searrow \\
\widetilde{P}_w/\sigma & & \widetilde{P}_v/\sigma
\end{array}
\]

\[
a y z^{-1} = 1 \qquad x^{-1} b z = 1 \qquad x y^{-1} = 1
\]

\[
\left.
\begin{array}{l}
a y z = 1 \\
x b z = 1
\end{array}
\right\rbrace
\overset{?}{\Downarrow}
\qquad a y = x b \qquad y = a^{-1} x b
\]

\[
\begin{array}{ccc}
& G & \\
& \big\uparrow & \\
& \Psi(G) & \\
\swarrow & & \searrow \\
G & & G
\end{array}
\]

\[
\begin{array}{l}
a_0, \bar r, \bar\pi, \bar r' \\
b_0, \tilde r', \tilde r'', \tilde\sigma \\
r, r', r'', \pi, \sigma, \rho
\end{array}
\]

\[
\begin{array}{ll}
\alpha_1 = a_0 & \alpha_2 = \bar r \\
\beta_2 = \tilde r' & \beta_1 = \tilde\sigma \\
\gamma_2 = r' & \gamma_1 = \sigma
\end{array}
\]

\[
\begin{array}{l}
\underline{a_0}, \bar r, \bar\pi, \bar r' \\
\underline{b_0}, \underline{\tilde r'}, \tilde r'', \tilde\sigma \\
\underline{r'}, \underline{\pi}, \underline{\sigma}, \underline{\rho}
\end{array}
\qquad
\left\lbrace
\begin{array}{c|c|c|c|c|c|c|c}
\bar\pi & \bar r' & a_0 & a_0 & \bar r & \bar r' & \bar\pi & \bar r \\
b_0 & b_0 & \tilde r' & \tilde\sigma & \tilde r' & \tilde r'' & \tilde r'' & \tilde\sigma \\
\pi & r' & r' & \sigma & \pi & \sigma & \rho & \rho
\end{array}
\right.
\]

\[
\forall\;
\begin{array}{l}
a_0, \bar r, \bar r' \in P \\
b_0, \tilde r'' \in Q
\end{array}
\quad
\exists\;
\begin{array}{l}
\bar\pi \in P \\
\tilde r', \tilde\sigma \in Q \\
r', \pi, \sigma, \rho \in R
\end{array}
\]

\[
\begin{array}{l}
\bar\alpha_0 \\
\Vert \\
\alpha_1\ \alpha_2\ \alpha_3\ \alpha_4 \\
\beta_1\ \beta_2\ \beta_3\ \beta_4 \\
\gamma_1\ \gamma_2\ \gamma_3\ \gamma_4
\end{array}
\qquad
\begin{array}{l}
\alpha_1\ \alpha_1\ \alpha_2\ \alpha_2\ \alpha_3\ \alpha_3\ \alpha_4\ \alpha_4 \\
\beta_1\ \beta_2\ \beta_1\ \beta_2 \\
\gamma_1\ \gamma_2\ \gamma_3\ \gamma_4
\end{array}
\]

\[
\left.
\begin{array}{llll}
x_i\ y_i & 1 \leq i \leq 7 & 14 & \\
x_{ih} & 1 \leq i < h \leq 7 & 21 & x_{ih} = -x_{hi} \\
y_{ih} & 1 \leq i < h \leq 7 & 21 & y_{ih} = -y_{hi}
\end{array}
\right\rbrace
\ 56 \text{ indéterminées}
\]

\[
\sum_{i,j,h,k=1}^{7} x_i y_j x_{ih} y_{jk}
+ \frac{1}{4} \sum_{\lambda,\mu,\nu,\rho=1}^{7} x_{\lambda\mu} x_{\nu\rho} y_{\mu\nu} y_{\lambda\rho}
+ \sum \bigl( x_i y_{\lambda\mu} y_{\nu\rho} y_{\sigma\tau}
+ y_i x_{\lambda\mu} x_{\nu\rho} x_{\sigma\tau} \bigr)
\]

\[
\sum (x_i\, dy_i - y_i\, dx_i) + \sum_{i,h=1}^{7} (x_{ih}\, dy_{ih} - y_{ih}\, dx_{ih})
\]

\[
\lbrace G, P_1, P_2, P_3, t \rbrace \longrightarrow (P_1, P_2, P_3, \Gamma)
\]

\[
P_1 \times P_2 \xrightarrow{\ \pi_3\ } P_3
\]

\[
\left\lbrace
\begin{array}{lll}
\forall x_1 \in P_1, & x_2 \mapsto \pi_3(x_1, x_2) & P_2 \xrightarrow{\sim} P_3 \\
\forall x_2 \in P_2, & x_1 \mapsto \pi_3(x_1, x_2) & P_1 \xrightarrow{\sim} P_3
\end{array}
\right.
\]

\[
P_1 \xrightarrow{\ h_{23}\ } \operatorname{Isom}(P_2, P_3)
\]

\[
P_1 \xrightarrow{\ h_{32}\ } \operatorname{Isom}(P_3, P_2)
\]

\[
\operatorname{Aut}(G) \cdot (G \times G \times G) \longrightarrow
\mathfrak{S}_G \times \mathfrak{S}_G \times \mathfrak{S}_G
\]

\[
(u, \alpha, \beta, \gamma) \longmapsto
(\tau_\beta \tau'_\gamma u,\ \tau_\gamma \tau'_\alpha u,\ \tau_\alpha \tau'_\beta u)
\]

\[
\bigl(\underbrace{u \operatorname{int}(g) u^{-1}}_{\operatorname{int} u(g)},\
\alpha u(g)^{-1} (u \operatorname{int}(g))(u^{-1}(\alpha^{-1})), \dots \bigr)
\]

\[
\alpha u(g^{-1})\, \underbrace{u(g u^{-1}(\alpha^{-1}) g^{-1})}_{\alpha\, \alpha^{-1} u(g)^{-1}}
\]

\[
(\operatorname{Int} G) \cdot (G \times G \times G) \big/ \lbrace \operatorname{int}(g), g^{-1}, g^{-1}, g^{-1} \rbrace \simeq \Gamma(G)
\]

\[
\operatorname{Aut} G \cdot (G \times G \times G) \big/ \lbrace \quad \rbrace \simeq \hat\Gamma(G)
\]

\[
\Gamma_0 = \lbrace (x, y, z) \in G \times G \times G \mid xyz = 1 \rbrace
\]

\[
\begin{array}{l}
x = z^{-1} y^{-1} \\
y = x^{-1} z^{-1} \\
z = y^{-1} x^{-1}
\end{array}
\]

\[
\Gamma = \lbrace (g_2 g_3^{-1}, g_3 g_1^{-1}, g_1 g_2^{-1}) \in G \times G \times G \mid g_1, g_2, g_3 \in G \rbrace
\]

\[
\gamma' = \sigma(\alpha_1, \alpha_2, \alpha_3)(\gamma)
= (\alpha_2 a_1 \alpha_3^{-1},\ \alpha_3 a_2 \alpha_1^{-1},\ \alpha_1 a_3 \alpha_2^{-1})
\]

\[
\left\lbrace
\begin{array}{l}
\tilde a'_1 = \operatorname{int}(\alpha_3)\, \tilde a_1\, \operatorname{int}(\alpha_2)^{-1} \\
\tilde a'_2 = \operatorname{int}(\alpha_1)\, \tilde a_2\, \operatorname{int}(\alpha_3)^{-1} \\
\tilde a'_3 = \operatorname{int}(\alpha_2)\, \tilde a_3\, \operatorname{int}(\alpha_1)^{-1}
\end{array}
\right.
\]

\[
\begin{array}{ll}
x_2 \mapsto -x_2 - a_1 & \\
x_3 \mapsto -x_3 - a_2 & = x_2 + a_1 - a_2 \\
x_1 \mapsto -x_1 - a_3 & = -x_2 - a_1 + a_2 - a_3
\end{array}
\qquad
\left.
\begin{array}{l}
a_1 + x_2 + x_3 = 0 \\
x_1 + a_2 + x_3 = 0 \\
x_1 + x_2 + a_3 = 0
\end{array}
\right| \Downarrow
\]

\[
\Delta \times \Delta \longrightarrow \Delta \qquad x \mapsto -x - y
\qquad
\begin{array}{l}
x + y + z = 0 \\
-x - y - z = 0
\end{array}
\]

\[
\begin{array}{c|cccc}
& 0 & 1 & 2 & 3 \\
\hline
u_0 & 0 & 1 & 2 & 3 \\
u_1 & 1 & 2 & 3 & 0 \\
u_2 & 2 & 3 & 0 & 1 \\
u_3 & 3 & 0 & 1 & 2
\end{array}
\]

\[
u_1 \qquad
\begin{array}{c|cccc}
& 0 & 1 & 2 & 3 \\
\hline
0 & 0 & 1 & 2 & 3 \\
1 & 1 & 0 & 3 & 2 \\
1 & 2 & 3 & 1 & 0 \\
2 & 3 & 2 & 0 & 1
\end{array}
\]

\[
\left\lbrace
\begin{array}{l}
x \ast y = y \ast x \\
0 \ast y = y \\
x \ast 0 = x \\
\tau_x \text{ bijectif}
\end{array}
\right.
\]

\[
(a_1, a_2, a_3) \in \Gamma
\qquad\qquad
x_1 \quad x_2 \quad x_3
\]

\[
\begin{array}{ll}
o_1 : & x_2 \in P_2 \xrightarrow[\sim]{\ \tilde o_1\ } P_3 \\
& \qquad o_2 \qquad\ o_3 \\
& x_1 \in P_1 \xrightarrow[\sim]{\ \tilde o_2\ } P_3 \\
& \qquad o_1 \qquad\ o_2
\end{array}
\qquad
\tilde o_1(x_1) = \xi_1
\]

\[
\begin{array}{l}
(o_1, o_2, o_3) \in \Gamma \\
(x_1, o_2, u_3) \in \Gamma \\
(o_1, x_2, v_3) \in \Gamma \\
\qquad w_3 \in P_3 \\
(x_1, x_2, y_3) \in \Gamma \\
(z_1, o_2, y_3) \in \Gamma \\
(z_1, w_3, p_3) \in \Gamma \\
(\ \cdot
\end{array}
\]

\[
P_3 \times P_3 \longrightarrow P_3 \qquad
P_3 \times P_3 \times \Gamma \qquad
P_3 \times \Gamma
\]

\[
P_1 \underset{G_3}{\wedge} P_2 \underset{G_1}{\wedge} P_3 \simeq \mathbb{1}_{G_2}
\qquad
P_2 \wedge P_3 \wedge P_1 \simeq
\]

\[
\begin{array}{c|ccccc}
& 0 & 1 & 2 & 3 & 4 \\
\hline
0 & 0 & 1 & 2 & 3 & 4 \\
1 & 1 & 2 & 3 & 4 & 0 \\
2 & 2 & 3 & 4 & 0 & 1 \\
3 & 3 & 4 & 0 & 1 & 2 \\
4 & 4 & 0 & 1 & 2 & 3
\end{array}
\]

\[
\begin{array}{lll}
P_1 & \text{bitorseur sous} & G_2, G_3 \\
P_2 & \text{———} & G_3, G_1 \\
P_3 & \text{———} & G_1, G_2
\end{array}
\qquad
\begin{array}{lll}
\mathring{P}_1 & \text{bitorseur sous} & G_3, G_2 \\
\mathring{P}_2 & \text{———} & G_1, G_3 \\
\mathring{P}_3 & \text{———} & G_2, G_1
\end{array}
\]

\[
\begin{array}{lcl}
P_1 \wedge P_2 \wedge P_3 \overset{\varphi_2}{\simeq} \underset{G_2\ G_2}{\mathbb{1}}
& & \mathring{P}_3 \wedge \mathring{P}_2 \wedge \mathring{P}_1 \overset{\mathring\varphi_2}{\simeq} \underset{G_2\ G_2}{\mathbb{1}} \\
P_2 \wedge P_3 \wedge P_1 \overset{\varphi_3}{\simeq} \underset{G_3\ G_3}{\mathbb{1}}
& \Updownarrow & \mathring{P}_1 \wedge \mathring{P}_2 \wedge \mathring{P}_3 \overset{\mathring\varphi_3}{\simeq} \underset{G_3\ G_3}{\mathbb{1}} \\
P_3 \wedge P_1 \wedge P_2 \overset{\varphi_1}{\simeq} \underset{G_1\ G_1}{\mathbb{1}}
& & \mathring{P}_2 \wedge \mathring{P}_1 \wedge \mathring{P}_3 \overset{\mathring\varphi_1}{\simeq} \underset{G_1\ G_1}{\mathbb{1}}
\end{array}
\]

\[
\Gamma \subset P_1 \times P_2 \times P_3 \quad \text{tel que } \Gamma \neq \emptyset
\]

\[
\left.
\begin{array}{l}
x = (x_1, x_2, x_3) \in P_1 \times P_2 \times P_3 \\
g_1 \in G_1,\ g_2 \in G_2,\ g_3 \in G_3
\end{array}
\right.
\Longrightarrow
\left(
\begin{array}{l}
(x_1, x_2, x_3) \in \Gamma \\
\iff (g_2 x_1 g_3^{-1}, g_3 x_2 g_1^{-1}, g_1 x_3 g_2^{-1}) \in \Gamma
\end{array}
\right)
\]

\[
\sigma(g_1, g_2, g_3) \cdot (x_1, x_2, x_3) = (g_2 x_1 g_3^{-1}, g_3 x_2 g_1^{-1}, g_1 x_3 g_2^{-1})
\]

\[
\Gamma \xrightarrow{\ p_i\ } P_i \quad \text{induit par } \mathrm{pr}_i
\]

\[
\begin{array}{ll}
G_2 \xrightarrow{\ \tilde a_1\ } G_3 & g_2 \text{ et } g_3 \text{ se correspondent ssi } g_2 a_1 = a_1 g_3 \\
G_3 \xrightarrow{\ \tilde a_2\ } G_1 & g_3 \text{ et } g_1 \ \text{---} \qquad g_3 a_2 = a_2 g_1 \\
G_1 \xleftrightarrow{\ \tilde a_3\ } G_2 & g_1 \text{ et } g_2 \ \text{---} \qquad g_1 a_3 = a_3 g_2
\end{array}
\]

\[
\tilde x, \tilde y \text{ liés ssi } (x \neq y)
\left|
\begin{array}{l}
\text{ou bien } \exists i \in I \text{ avec } x, y \in \tilde P_i,\ y = \sigma x \text{ i.e. } \pi_i(x) = \pi_i(y) \\
\text{ou bien } x \in \tilde P_i,\ y \in \tilde P_j \text{ avec } i \neq j, \text{ et } \exists \gamma \in \tilde\Gamma \text{ avec} \\
\qquad x = \tilde p_i(\gamma),\ y = \tilde p_j(\gamma)
\end{array}
\right.
\]

\[
\begin{array}{cc}
\tilde P_i & \tilde P_j \\
\downarrow & \downarrow \\
P_i & P_j
\end{array}
\]

\[
\Gamma(P_j) = \operatorname{Ker}\bigl(\mathbb{F}_2^{P_j} \xrightarrow[\text{somme}]{} \mathbb{F}_2\bigr)
\text{ de } \mathbb{F}_2^{P_j}
\]

\[
\begin{array}{ccc}
& P_3 & \\
& \big\uparrow & \\
& \Gamma & \\
\swarrow & & \searrow \\
P_1 & & P_2
\end{array}
\qquad
\begin{array}{ccc}
& P_1 \times P_2 & \\
\swarrow & & \searrow \\
P_1 & & P_2 \\
a_1 & & a_2
\end{array}
\qquad
(x_1, x_2)
\begin{array}{l}
\xrightarrow{\ \sim\ } x_1 \\
\xrightarrow{\ \sim\ } x_2
\end{array}
\]

\[
\begin{array}{ccccc}
& (x_1, x_2) & \xrightarrow{\ \sim\ } & (a_1, x_2) & \\
\swarrow{\scriptstyle \sim} & \big\downarrow{\scriptstyle \wr} & \swarrow & \big\downarrow{\scriptstyle \wr} & \\
x_1 & (x_1, a_2) & \xrightarrow{\ \sim\ } & (a_1, a_2) & \\
\Vert \ \swarrow{\scriptstyle \sim} & & & & \\
x_1 & & & &
\end{array}
\]

\[
\begin{array}{ccc}
& X \times Y & \\
\swarrow & & \searrow \\
X & & Y \\
a & & b
\end{array}
\qquad
x \xrightarrow[\sim]{\ \alpha_{x,y}\ } y
\qquad
\begin{array}{l}
x \simeq b \\
y \simeq a
\end{array}
\qquad
\begin{array}{l}
E_a \simeq E_b \\
E_y \simeq E_a
\end{array}
\qquad
E_a \simeq E_b \simeq E
\]

\[
\frac{\Gamma.16.2}{N}
\qquad\qquad
x
\qquad\qquad
\begin{array}{l}
E_x \simeq E \\
E_y \simeq E
\end{array}
\]

\[
X \times Y \xrightarrow{\ \pi\ } Z
\qquad
E_{x,y} \simeq E_{\pi(x,y)}
\qquad
xy = x'y' = \pi
\]

\[
\begin{array}{ccc}
E_{e,\pi} & \overset{\varphi_{\pi;x}}{\simeq} & E_{x, x^{-1}\pi} \\
\wr & & \wr \\
E & & E
\end{array}
\qquad \notin \quad x \neq e
\]

\[
\left.
\begin{array}{r}
n^2 - 2n + 1 \\
+ (n-1) n
\end{array}
\right.
= \underbrace{2n^2 - 3n + 1}
\qquad\Big|\qquad
\gamma \in \Gamma \qquad 21 / 42
\]

\[
\begin{array}{l}
E_{x_1} \\
\ \wr \\
E_\gamma
\end{array}
\]

\[
\begin{array}{ccc}
& P_3 & \\
& \big\uparrow & \\
& \Gamma \quad s & \\
\swarrow & & \searrow \\
P_1 & & P_2
\end{array}
\qquad
p_i(s) = a_i
\qquad
E = E_s \simeq E_{a_i}
\]

\[
\gamma \longmapsto (x_1, a_2)
\qquad
\begin{array}{c}
E_\gamma \simeq E_{x_1} \\
\wr \\
E_{a_2} \\
\big\uparrow \wr \\
E = E_s
\end{array}
\qquad
\begin{array}{l}
E_{x_1} \simeq E \\
E_{x_2} \simeq E \\
E_{x_3} \simeq E
\end{array}
\]

\[
E \xleftarrow{\ u\ } E_\gamma \xrightarrow{\ v\ } E
\qquad
E_\gamma \xrightarrow{\ w\ } E
\qquad\qquad
E_\gamma \to E \quad \ill{}
\]

\[
\begin{array}{c}
E_{x_2} \\
\nwarrow{\scriptstyle \sim} \\
E_\gamma \simeq E_{x_2} \\
\wr \\
E_{a_1} \\
\wr \\
E
\end{array}
\qquad
\gamma \mapsto a_1, x_2, x_3
\]

\[
\begin{array}{l}
\tilde\Gamma \\
\big\downarrow \\
\Gamma \subset G \times G \times G
\end{array}
\qquad
g_1 + g_2 + g_3 = 0
\qquad
p_i^{-1}(a_i)
\]

\[
\Gamma = \underset{1}{\lbrace s \rbrace} \cup \underset{n^2 - 1}{\Gamma_1} \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma^{*}
\qquad
n^2 - 3(n-1) - 1 = n^2 - 3n + 2
\]

\[
\Gamma_i = p_i^{-1}\lbrace a_i \rbrace - \lbrace s \rbrace
\qquad
\Gamma^{*} = \bigcap_i p_i^{-1}(P_i - \lbrace a_i \rbrace) = \Gamma - \bigcup \Gamma_i - \lbrace s \rbrace
\]

\[
\begin{array}{ll}
x \xrightarrow{\ \sim\ } y & \\
0 \to y - x & \\
0 \to a & \\
0 \to b & a \to a + b
\end{array}
\qquad
\boxed{3 \cdot (n-1) + 2(n^2 - 3n + 2)}
\]

\[
\begin{array}{l}
v u^{-1} = \xi \\
w u^{-1} = \eta
\end{array}
\qquad
\left\lbrace
\begin{array}{l}
u v^{-1} = \xi^{-1} \\
w v^{-1} = \eta \xi^{-1}
\end{array}
\right.
\]

\[
\begin{array}{ccc}
& P_1 & \\
& \big\uparrow & \\
& \Gamma & \\
\swarrow & & \nwarrow \\
P_2 & & P_3
\end{array}
\qquad
P_2 \times P_3 \text{ — } P_1
\qquad
\begin{array}{ccccc}
G & \times & G & \to & G \\
\big|{\scriptstyle \varphi} & & \big|{\scriptstyle \psi} & & \big|{\scriptstyle \chi} \\
e & \times & G & \to & G
\end{array}
\]

\[
\varphi(x) \psi(y) = \chi(xy)
\qquad
\varphi(1) = a,\ \psi(1) = b,\ \chi(1) = c
\qquad
ab = c
\]

\[
\begin{array}{ccc}
\beta\gamma^{-1} & \gamma\alpha^{-1} & \alpha\beta^{-1}
\end{array}
\qquad
\begin{array}{l}
abc = 1 \\
a = \beta\gamma^{-1} \quad b = \gamma\alpha^{-1} \\
abc = \beta\alpha^{-1} c
\end{array}
\]

\[
\begin{array}{c|c|c}
\tau_\beta^{-1} \varphi \tau_{\gamma^{-1}}' & \tau_{\gamma^{-1}} \psi \tau_\alpha' & \tau_{\alpha^{-1}} \chi \tau_\beta' \\
\Vert & \Vert & \Vert \\
\varphi' & \psi' & \chi'
\end{array}
\]

\[
\bigl(\beta^{-1} \varphi(x) \gamma\bigr) \bigl(\gamma^{-1} \psi(y) \alpha\bigr) \bigl(\alpha^{-1} \chi(z) \beta\bigr) = 1
\]

\[
(u \times u \times u)
\qquad
(\tau_\beta u \tau_\gamma'^{-1},\ \tau_\gamma u \tau_\alpha'^{-1},\ \tau_\alpha u \tau_\beta'^{-1})
\]

\[
\left\lbrace
\begin{array}{lll}
\alpha_1, \alpha_2, \alpha_3, \alpha_4 & \text{liés à} & \beta_4 \\
\beta_1, \beta_2, \beta_3, \beta_4 & \text{———} & \gamma_4 \\
\gamma_1, \gamma_2, \gamma_3, \gamma_4 & \text{———} & \alpha_4
\end{array}
\right.
\]

\[
12 \text{ arêtes}
\left\lbrace
\begin{array}{l}
\alpha_i \text{ lié à } \alpha'_i \\
\beta_i \text{ lié à } \beta'_i \\
\gamma_i \text{ lié à } \gamma'_i
\end{array}
\right.
\quad 1 \leq i \leq 4
\qquad\qquad
12
\left\lbrace
\begin{array}{lll}
\alpha'_i & \text{liés à} & \beta'_4 \\
\beta'_i & \text{———} & \gamma'_4 \\
\gamma'_i & \text{———} & \alpha'_4
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{lll}
\alpha_1 & \text{lié à} & \beta'_1, \beta_2, \beta_3, \beta_4 \\
\alpha_2 & \text{———} & \beta_1, \beta'_2, \beta_3, \beta_4 \\
\alpha_3 & \text{———} & \beta_1, \beta_2, \beta'_3, \beta_4 \\
\alpha_4 & \text{———} & \beta'_1, \beta'_2, \beta'_3, \beta_4
\end{array}
\right.
\qquad
\alpha_1 \text{ lié à } \beta_1, \beta'_2, \beta'_3, \beta'_4
\]

\[
\boxed{
\begin{array}{l}
\alpha_1 \text{ lié à } \beta'_1 \\
\alpha_2 \text{ lié à } \beta_2
\end{array}
}
\quad \longleftarrow \quad
\boxed{\alpha_1 \text{ lié à } \gamma'_4}
\]

\[
\begin{array}{l}
\alpha_1 \text{ lié à } \beta_4 \text{ et } \gamma'_4 \\
\beta_1 \text{ lié à } \gamma_4 \text{ et } \beta'_4 \\
\text{or} \ (\alpha_1 \text{ lié à } \beta_1) \Longrightarrow
\end{array}
\]

\[
\left\lbrace
\begin{array}{l}
\alpha_1, \beta'_1, \gamma'_4 \\
\alpha_1, \beta_2, \gamma'_3 \\
\alpha'_2, \beta'_1, \gamma'_3 \\
\alpha'_2, \beta_2, \gamma_4
\end{array}
\right.
\]

\[
P_1 \underset{G_3}{\wedge} (P_2 \underset{G_3}{\wedge} P_3)
\]

\[
\tilde A_1(g_2)\, \alpha_3 = \tilde a_1(g_2)\, \tilde a_1(\alpha_2)
\qquad
\tilde A_1(g_2) = \tilde a_1(g_2)\, \tilde a_1(\alpha_2)\, \alpha_3^{-1}
\]

\[
(g_1, g_2, g_3)(a_1, a_2, a_3) = (a_1, a_2, a_3)
\]

\[
\left.
\begin{array}{ll}
g_2 a_1 = a_1 g_3 = \cdots & \text{i.e. } g_2 a_1 g_3^{-1} = a_1 \\
g_3 a_2 = a_2 g_1 = b_2 & \text{i.e. } g_3 a_2 g_1^{-1} = a_2 \\
g_1 a_3 = a_3 g_2 &
\end{array}
\right.
\qquad \overset{?}{\Downarrow}
\qquad a_1, a_2,
\]

\[
\begin{array}{l}
\varphi(1) = \beta\gamma^{-1} \\
\varphi'(x) = \beta^{-1} \varphi(x) \gamma \\
\psi'(x) = \gamma^{-1} \psi(x) \alpha \\
\chi'(x) = \alpha^{-1} \chi(x) \beta
\end{array}
\qquad
\varphi'(x) \psi'(x) \chi'(x) = 0
\qquad
\varphi = \beta u(x) \gamma^{-1}
\]

\[
(u, \alpha, \beta, \gamma)
\begin{array}{l}
\nearrow \\
\longrightarrow \tau_\gamma \tau'_\alpha u \\
\searrow \tau_\alpha \tau'_\beta u
\end{array}
\qquad
\begin{array}{l}
\beta u(x) \gamma^{-1} = x \\
\beta = \gamma
\end{array}
\]

\[
(u, \alpha, \beta, \gamma)(u', \alpha', \beta', \gamma') = (u u',\ \alpha u(\alpha'),\ \beta u(\beta'),\ \gamma u(\gamma'))
\]

\[
\bigl((u, \alpha, \beta, \gamma)(u', \alpha', \beta', \gamma')\bigr)(u'', \alpha'', \beta'', \gamma'')
= (u u' u'',\ \alpha u(\alpha') u u'(\alpha''), \dots)
\]

\[
(u, \alpha, \beta, \gamma)\bigl(\underbrace{\quad\quad\quad}_{u' u'',\ \alpha' u'(\alpha'') \dots}\bigr)
\qquad
(u u' u'',\ \alpha u(\alpha' u'(\alpha'')))
\]

\[
S^2 \times S^2 - \text{diag} - \text{antidiag} = \mathbb{P}
\qquad
\begin{array}{c}
\mathbb{C}^{*} \\
\big\downarrow \\
\mathbb{P} \\
\big\downarrow \\
S^2
\end{array}
\qquad
\begin{array}{c}
\pi_1(S^1) = \mathbb{Z} \\
\big\downarrow{\scriptstyle 2} \\
\pi_1(\mathbb{C}^{*}) = \mathbb{Z} \\
\big\downarrow \\
\pi_1(\mathbb{P}) \\
\big\downarrow \\
\pi_1(S^2) = 0
\end{array}
\]

\[
X \times
\qquad
\begin{array}{l}
E \to L \\
E \to L' \\
F \xrightarrow{\ \sim\ } L'
\end{array}
\qquad
E \simeq L \oplus L'
\qquad
\boxed{p^2 = p \quad \operatorname{Tr} p = 1}
\]

\[
0 \to F \to E_p \to \mathcal{O}(1) \to 0
\qquad
F \otimes \mathcal{O}(1) \simeq \mathcal{O}
\qquad
F \simeq \mathcal{O}(-1)
\qquad
\mathcal{O}(-2)
\]