\[
      \mathbf{Z}/6\mathbf{Z} \wedge \omega_H \simeq
      (\mathbf{Z}/3\mathbf{Z} \wedge \omega_H) \times
      \underbrace{(\mathbf{Z}/2\mathbf{Z} \wedge \omega_H)}_{\mathbf{Z}/2\mathbf{Z}}
    \]

\[
      \begin{cases}
        \Delta \in \mathrm{Ens}_3 & \text{avec iso} \\
        \omega_\Delta \simeq \omega_H
      \end{cases}
    \]

\[
  S \simeq \Delta \times T
\]

\[
  S \simeq \coprod_{i \in T} S_i \qquad \text{avec}\quad
  \begin{array}{l} \operatorname{card} T = 2 \\
  \operatorname{card} S_i = 3 \quad \forall\, i \in T \end{array}
\]

\[
  S_i = \Delta \times \{i\},
\]

\[
  \bigwedge_{i \in T} \omega_{S_i} \simeq \mathbf{1}.
  \quad (\text{torseur } \uncertain{\text{neutre}} \text{ sous } \mathbf{Z}/2\mathbf{Z})
\]

\[
  S = \coprod_{i \in T} S_i
\]

\[
  (\mathrm{A}) \qquad \bigwedge_{i \in T} \omega_{S_i} \simeq \mathbf{1}.
\]

\[
  S_i \simeq S_j \qquad (j \in T,\ j \neq i)
\]

\[
  I = \{i, j, k\}
\]

\[
  \sigma_k = \sigma_i \sigma_j \sigma_i
\]

\[
  \sigma_i = \sigma_j \sigma_k \sigma_j = \sigma_j (\sigma_i \sigma_j \sigma_i) \sigma_j
  \qquad \text{i.e.} \qquad
  (\sigma_i \sigma_j)^3 = 1
\]

\[
  \struck{\sigma_i^2 = 1 \quad (i \in I)}
\]

\[
  G' \xrightarrow{\ \sim\ } \mathfrak{S}_I
\]

\[
  \sigma_i^2 = 1 \qquad \sigma_k = \sigma_i \sigma_j \sigma_i
  \qquad (\text{si } I = \{i, j, k\})
\]

\[
  \struck{\sigma_i^2 = \sigma_j^2 =} \qquad
  \sigma^2 = \sigma'^2 = (\sigma\sigma')^3 = 1 .
\]

\[
  \mathsf{T} = \struck{\prod_{i \in T}} \bigwedge_{i \in T} S_i
\]

\[
  S_i \wedge S_j \xrightarrow{\ \sim\ } S_j \wedge S_i
  \overset{\text{sym}}{\simeq} S_i \wedge S_j
  \qquad (x \wedge y \mapsto (u^{-1} y \wedge u x))
\]

\[
  x = \gamma a, \quad y = \gamma' b
\]

\[
  \sigma(\underbrace{\gamma a \wedge \gamma' b}_{\gamma\gamma'\, a \wedge b})
  = (\underbrace{u^{-1} \gamma' b}_{\gamma' a}) \wedge (\underbrace{u \gamma a}_{\gamma b})
  = \struck{\gamma\gamma'}\ \gamma' a \wedge \gamma b
  = \underset{}{\gamma'\gamma\, a \wedge b}
\]

\[
  \mathrm{Isom}(S_i, S_j) \qquad \text{où } j \in T,\ j \neq i,
\]

\[
  S' \longrightarrow T' \overset{\text{déf}}{=} \bigwedge_i \omega_{S_i}
\]

\[
  (1) \qquad V = \bigoplus_{i \in T} D_i \simeq \prod_{i \in T} D_i
\]

\[
  (2) \qquad V = \bigoplus_{i \in T'} D_i \simeq \prod_{i \in T'} D_i
\]

\[
  (3) \qquad
  \begin{cases}
    T' \simeq \bigwedge_{i \in T} D_i^{*} \\
    T \simeq \bigwedge_{i \in T'} D_i^{*}
  \end{cases}
\]

\[
  \prod_{i \in T} D_i^{*} \longrightarrow T'
\]

\[
  (4) \qquad (u_i)_{i \in T} \longmapsto \mathbf{F}_3 \cdot \Bigl(\sum_{i \in T} u_i\Bigr).
\]

\[
  (5) \qquad P \simeq \prod_{i \in T} P_i\, , \qquad P \simeq \prod_{i \in T'} P_i
\]

\[
  (6) \qquad P_i = P / D_{\bar\imath} = P \wedge_V (D_i \simeq V / D_{\bar\imath})
\]

\[
  (\ast) \qquad P \simeq \prod_{i \in T} P_i
\]

\[
  (\ast') \qquad P \simeq \prod_{i \in T'} P_i
\]

\[
  \omega_E \simeq D^{*}
\]

\[
  (P_i)_{i \in T}
\]

\[
  \omega_{P_i} \simeq D_i^{*} \qquad (i \in T)
\]

\[
  (8) \qquad
  \begin{array}{l}
    \text{Catégorie des ens. } E \text{ de} \\
    \text{cardinal } 6, \text{ munis d'une} \\
    \text{partition } (P_i)_{i \in T} \text{ du type } (3,3), \\
    \text{indexée par } T, \text{ et d'une} \\
    \struck{\text{famille}}\ \text{paire de bijections} \\
    \qquad D_i^{*} \simeq \omega_{P_i}
  \end{array}
  \quad \xrightarrow{\ \approx\ } \quad
  \begin{array}{l}
    \text{Catégorie analogue} \\
    \text{définie en termes} \\
    \text{de } T' \text{ et de} \\
    \quad (D_i^{*})_{i \in T'}
  \end{array}
\]

\[
  \mathbf{F}_3^{*} = \{\pm 1\} \quad ).
\]

\[
  V = \prod_{i \in T} D_i
\]

\[
  (9) \qquad
  \begin{array}{c} \text{Carrés combinatoires} \\ Q \end{array}
  \quad \overset{\approx}{\longleftrightarrow} \quad
  \begin{array}{l}
    \text{plans vectoriels } V \text{ sur } \mathbf{F}_3, \\
    \text{munis d'une paire} \\
    \text{de droites vectorielles} \\
    \quad (D_i)_{i \in T}
  \end{array}
\]

\[
  (10) \qquad
  \left\{
  \begin{array}{l}
    T \simeq \operatorname{diag} Q
    \quad \text{et} \quad D_i^{*} \simeq d_i \quad \text{pour } i \in T \\[4pt]
    S_Q = \bigcup_{i \in T} D_i^{*} = \coprod_{i \in T} D_i^{*} \\[4pt]
    A_Q = \prod_{i \in T} D_i^{*} \simeq V^{*} \smallsetminus \bigcup_{i \in T} D_i^{*}
    = \bigcup_{i \in T'} D_i^{*}
  \end{array}
  \right.
\]

\[
  \underset{\substack{\text{ens. d'arêtes} \\ \text{de } Q}}{A_Q}
  \simeq
  \underset{\substack{\text{produit} \\ \text{des deux} \\ \text{diagonales de } Q}}{\prod_{i \in T} D_i^{*}}
  \simeq V^{*} - \bigcup_{i \in T} D_i^{*} = \bigcup_{i \in T'} D_i^{*}
\]

\[
  (11) \qquad V(DQ) \xrightarrow[\ \sim\ ]{\ \alpha_Q\ } V(Q)
\]

\[
  (\uncertain{12}) \qquad
  \underset{\substack{\wr \\ DDQ}}{V(Q)} \xrightarrow{\ \alpha_{DQ}\ } V(DQ)
\]

\[
  (13) \qquad \alpha_Q\, \alpha_{DQ} = -\mathrm{id}_{V(Q)}
  = V(\underset{\substack{\text{antipodisme} \\ \text{de } Q}}{a_Q})
\]

\[
  \alpha_{DQ}\, \alpha_Q = -\mathrm{id}_{V(DQ)} = V(a_{DQ})
\]

\[
  (3) \qquad T_x = D_x^{*} \qquad (\Lambda^{*}\text{-torseur})
\]

\[
  (4) \qquad \omega_V = \det(V)^{*} \qquad (\text{un } \Lambda^{*}\text{-torseur})
\]

\[
  x, y \in P(V)^! \ \text{« \textbf{disjoints} »} \ \text{ssi} \ V \simeq D_x \oplus D_y
\]

\[
  \begin{array}{c}
    (u, v) \longmapsto u \wedge v \\
    D_x \times D_y \longrightarrow \det V
  \end{array}
\]

\[
  T_x \times T_y \longrightarrow \omega_V
\]

\[
  (5) \qquad
  \underset{\substack{\text{produit contracté} \\ \text{de } \Lambda^{*}\text{-torseurs}}}{T_x \wedge T_y}
  \xrightarrow{\ \sim\ } \omega_V
\]

\[
  (6) \qquad u \wedge v = - v \wedge u
\]

\[
  u \in T_x, \quad v \in T_y, \quad w \in T_z
\]

\[
  (7) \qquad u \wedge v = v \wedge w = w \wedge u
\]

\[
  (7') \qquad w \wedge v = v \wedge u = u \wedge w \qquad )
\]

\[
  (8) \qquad u + v + w = 0
\]

\[
      T_x \wedge T_y \simeq \omega \qquad \text{noté}\quad
      u \mathbin{\underset{\Lambda^{*}}{\wedge}} v \longmapsto u \wedge v .
    \]

\[
  u + v + w = 0 \iff u \wedge v = v \wedge w = w \wedge u
\]

\[
  (u, v) \longmapsto u + v \quad \text{sur } V^{*} \text{ pour } u, v
  \ \textbf{disjoints}.
\]

\[
  \begin{cases}
    V = V^{*} \amalg \{0\} \\
    D_x = T_x \cup \{0\} \subset V
  \end{cases}
\]

\[
  u \wedge v = v \wedge w = w \wedge u .
\]

\[
      u \wedge v = v \wedge w = w \wedge u \qquad (\ill{})
    \]

\[
  A \hookrightarrow \mathfrak{P}_2(S) \quad
  \text{structure ens. de parties à deux éléments de } S
  \ (\text{appelées \textbf{arêtes}})
\]