\[
  \begin{array}{l}
  x \preceq y \ \text{ssi} \\
  x \in F_\alpha,\ y \in F_\beta
  \end{array}
  \quad
  \left\lbrace
  \begin{array}{l}
  \text{ou bien } x = y \ (\text{donc } \alpha = \beta) \\
  \text{ou bien } \alpha < \beta \text{ et } \exists\, z_\alpha = x \in
  F_\alpha,\ z_{\alpha+1} \in F_{\alpha+1}, \dots, z_{\beta-1} \in
  F_{\beta-1}, \\
  \qquad z_\beta = y \in F_\beta \text{ tels que } \forall\, \alpha
  \leqslant i < \beta,\ z_i \text{ et } z_{i+1} \text{ soient incidents}
  \end{array}
  \right.
\]

\begin{align*}
  F_0 &= \text{ens.\ des él.\ minimaux de } F_* \\
  F_1 &= \text{\quad''\quad''\quad''\quad} F_* - F_0 \\
  &\ \ \vdots \\
  F_i &= \text{\quad''\quad''\quad''\quad} F_* - F_0 - \dots - F_{i-1}
  \qquad (i \leqslant n) \\
  I_\alpha &\subset F_\alpha \times F_{\alpha+1}, \quad I_\alpha = \lbrace
  (x,y) \mid x \in F_\alpha,\ y \in F_{\alpha+1},\ x \preceq y \rbrace
\end{align*}

\[
  \sigma_0(d) = (x_0', x_1, \dots, x_\nu)
\]

\[
  \sigma_\alpha(d) = (x_0, \dots, x_{\alpha-1}, x_\alpha', x_{\alpha+1},
  \dots)
\]

\[
  \sigma_n(d) = (x_0, \dots, x_{n-1}, x_n') \quad \text{caractérisé par }
  x_n' \neq x_n \quad (\text{etc.}\dots)
\]

\[
  \sigma_\alpha \sigma_\beta = \sigma_\beta \sigma_\alpha \quad \text{si }
  |\beta - \alpha| \neq 1 \ \text{i.e.\ } \alpha, \beta \text{ non
  consécutifs.}
\]

\[
  \mathcal{A}/G_n^{(\alpha)} \longrightarrow F_\alpha \qquad (0 \leqslant
  \alpha \leqslant n-1)
\]

\[
  \mathcal{A}/G_n^{(\delta)} \longrightarrow \mathcal{A}_\delta \quad
  (\text{drapeaux de type } \delta)
\]

\[
  1 \to \underline{\mathrm{Aut}}(X, \mathrm{id}_{\partial X}) \to
  \mathrm{Aut}(X) \to \mathrm{Aut}(\partial X) \to 1
\]

\[
  \widetilde{\mathrm{Aut}}(X) = \lbrace (u, \tilde{v}) \mid u \in
  \mathrm{Aut}(X),\ \tilde{v} \in \mathrm{Aut}(\widetilde{\partial X}),\ \ 
  (u \mid \partial X) \circ p = p \circ \tilde{v} \rbrace \subset
  \mathrm{Aut}(X) \times \mathrm{Aut}(\widetilde{\partial X})
\]

\[
  \begin{array}{ccccccccc}
    1 & \to & S\mathcal{A}(X) & \to & \mathcal{A}(X) & \to &
    \mathcal{A}(\partial X) & \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow & & \\
    1 & \to & S\widetilde{\mathcal{A}}(X) & \to &
    \widetilde{\mathcal{A}}(X) & \to & \mathcal{A}(\widetilde{\partial X})
    & \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow & & \\
    1 & \to & \mathbf{Z}^I & \Rightarrow & \mathbf{Z}^I & \to & 1 & \to &
    1
  \end{array}
  \qquad I = \pi_0(\partial X)
\]

\[
  \begin{array}{ccccccccc}
    1 & \to & ST(X) & \to & \mathcal{A}(X)/S\mathcal{A}^\circ(X) & \to &
    \mathcal{A}(\partial X) & \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow\!\wr & & \\
    1 & \to & \widetilde{ST}(X) & \to & \widetilde{\mathcal{A}}(X)/
    S\mathcal{A}^\circ(X) & \to & \mathcal{A}(\widetilde{\partial X}) &
    \to & 1 \\
    & & \uparrow & & \uparrow & & \uparrow & & \\
    1 & \to & \mathbf{Z}^I & \Rightarrow & \mathbf{Z}^I & \to & 1 & \to &
    1
  \end{array}
\]

\[
  X_0 \subset X_1 \subset \dots \subset X_n = X .
\]

\[
  D_n \longrightarrow D_{n-1} \longrightarrow D_{n-2} \longrightarrow
  \cdots \longrightarrow D_1 \longrightarrow D_0
\]

\begin{align*}
  \delta_{n-1} &= (D_n;\ \sigma_0, \dots, \sigma_{n-1}) \\
  \delta_{n-2} &= (D_n/\sigma_n;\ \sigma_0, \dots, \sigma_{n-2}) \\
  \delta_{n-3} &= (D_n/(\sigma_n, \sigma_{n-1});\ \sigma_0, \dots,
  \sigma_{n-3}) \\
  &\ \ \cdots \\
  \delta_{i-1} &= (D_n/(\sigma_n, \dots, \sigma_{i+1});\ \sigma_0, \dots,
  \sigma_{i-1})
\end{align*}

\[
  X_0 \subset X_1 \subset \dots \subset X_{n-1} \subset X_n = X
\]

\[
  \dot{Z}_1 = \coprod_\alpha \dot{Z}_1^\alpha \to X_0
\]

\[
  \begin{array}{ccccccc}
    x_{i+1} & & \dot{Z}_{i+1} \subset Z_{i+1} & & \dot{Z}_{i+2} \subset
    \overline{Z}_{i+2} & & \\
    & & \downarrow & & \downarrow & & \\
    X_{i-1} & \subset & X_i \subset X_{i+1} & & \subset & & X_{i+2}
  \end{array}
\]

\[
  \begin{array}{ccccc}
    Z_{i+1} \mid X_{i-1} & \subset & Z_{i+1} \mid X_i & \to &
    \overline{Z}_{i+2} \\
    \downarrow & & \downarrow & & \downarrow \\
    X_{i-1} & \subset & X_i & & X_{i+2}
  \end{array}
\]

\[
  \begin{array}{c}
    \overline{X}_{i+1} \\
    \mid \\
    X_{i-1}\ X_i
  \end{array}
  \qquad X_i
\]

\[
  D_{01\dots i} \to D_I \ \text{\emph{épi}} \quad \text{si } i = \sup I
  \quad \text{i.e.\ } i \in I \subset [0,i]
\]

\begin{align*}
  D_{01} &\to D_1 & D_{013} &\to D_{13} \\
  D_{012} &\to D_{12} & D_{0123} &\to D_{123} \\
  D_{12} &\to D_2 & D_{123} &\to D_{23} \\
  D_{02} &\to D_2 & D_{23} &\to D_3
\end{align*}

\[
  n_i = \sum_{j < i} n_j + 1
\]

\begin{align*}
  \text{type } 0 &: F_0 \\
  \text{type } 1 &: F_1 \\
  \text{type } (0,1) &: R_1
\end{align*}

\[
  \Phi(\mathcal{C}) \xrightarrow{\ \Phi(\varphi)\ } \mathrm{Top}
\]

\[
  (X_i \xrightarrow{f_i} X)_{i \in I} \longmapsto
  \coprod_i \mathrm{C\hat{o}ne}\, \varphi(X_i)
  \ \amalg_{\coprod \varphi(X_i)}\ \varphi(X)
\]

\[
  \Phi(\Phi(\mathcal{C})) \to \mathrm{Top}
\]

\[
  (\text{parties finies} \neq \emptyset \text{ de } \Phi)^\circ
  \xrightarrow{\ D\ } (\mathrm{Ens})
  \qquad \text{tel que } \mathrm{card}\, D(\lbrace i \rbrace) = 1 \quad
  \forall\, i \in \Phi
\]

\[
  D \simeq \coprod_{\sigma \in \mathfrak{P}_f^*(\Phi)} D(\sigma)
\]

\[
  \left\lbrace
  \begin{array}{l}
    \text{géométries de drapeaux} \\
    \text{(ens.\ ordonnés ainsi)}
  \end{array}
  \right.
  \Longleftrightarrow
  \begin{array}{l}
    \text{Couples } (\Phi, D) \text{ d'un ens.\ } \Phi \text{ et d'un
    foncteur} \\
    \mathfrak{P}_f^*(\Phi)^\circ \longrightarrow (\mathrm{Ens}) \\
    \text{tels que } \mathrm{card}\, D(\lbrace i \rbrace) = 1\ \forall\, i
    \in \Phi
  \end{array}
\]

\[
  d = \sup_{i \in I_d} i
\]

\[
  \left(
  \begin{array}{c}
    \text{Géom.\ de drapeaux} \\
    \text{strictes}
  \end{array}
  \right)
  \Longleftrightarrow
  \left(
  \begin{array}{c}
    \text{ensembles} \\
    \text{simpliciaux}
  \end{array}
  \right)
\]

\[
  \mathfrak{P}_f^*(I)^\circ \xrightarrow{\ D'\ } (\mathrm{Ens})
\]

\[
  D = \coprod_{\sigma \in \mathfrak{P}_f^*(I)} D'(\sigma)
\]

\[
  \Phi = \coprod_{i \in I} D'(\lbrace i \rbrace)
\]

\[
  \Phi \xrightarrow{\ \delta\ } I
\]

\[
  \delta \mid I_d : I_d \to I
\]

\[
  D' : \mathfrak{P}_f^*(I) \to (\mathrm{Ens}) \quad \text{par}
\]

\[
  D'(\sigma) = \coprod_{\substack{\sigma' \in \mathfrak{P}_f^*(\Phi) \\
  \text{t.q.\ } \delta \mid \sigma' : \sigma' \xrightarrow{\sim} \sigma}}
  D(\sigma')
\]

\[
  \left.
  \begin{array}{l}
    \text{Géom.\ de drapeaux } D \\
    + \text{« fonction dimension »} \\
    \delta : \Phi \longrightarrow I \\
    \quad \text{\scriptsize él.\ min.}
  \end{array}
  \right|
  \Longleftrightarrow
  \left|
  \begin{array}{l}
    \text{foncteur } D' : \mathfrak{P}_f^*(I)^\circ \to (\mathrm{Ens}) \\
    \text{\emph{NB} Posant } I_0 = \lbrace i \in I \mid D'(\lbrace i
    \rbrace) \neq \emptyset \rbrace \\
    \text{on a sur } I_0 \text{ une structure } \dots
  \end{array}
  \right.
\]

\[
  I = \mathbf{N}
\]

\begin{align*}
  D[n] &= \lbrace d \in D \mid \mathrm{card}\, I_d = n+1 \rbrace \\
  &= \coprod_{\sigma \in \mathfrak{P}_{n+1}(\Phi)} D(\sigma) \\
  & \Big( \simeq \text{aussi } \coprod_{\sigma \in
  \mathfrak{P}_{n+1}(I)} D'(\sigma) \Big)
\end{align*}

\[
  \forall\, n, m \in \mathbf{N} \text{ avec } n < m, \text{ et } d_m \in
  D[m], \text{ l'application}
\]

\[
  \mathrm{Hom}_\Delta(\Delta_n, \Delta_m) \to D_n, \qquad u \mapsto
  u^*(d_m)
\]

\[
  \widehat{\Delta}/D[\,]
\]

\[
  \begin{array}{l}
    \text{géométrie des drapeaux } D \\
    \text{avec syst.\ d'ordres totaux sur} \\
    \text{les simplexes de } \Phi \text{ (ens.\ des} \\
    \text{él.\ minimaux de } D)
  \end{array}
  \Longleftrightarrow
  \begin{array}{l}
    \text{ens.\ quasi-} \\
    \text{simpliciaux} \\
    \text{satisfaisant la} \\
    \text{condition dess.}
  \end{array}
\]

\[
  X = |D|
\]

\[
  X(d) \qquad d \in D
\]

\[
  \left\lbrace
  \begin{array}{l}
    |F| = \displaystyle\bigcup_{\substack{d \in D_* \\ \text{t.q.\ }
    \sup(I_d) = F}} X(d) \\[3ex]
    |F|^\circ = \displaystyle\bigcup_{\text{idem}} X(d)^\circ
  \end{array}
  \right.
\]

\[
  D_* \in \mathrm{Ob}\, \widehat{\Delta} \qquad D_* : \Delta^\circ
  \longrightarrow (\mathrm{Ens})
\]

\[
  \mathrm{Hom}_\Delta(\Delta_n, \Delta_m) \longrightarrow D_n, \qquad u
  \longmapsto u^*(d_m)
\]

\[
  \underline{D}_* \longrightarrow \Delta
\]

\[
  D_* \to \mathfrak{P}(|D_*|)
\]

\[
  |D_*| = \varinjlim_{d \in D_*} |d| = \varinjlim_{d \in D_*} |\Delta_m| =
  \varinjlim_{\Delta \in \Delta/D} |\Delta|
\]

\[
  D_0 \;\overset{\partial_0}{\underset{\partial_1}{\leftleftarrows}}\; D_1
  \;\overset{\partial_0,\ \partial_1,\ \partial_2}{\Lleftarrow}\; D_2
\]

\[
  x_0 \prec y_0 \;\overset{\text{déf}}{\Longleftrightarrow}\; \exists\, z_1
  \in D_1 \text{ t.q. } x_0 = \delta_0(z_1),\ y_0 = \delta_1(z_1)
\]

\[
  D_2 \xrightarrow{(\delta_2, \delta_0)} (D_1, \delta_1) \times_{D_0}
  (D_1, \delta_0)
\]

\[
  x_0 = \delta_0(z_1),\ y_0 = \delta_1(z_1),\ \struck{y_0} = \delta_0(z'_1),\
  z_0 = \delta_1(z'_1),
\]

\[
  \delta_0(z''_1) = \delta_0\delta_1(u_2) = \delta_0\delta_2 u_2 =
  \delta_0(z_1) = x_0, \quad \delta_1(z''_1) = \delta_1\delta_1(u_2) =
  \delta_1\delta_0(u_2) = \delta_1(z'_1) = z_0,
\]

\[
  x_0 \preccurlyeq y_0 \;\overset{\text{déf}}{\Longleftrightarrow}\;
  x_0 = y_0 \text{ ou } x_0 \prec y_0
\]

\[
  I_d = \lbrace x \in D_0 \mid x < d \rbrace
\]

\[
  D_n = \lbrace d \in D_* \mid \operatorname{card} I_d = n+1 \rbrace
\]

\[
  |D_*^\omega| \simeq |D_*^{\omega'}|
\]

\[
  |D_*| \simeq |D_*| \simeq |\mathfrak{X}|
\]

\[
  D(H) = X / N_H \qquad \text{(drapeaux de type } H\text{)}
\]

\[
  H \subset H' \Longrightarrow i^*_{H,H'} : D(H') \to D(H),
\]

\[
  \underline{\mathfrak{P}(\Delta_n)}^\circ \longrightarrow (\mathrm{Ens})
\]

\[
  D_* = \coprod_{H \in \mathfrak{P}(\Delta_n)} D(H)
\]

\[
  d \leqslant d' \;\overset{\text{déf}}{\Longleftrightarrow}\; H \subset
  H' \text{ et } d = i^*_{H,H'}(d').
\]

\[
  D_0 = \coprod_{0 \leqslant i \leqslant n}
  \underbrace{D(\lbrace i \rbrace)}_{X / (\sigma_0 \cdots \widehat{\sigma}_i
  \cdots \sigma_n)}
\]

\[
  D(\lbrace i, j \rbrace) \times_{D(\lbrace j \rbrace)} D(\lbrace j, k
  \rbrace)
\]

\[
  D(\lbrace i, j, k \rbrace) \xrightarrow{\text{épi}} D(\lbrace i, j
  \rbrace) \times_{D(\lbrace j \rbrace)} D(\lbrace j, k \rbrace).
\]

\[
  D(H \cup H') \longrightarrow D(H) \times_{D(H \cap
  H')} D(H')
\]

\[
  X / G_{K \cap K'} \xrightarrow{\;p\;} X/G_K \times_{X/G_{K \cup K'}}
  X/G_{K'}
\]

\[
  G_{K \cup K'} = G_K \cdot G_{K'}
\]

\[
  K \cup K' = (K \cup K') \cap \underbrace{\complement \lbrace i
  \rbrace}_{[0, i-1] \sqcup [i+1, n]} = \overbrace{(K \cup K') \cap [0,
  i-1]}^{K'_0} \sqcup \underbrace{(K \cup K') \cap [i+1, n]}_{K_0}
\]

\[
  G_{K \cup K'} \subset \underbrace{G_{K_0} \cdot G_{K'_0}}_{\text{ss-gr}}
  \subset G_K \cdot G_{K'} \quad \text{cqfd.}
\]

\[
  D(\Delta_n) \longrightarrow D(H) \times_{D(H \cap H')} D(H')
\]

\[
  \Theta(D) = \coprod_{\substack{A \subset \mathbf{N} \\ \text{et } A
  \text{ fini}}} D(A)
\]

\[
  \Theta_n(D) = \coprod_{\substack{A \subset \mathbf{N} \\
  \operatorname{card} A = n+1}} D(A) \qquad n \in \mathbf{N}
\]

\[
  \Theta(D) = \coprod_{n \in \mathbf{N}} \Theta_n(D)
\]

\[
  \underset{D(A)}{d} \;\leqslant\; \underset{D(A')}{d'}
  \;\overset{\text{déf}}{\Longleftrightarrow}\; A \subset A' \text{ et } d =
  i^*_{A,A'}(d')
\]

\[
  \Theta(D) \longrightarrow \mathfrak{P}_f(\mathbf{N})
\]

\[
  A \not\subset [0, N] \Longrightarrow D(A) = \emptyset.
\]

\[
  \sigma : \Delta_n \hookrightarrow \Delta_m
\]

\[
  \Delta_m \xrightarrow[\sim]{\;u_A\;} A
\]

\[
  \sigma^* : \lbrace D(A) \to D(B) \hookrightarrow \Theta_n(D) \qquad (B =
  u_A \sigma(\Delta_n))
\]

\[
  \sigma^* : \Theta_m(D) \longrightarrow \Theta_n(D)
\]

\[
  \underbrace{(\tfrac{1}{2}\mathrm{simpl}^*)^\circ}_{\substack{\text{ens.\
  finis tot}^{\text{t}} \\ \text{ordonnés, avec} \\ \text{appl.\ str.\
  cr.}}} \longrightarrow (\mathrm{Ens})
\]

\[
  (\tfrac{1}{2}\mathrm{simpl}^*) \xrightarrow{\;\varphi_N\;} \mathrm{Ens}
\]

\[
  \varphi_N(\Delta_n) = \mathrm{Mon.croiss}(\Delta_n, N)
\]

\[
  D : \underline{\mathrm{Dr}(N)}^\circ \longrightarrow \mathrm{Ens},
\]

\[
  X_{-1} = \emptyset \subset X_0 \subset X_1 \subset \cdots \subset
  X_{n-1} \subset X_n = X \qquad \text{« filtration } n\text{-admissible »}
\]

\[
  \emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_{i-1}
\]

\[
  \emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_{n-1}
\]

\[
  \partial\widetilde{X}_n \xrightarrow{\;q_n\;} X_{n-1}
\]

\[
  q_n^{-1}(X_i) - q_n^{-1}(X_{i-1}) \longrightarrow X_i - X_{i-1} \qquad (0
  \leqslant i \leqslant n-1)
\]

\[
  f : \underset{\substack{\| \\ X_n \\ \cup \\ \vdots}}{X} \longrightarrow
  \underset{\substack{\| \\ Y_n \\ \cup \\ \vdots}}{Y}
\]

\[
  E_1(X_1) = \partial\widetilde{X}_1 \simeq \pi_0(\partial\widetilde{X}_1)
  \longrightarrow \pi_0(\widetilde{X}_1) \simeq \operatorname{arcs}(X_1)
\]

\[
  E_n(X_n) = E^\circ_{n-1}(\partial\widetilde{X}_n) \quad (=
  E_{n-1}(\partial\widetilde{X}_n))
\]

\[
  \mathcal{F}_n^\circ \longrightarrow (G_n\text{-ens})
\]

\[
  \begin{array}{lll}
    R_n(X_n) \overset{\text{déf}}{=} E_n(X_n) & G_{n-1}\text{-ens} &
    \sigma_0 \cdots \sigma_{n-1} \\
    R_{n-1}(X_n) = E_{n-1}(X_{n-1}) & G_{n-2}\text{-ens} & \sigma_0 \cdots
    \sigma_{n-2} \\
    \cdots & & \\
    R_1(X_n) = E_1(X_1) & G_0\text{-ens} & \sigma_0 \\
    R_0(X_n) = E_0(X_0) \overset{\text{déf}}{=} X_0 & G_{-1}\text{-ens} &
    \text{i.e.\ ensemble}
  \end{array}
\]

\[
  R_n(X_n) \to R_{n-1}(X_n) \to \cdots \to R_0(X_n)
\]

\[
  \begin{array}{ccc}
    R_i(X_n) & \longrightarrow & R_{i-1}(X_n) \\
    \| & & \| \\
    E_i(X_i) & & E_{i-1}(X_{i-1}) \\
    \| & & \\
    E^\circ_{i-1}(\partial\widetilde{X}_i) = E_{i-1}(\partial\widetilde{X}_i)
    & &
  \end{array}
\]

\[
  \underset{\substack{| \\ \text{catégorie isotopique} \\ \text{déduite de }
  \mathcal{F}_n}}{\mathcal{F}_n} \xrightarrow{\;\varphi_n\;}
  (\mathrm{Casc})_n
\]

\[
  (G_n\text{-ens}) \xrightarrow{\;\psi_n\;} (\mathrm{Casc})_n
\]

\[
  \left\lbrace
  \begin{array}{ll}
    R_n(X_n) \simeq E_n^\circ(X_n) & G_{n-1}\text{-ens} \\
    R_{n-1}(X_n) \simeq E_n^\circ(X_n)/\sigma_n & G_{n-2}\text{-ens} \\
    R_{n-2}(X_n) \simeq E_n^\circ(X_n)/(\sigma_n, \sigma_{n-1}) &
    G_{n-3}\text{-ens} \\
    \vdots & \\
    R_1(X_n) \simeq E_n(X_n)/(\sigma_n, \ldots, \sigma_2) & \\
    R_0(X_n) \simeq E_n(X_n)/(\sigma_n, \ldots, \sigma_2, \sigma_1) &
  \end{array}
  \right.
\]

\[
  \overbrace{E_n \to \underbrace{E_{n-1} \to \cdots \to E_1 \to
  E_0}_{\substack{\text{définit } X_{n-1} \supset X_{n-2} \supset \cdots \\
  \text{par hyp.\ de réc.} \\ X_{n-1} = \psi_{n-1}(C_{n-1})}}}^{C_n}
\]

\[
  X_{n-1} = \psi_{n-1}(C_{n-1}), \qquad \partial\widetilde{X}_n =
  \psi_{n-1}(\rho_{n-1}(E_n))
\]

\[
  \rho_{n-1}(E_n) \longrightarrow C_{n-1}
\]

\[
  \psi_n(C_n) = \text{multicône de } \psi_{n-1}(\rho_{n-1}(E_n) \to C_{n-1})
\]

\[
  \widetilde{X}_i^\alpha \xrightarrow{\ \sim\ }
  \mathrm{C\hat{o}n}(\partial\widetilde{X}_i^\alpha)
\]

\[
  \boxed{\ \mathrm{card}\, G(\mathbf{F}_q) = q^N (q-1)^r
  \prod_{1 \leqslant \lambda \leqslant r}
  \frac{q^{d_\lambda} - 1}{q - 1}\ }
\]

\[
  P_{G/B} = \frac{(1 - T^2)^r}{\prod_\lambda (1 - T^{2d_\lambda})}
\]

\[
  Sl(3, \mathbf{F}_q) \quad \mathrm{card} \ \sum_i b\, q^{d(i)} \qquad
  b = q^3 (q-1)^2 \sum_{0 \leqslant i \leqslant 3} b_{2i}\, q^i
\]

\[
  \Bigl(1 : P_{G/B}(\sqrt{q})\Bigr) = \frac{(1-q)^r}{\prod_\lambda (1 -
  q^{d_\lambda})} = \frac{(q-1)^r}{\prod_\lambda (q^{d_\lambda} - 1)}
\]

\[
  \mathrm{card}\, G(\mathbf{F}_q) = q^N (q-1)^r\, \mathrm{card}\,
  G/B(\mathbf{F}_q)
\]

\[
  \mathrm{card}\, G/B(\mathbf{F}_q) = \sum_{\substack{\alpha \text{
  cellules} \\ (d(\alpha) = \text{dim})}} q^{d(\alpha)} = \sum_{0
  \leqslant i \leqslant N} b_{2i}\, q^i = P_{G/B}(\sqrt{q})
\]

\[
  B \supset T, \quad G/T \simeq G/B \qquad P_{B_G} \cdot P_{G/B} = P_{B_T}
\]

\[
  \begin{array}{c|cccc|l}
    q & 2 & 3 & 4 & 5 & \\
    \hline
    & 1 & 2 & 3 & 4 & Gl(1, \mathbf{F}_q) \\
    & 2 & 6 & 12 & 20 & \mathrm{Aff}(1, \mathbf{F}_q) \\
    & 6 & 24 & 60 & 120 & Sl(2, \mathbf{F}_q) \\
    & 6 & 24 & 60 & 120 & GP(1, \mathbf{F}_q) \\
    & 6 & 48 & 180 & 480 & Gl(2, \mathbf{F}_q) \\
    & 24 & 9\cdot 48 & 16 \cdot 180 & 25 \cdot 480 & \mathrm{Aff}(2,
    \mathbf{F}_q)
  \end{array}
\]

\begin{align*}
  Gl(1, \mathbf{F}_q) &\simeq \mathbf{F}_q^* \simeq \mathbf{Z}/(q-1),
  && \mathrm{card}\ q - 1 \\
  \mathrm{Aff}(1, \mathbf{F}_q) &\simeq \mathbf{F}_q^* \cdot \mathbf{F}_q
  && \mathrm{card}\ q(q-1) \\
  GP(1, \mathbf{F}_q) &\simeq Gl(2, \mathbf{F}_q)/\mathbf{F}_q^*
\end{align*}

\[
  \mathrm{card}\ Sl(2, \mathbf{F}_q) = \sum_{\text{cell.\ } i} b\,
  q^{d(i)} \qquad b = q(q-1) \qquad \mathrm{card}\ Sl(2, \mathbf{F}_q) =
  q(q-1)(q+1)
\]

\[
  \mathrm{card}\ GP(1, \mathbf{F}_q) = (q+1)q(q-1), \qquad \mathrm{card}\
  \mathrm{Aff}(2, \mathbf{F}_q) = q^3 (q-1)^{2} (q+1)
\]

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

\[
  \downarrow
\]

\begin{align*}
  E &= E(\mathbb{B}, \sigma) = \mathrm{Hom}_{\lbrace \pm 1
  \rbrace}(\mathbb{B}, \mathbf{R}) \\
  C &= C(\mathbb{B}, \sigma) = \lbrace \varphi \in E \mid |\varphi(j)|
  \leqslant 1 \ \forall j \in \mathbb{B} \rbrace
  \qquad \mathbb{B} \hookrightarrow C \hookrightarrow E
\end{align*}

\[
  F(I') = \lbrace \varphi \in C \mid \varphi(j) = 1 \ \forall j \in I',\
  \varphi(j) \neq \pm 1 \ \forall j \in \mathbb{B} - I' - \sigma I'
  \rbrace
\]

\[
  \mathrm{Spd}(\mathbb{B}/I) \xrightarrow{\ \sim\ } \bigl(\mathrm{Fac}(C)
  - \lbrace \emptyset \rbrace\bigr)
\]

\[
  \begin{array}{ccc}
    \mathbb{B} & \xrightarrow{\ \sim\ } & \mathbb{B}' \\
    \downarrow p & & \downarrow p' \\
    I & \xrightarrow{\ \sim\ } & I'
  \end{array}
\]

\[
  \mathbb{B} \hookrightarrow \mathcal{P}
\]

\[
  p(b) = p(b') \Longleftrightarrow (b = b' \text{ ou } C_b \cap C_{b'} =
  \emptyset),
\]

\[
  \mathrm{Isom}\bigl((\mathbb{B}, \sigma), (\mathbb{B}', \sigma')\bigr)
  \xrightarrow{\ \alpha\ } \mathrm{Isom}_{\mathrm{aff}}(C, C')
  \xrightarrow{\ \beta\ } \mathrm{Isom}_{\mathrm{ord}}\bigl(\mathrm{Comb}(C),
  \mathrm{Comb}(C')\bigr)
\]

\[
  \mathrm{Aut}\bigl((\mathbb{B}, \sigma)\bigr) \xrightarrow{\ \sim\ }
  \mathrm{Aut}_{\mathrm{aff}}(C) \xrightarrow{\ \sim\ }
  \mathrm{Aut}_{\mathrm{ord}}\bigl(\mathrm{Comb}(C)\bigr)
\]

\[
  \begin{array}{l}
    \quad \| \\
    \text{commutant de } \sigma \text{ dans } \mathfrak{S}_{\mathbb{B}} \\
    \quad \cdots \\
    \mathfrak{S}_I \cdot \lbrace \pm 1 \rbrace^I
  \end{array}
\]

\[
  2^n \cdot n \cdot (n-1) \cdots 2 \cdot 1 = 2^n n!
\]

\[
  \mathrm{Isom}(C_0, C) \xrightarrow{\ \sim\ } \mathrm{Drap}(C)
\]

\[
  \begin{array}{ccc}
    C & \longmapsto & \mathrm{Drap}(C) \\
    n\text{-Cubes} & \longrightarrow & G_0\text{-tors.\ à dr.}
  \end{array}
\]

\[
  G_0 \simeq \lbrace \pm 1 \rbrace^n \cdot \mathfrak{S}_n \simeq
  \mathrm{Aut}(C_n)
\]

\[
  c_d = \binom{n}{d} 2^{n-d}
\]

\begin{align*}
  \sum_0^n (-1)^d c_d &= \sum (-1)^d \binom{n}{d} 2^{n-d} = (-1)^n \sum
  \binom{n}{d} 1^d (-2)^{n-d} \\
  &= (-1)^n (1-2)^n = 1 \qquad \text{i.e.}
\end{align*}

\[
  \boxed{\ \sum_0^n (-1)^d c_d = 1\ }
\]

\[
  \omega_C \overset{\text{déf}}{=} \omega_E \simeq \omega_I \wedge
  \bigwedge_{i \in I} \mathbb{B}_i
\]

\[
  \mathrm{Aut}(C) \xrightarrow{\ \alpha\ }
  \mathrm{Aut}_{\mathrm{ord}}(\mathrm{Fac}(C)) \xrightarrow{\ \beta\ }
  \mathrm{Aut}_{\mathrm{ens}}\, \mathrm{Rep}(C)
\]

\[
  \mathrm{card}\, \mathrm{Rep}(C) = d_0 d_1 \cdots d_{n-1} = d_0 \cdots
  d_{n-1}
\]

\begin{align*}
  N &= \mathrm{card}\, \mathrm{Aut}(C) \\
  N_c &= \mathrm{card}\, \mathrm{Aut}(\mathrm{Fac}(C)) \\
  R &= \mathrm{card}\, \mathrm{Rep}(C),
\end{align*}

\[
  N \mid N_c \mid R
\]

\[
  N = N_c = d_0 \cdots d_{n-1}
\]

\[
  \mathrm{Isom}_{\mathrm{ord}}\bigl(\Phi(C), \Phi(C')\bigr) \longrightarrow
  \mathrm{Isom}\bigl(\omega(C), \omega(C')\bigr)
\]

\[
  \mathrm{Aut}_{\mathrm{ord}}\bigl(\Phi(C)\bigr) \longrightarrow \lbrace
  \pm 1 \rbrace = \mathrm{Aut}_{\mathrm{ens}}\bigl(\omega(C)\bigr) \qquad
  (\text{hom.\ de groupes})
\]

\[
  I = \coprod_{\alpha} I_\alpha
\]

\[
  \mathbb{B} \simeq \varepsilon \times I, \qquad u =
  \mathrm{id}_\varepsilon \times u_I
\]

\[
  \begin{array}{c}
    \phantom{u b_n =}\ \overset{u b_1}{\|} \qquad\qquad
    \overset{u^{n-1} b_n}{\|} \\
    \begin{array}{r|cccc}
      & b_1 & b_2 & \cdots & b_n \\
      u b_n = & b'_1 & b'_2 & \cdots & b'_n \\
      \hline
      & i_1 & i_2 & \cdots & i_n
    \end{array} \\
    \phantom{u b_n = i_1}\ \underset{u i_1}{\|} \qquad\quad
    \underset{u^{n-1} i_1}{\|}
  \end{array}
\]

\[
  \mathrm{Aut}(C_n) = \lbrace \pm 1 \rbrace^n \cdot \mathfrak{S}_n\,)
\]

\[
  \left.
  \begin{array}{l}
    \alpha_1^+ \ \cdots\ \alpha_n^+ \in \mathbf{N} \\
    \alpha_1^- \ \cdots\ \alpha_n^- \in \mathbf{N}
  \end{array}
  \right| \quad
  \begin{array}{l}
    \alpha_i^+ \ (\alpha_i^-) = \text{nb des cycles de long.\ } i \\
    \text{de } u_I \text{ au-dessus desquels il y a} \\
    \text{un cycle de long.\ } i \ (\text{resp.\ } 2i)
  \end{array}
\]

\[
  \sum_i i \underbrace{(\alpha_i^+ + \alpha_i^-)}_{\alpha_i} = n
\]

\[
  3^{\sum \alpha_i^+}
\]

\[
  \boxed{\ \sum_{i=0}^n f_d\, t^d = \prod_{i=1}^n (1 + 2t^i)^{\alpha_i^+}\ }
\]

\[
  \left\lbrace
  \begin{array}{lll}
    1^\circ)\ \text{non tordu :} & e_1 \to e_2 \to e_3 \to e_1 & (+) \\
    2^\circ)\ \text{tordu} & e_1 \to e_2 \to e_3 \to -e_1 & (-)
  \end{array}
  \right.
\]

\[
  \mathrm{Aut}(C)/(\pm 1) \hookrightarrow
  \mathrm{Aut}\bigl(\Phi_0(C)/\sigma\bigr)
\]

\[
  \mathrm{Aut}^+(C) \hookrightarrow
  \mathrm{Aut}\bigl(\Phi_0(C)\bigr)
\]

\[
  \mathrm{Aut}^+(C) \xrightarrow{\ \sim\ }
  \mathrm{Aut}_{\mathrm{ens}}\bigl(\underbrace{\Phi_0(C)/\sigma}_{\text{ens.\ de card.\ } 4}\bigr)
\]

\[
  C = C(\mathbb{B}, \sigma) \subset E = E_C = E(\mathbb{B}, \sigma)
\]

\[
  \partial\hat{C}/\sigma \simeq E^*/\mathbf{R}^* = P(E)
\]

\[
  \widetilde{C} = \partial\hat{C}/\sigma
  = \coprod_{F \in \Phi^{**}} p(F)
  = \coprod_{F \in \Phi^{**}} p(\overline{F})
\]

\[
  p(\overline{F}) \cap p(\overline{F'}) =
  \left\lbrace
  \begin{array}{ll}
    \varnothing & \text{si } \overline{F} \cap \overline{F'} = \varnothing
      \text{ et } \overline{F} \cap \sigma\overline{F'} = \varnothing \\
    \bigl[p(\overline{F}) \cap p(\overline{F'})\bigr] \cup
      \bigl[p(\overline{F}) \cap p(\overline{F'})\bigr] & \text{sinon}
  \end{array}
  \right.
\]

\[
  \bigl[p(\overline{F}) \subset p(\overline{F'})\bigr] \Longleftrightarrow
  \overline{F} \subset \overline{F'}
\]

\[
  \mathrm{Aut}(C)/\sigma \xrightarrow{\ \sim\ }
  \mathrm{Aut}_{\mathrm{proj}}\,\widetilde{C} \xrightarrow{\ \sim\ }
  \mathrm{Aut}_{\mathrm{ord}}\bigl(\Phi(\widetilde{C})\bigr) \quad ?
\]