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Formulas from Ambjörn’s presentation at SIGGRAPH 2000:

$\, \mathrm{V}^n \, , \, e_1 \, , e_2 \, , \dots , e_n \,$

$\, \mathrm{G} = \mathrm{G}_n = \mathrm{G}(\mathrm{V}^n) \,$

$\, M = \sum\limits_{k = 0}^{n} {\langle \, M \, \rangle}_k \,$

$\, {\langle \, M \, \rangle}_k = B_1 + B_2 + \cdots \,$

$\, A_k = a_1 \wedge a_2 \wedge \cdots \wedge a_k \,$

$\, A_k \neq 0 \, \iff \, \{ a_1, a_2, \dots , a_k \} \,$

$\, P \in \mathrm{G}_n \,$

$\, P = p_1 \wedge p_2 \wedge \cdots \wedge p_n \,$

$\, I = e_1 \wedge e_2 \wedge \cdots \wedge e_n \,$

$\, [P] = PI^{-1} \,$

$\, \text{dual}(X) = XI^{-1} \,$

$\, \text{dual}(X) = X^* \,$

$\, B = b_1 \wedge b_2 \wedge \cdots \wedge b_m \,$

$\, \overline{B} \subseteq \mathrm{V}^n \,$

$\, \overline{B} = \text{Linspan}\{ b_1, b_2, \dots , b_m \} = \,$

$\, = \text{Linspan}\{ b \in \mathrm{G}_n : b \wedge B = 0 \} \,$

$\, \{ e_1, \dots , e_m \} \,$

$\, b_i = \sum\limits_{k = 0}^{m} b_{ik}e_k \; \text{for} \; i = 1, \, \dots \, , m \; , \,$

$\, B =( \det{b_{ik}}) \, e_1 \wedge e_2 \wedge \cdots \wedge e_m = \,$

$\, = (\det{b_{ik}}) \, e_1 e_2 \cdots e_m = \,$

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