Vinberg's algorithm

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In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. (Conway 1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice.

Description of the algorithm

Let [math]\displaystyle{ \Gamma \lt \mathrm{Isom}(\mathbb{H}^n) }[/math] be a hyperbolic reflection group. Choose any point [math]\displaystyle{ v_0 \in \mathbb{H}^n }[/math]; we shall call it the basic (or initial) point. The fundamental domain [math]\displaystyle{ P_0 }[/math] of its stabilizer [math]\displaystyle{ \Gamma_{v_0} }[/math] is a polyhedral cone in [math]\displaystyle{ \mathbb{H}^n }[/math]. Let [math]\displaystyle{ H_1,...,H_m }[/math] be the faces of this cone, and let [math]\displaystyle{ a_1,...,a_m }[/math] be outer normal vectors to it. Consider the half-spaces [math]\displaystyle{ H_k^- = \{x \in \R^{n,1} |(x,a_k) \le 0\}. }[/math]

There exists a unique fundamental polyhedron [math]\displaystyle{ P }[/math] of [math]\displaystyle{ \Gamma }[/math] contained in [math]\displaystyle{ P_0 }[/math] and containing the point [math]\displaystyle{ v_0 }[/math]. Its faces containing [math]\displaystyle{ v_0 }[/math] are formed by faces [math]\displaystyle{ H_1,...,H_m }[/math] of the cone [math]\displaystyle{ P_0 }[/math]. The other faces [math]\displaystyle{ H_{m+1},... }[/math] and the corresponding outward normals [math]\displaystyle{ a_{m+1}, ... }[/math] are constructed by induction. Namely, for [math]\displaystyle{ H_j }[/math] we take a mirror such that the root [math]\displaystyle{ a_j }[/math] orthogonal to it satisfies the conditions

(1) [math]\displaystyle{ (v_0,a_j) \lt 0 }[/math];

(2) [math]\displaystyle{ (a_i, a_j ) \le 0 }[/math] for all [math]\displaystyle{ i \lt j }[/math];

(3) the distance [math]\displaystyle{ (v_0 , H_j) }[/math] is minimum subject to constraints (1) and (2).


References