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. used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II 25,1 in terms of the Leech lattice.
In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II 25,1 in terms of the Leech lattice.
Description of the algorithm
Let \Gamma be a hyperbolic reflection group. Choose any point v_0 \in \mathbb{H}^n ; we shall call it the basic (or initial) point. The fundamental domain P_0 of its stabilizer \Gamma_{v_0} is a polyhedral cone in \mathbb{H}^n . Let H_1,...,H_m be the faces of this cone, and let a_1,...,a_m be outer normal vectors to it. Consider the half-spaces H_k^- = \{x \in \R^{n,1} |(x,a_k) \le 0\}. There exists a unique fundamental polyhedron P of \Gamma contained in P_0 and containing the point v_0 . Its faces containing v_0 are formed by faces H_1,...,H_m of the cone P_0 . The other faces H_{m+1},... and the corresponding outward normals a_{m+1}, ... are constructed by induction. Namely, for H_j we take a mirror such that the root a_j orthogonal to it satisfies the conditions (1) (v_0,a_j) ; (2) (a_i, a_j ) \le 0 for all i ; (3) the distance (v_0 , H_j) is minimum subject to constraints (1) and (2).
References
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