Vinberg's algorithm

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 II_25,1 in terms of the Leech lattice.

Description of the algorithm

Let $Γ < I s o m (ℍ^{n})$ be a hyperbolic reflection group. Choose any point $v_{0} \in ℍ^{n}$ ; we shall call it the basic (or initial) point. The fundamental domain $P_{0}$ of its stabilizer $Γ_{v_{0}}$ is a polyhedral cone in $ℍ^{n}$ . 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 $H_{k}^{-} = {x \in ℝ^{n, 1} | (x, a_{k}) \leq 0} .$

There exists a unique fundamental polyhedron $P$ of $Γ$ 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) $(v_{0}, a_{j}) < 0$ ;

(2) $(a_{i}, a_{j}) \leq 0$ for all $i < j$ ;

(3) the distance $(v_{0}, H_{j})$ is minimum subject to constraints (1) and (2).

References

Conway, John Horton (1983), "The automorphism group of the 26-dimensional even unimodular Lorentzian lattice", Journal of Algebra, 80 (1): 159–163, doi:10.1016/0021-8693(83)90025-X, ISSN 0021-8693, MR 0690711
Vinberg, È. B. (1975), "Some arithmetical discrete groups in Lobačevskiĭ spaces", in Baily, Walter L. (ed.), Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Oxford University Press, pp. 323–348, ISBN 978-0-19-560525-9, MR 0422505