Vinberg's algorithm

Summary

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

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Let   be a hyperbolic reflection group. Choose any point  ; we shall call it the basic (or initial) point. The fundamental domain   of its stabilizer   is a polyhedral cone in  . Let   be the faces of this cone, and let   be outer normal vectors to it. Consider the half-spaces  

There exists a unique fundamental polyhedron   of   contained in   and containing the point  . Its faces containing   are formed by faces   of the cone  . The other faces   and the corresponding outward normals   are constructed by induction. Namely, for   we take a mirror such that the root   orthogonal to it satisfies the conditions

(1)  ;

(2)   for all  ;

(3) the distance   is minimum subject to constraints (1) and (2).


References

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  • 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