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 Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
of a
hyperbolic reflection group Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
. 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 In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by E ...
.


Description of the algorithm

Let \Gamma < \mathrm(\mathbb^n) be a hyperbolic reflection group. Choose any point v_0 \in \mathbb^n; we shall call it the basic (or initial) point. The fundamental domain P_0 of its stabilizer \Gamma_ is a polyhedral cone in \mathbb^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^- = \. 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_,... and the corresponding outward normals a_, ... 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 ) \le 0 for all i < j; (3) the distance (v_0 , H_j) is minimum subject to constraints (1) and (2).


References

* *{{Citation , last1=Vinberg , first1=È. B. , editor1-last=Baily , editor1-first=Walter L. , title=Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) , url=https://books.google.com/books?id=7_g_AQAAIAAJ , publisher=
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
, isbn=978-0-19-560525-9 , mr=0422505 , year=1975 , chapter=Some arithmetical discrete groups in Lobačevskiĭ spaces , pages=323–348 Hyperbolic geometry Reflection groups