Vinberg's algorithm explained

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 < Isom ( 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 \Gamma _{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)} \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)} < 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)