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.
<
)
{\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})}
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
be the faces of this cone, and let
be outer normal vectors to it.
Consider the half-spaces
There exists a unique fundamental polyhedron
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.
we take a mirror such that the root
orthogonal to it satisfies the conditions (1)
; (3) the distance
is minimum subject to constraints (1) and (2).