Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by

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 ${\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})}$ be a hyperbolic reflection group. Choose any point ${\displaystyle v_{0}\in \mathbb {H} ^{n}}$; we shall call it the basic (or initial) point. The fundamental domain ${\displaystyle P_{0}}$ of its stabilizer ${\displaystyle \Gamma _{v_{0}}}$ is a polyhedral cone in ${\displaystyle \mathbb {H} ^{n}}$. Let ${\displaystyle H_{1},...,H_{m}}$ be the faces of this cone, and let ${\displaystyle a_{1},...,a_{m}}$ be outer normal vectors to it. Consider the half-spaces ${\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.}$

There exists a unique fundamental polyhedron ${\displaystyle P}$ of ${\displaystyle \Gamma }$ contained in ${\displaystyle P_{0}}$ and containing the point ${\displaystyle v_{0}}$. Its faces containing ${\displaystyle v_{0}}$ are formed by faces ${\displaystyle H_{1},...,H_{m}}$ of the cone ${\displaystyle P_{0}}$. The other faces ${\displaystyle H_{m+1},...}$ and the corresponding outward normals ${\displaystyle a_{m+1},...}$ are constructed by induction. Namely, for ${\displaystyle H_{j}}$ we take a mirror such that the root ${\displaystyle a_{j}}$ orthogonal to it satisfies the conditions

(1) ${\displaystyle (v_{0},a_{j})<0}$;

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

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

References

• 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),
  MR 0422505