Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group.

Template:Harvtxt 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 $\mathrm{\Gamma }<\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}({ℍ}^n)$ 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 ${\mathrm{\Gamma }}_{v_0}$ 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 $\mathrm{\Gamma }$ 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).

References

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