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.

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.

Description of the algorithm
Let $$\Gamma < \mathrm{Isom}(\mathbb{H}^n)$$ be a hyperbolic reflection group. Choose any point $$v_0 \in \mathbb{H}^n$$; 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 $$\mathbb{H}^n$$. 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).