Dual Cones and the Voronoi Algorithm

Almost a century ago Voronoi [1908] formulated his fundamental algorithm to find the perfect (real positive definite quadratic) forms in n variables. Among these one finds the forms representing the locally extreme lattice packings of spheres. The subject was taken up by M. Koecher [1960], who gave an axiomatic treatment of self-dual cones and a corresponding Voronoi algorithm in this situation aiming at the application of finding generators for certain arithmetic groups. Independently, the subject of extreme forms was taken up in [Berge et al. 1992] to adjust Voronoi's algorithm to find the G-perfect forms, where G is a finite unimodular group and the forms under consideration are G-invariant. The key observation of the present paper is that a wide range of applications can be made if one generalizes Koecher's axioms to a pair of dual cones. The theory has here its natural setting and becomes more transparent; see Section 2. In particular, the perfect points live in one cone and the associated tessellation, which leads to Voronoi's neighbouring graph in the classical situation, lives in the dual cone. The context of discontinuously acting groups is treated in Section 3, where the quotient of the resulting Voronoi graph modulo this group action leads to a generating set for the group considered. As an application, the last section gives an algorithm to calculate normalizers of finite subgroups of GLn(Z). It turns out that the natural setup for a finite unimodular group G is not just to look at the cone