Web Pages on the Voronoi Algorithm and Minimal Classes

The first 2 PARI-GP files below give the sets of vertices and edges of the Voronoi graph first in dimensions 2 to 6, then in dimension 7. The numerical data are extracted from Jaquet’s thesis [Ja]. The third file, based on Chapters 9 and 14 of [Mar], is devoted to minimal classes in dimensions 2 to 4. We present below a short account of Voronoi’s theory and minimal classes. 1. The perfection rank Let S ⊂ R (n ≥ 2) be a set of non-zero vectors viewed as columnmatrices, invariant under the symmetry x 7→ −x. Unless otherwise stated, we assume that S is finite, and we set s = 1 2 |S|. With S we associate the set of the s matrices M = X X for X ∈ S (note that X and −X define the same matrix M) and its span VS in the space Symn(R) of real, n× n symmetric matrices [The symmetric matrices XtX may be viewed as projection matrices: indeed, let E be an n-dimensional Euclidean vector space equipped with a basis B ; if X is the set of components on B of some vector x ∈ E, then XtX is the matrix in the pair of bases (B∗,B) of the orthogonal projection px ∈ End(E) to x.] Definition 1.1. The rank in Symn(R) of the set {X X}X∈S is called the perfection rank of S and denoted by perf S The perfection rank of a lattice Λ or a positive definite quadratic form Q on R (or the symmetric matrix A such that Q(X) = XAX) is the perfection rank of its set of minimal vectors. We say that Λ or Q (or A) is perfect if its perfection rank attains its maximal possible value, namely dim Symn(R) = n(n+1) 2 .