SUBLATTICES OF CERTAIN COXETER LATTICES

Abstract. In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs. Résumé. Dans cet article, nous décrivons les sous-réseaux de certains réseaux de Coxeter, prolongeant les résultats de [Ber]. Notre description utilise des graphes. In his 1951 paper Extreme forms ([Cox]), the late Coxeter studied lattices Λ which satisfy inclusions Λ0 ⊂ Λ ⊂ Λ ∗ 0 for some root lattice L. (As usual, L ∗ stands for the dual of a lattice L, and a root lattice is an integral lattice generated by norm 2 vectors.) We shall more specially consider the root lattice Λ0 = An, the section of Z n+1 endowed with its canonical basis B0 = (ε0, ε1,..., εn) by the hyperplane x0 + x1 + · · · + xn = 0. Then Λ ∗ 0/Λ0 is cyclic of order n + 1, so that there exists for every divisor r of n+1 a well defined lattice A r n such that An ⊂ A r n ⊂ A ∗ n and [A r n: An] = r. We denote by Coxn (n odd) the lattice A (n+1)/2 n scaled to the minimum which makes it integral and primitive. The lattices Coxn are perfect lattices which have various curious properties. In particular, they are hollow in the sense of [Ber], and their minimum is odd whenever n ≡ 3 mod 4. This paper is a continuation of [Ber], which was devoted to the classification of cross-sections of Coxn, but this time we consider sublattices of finite index (or more generally, of finite index in a cross-section). As [Ber], our paper makes intensive use of graphs to describe the possible sublattices of Coxn. As in [Ber], Euclidean properties play only a modest rôle: the minimal vectors of A ∗ n are vectors ±v0, ±v1,..., ±vn where the vi satisfy the unique (up to proportionality) dependence relation v0+v1+ · · ·+vn = 0. We set S = {vi + vj, 0 ≤ i < j ≤ n}. Key words and phrases. eutactic lattices, spherical designs, kissing number, bipartite graph