Bergé, Anne-Marie; Martinet, Jacques Sublattices of certain Coxeter lattices. (English) Zbl 1093.11050 J. Théor. Nombres Bordx. 17, No. 2, 455-465 (2005). Consider the root lattice \({\mathbb A}_n\) and its dual \({\mathbb A}_n^*\). Since \({\mathbb A}_n^*/{\mathbb A}_n\) is cyclic of order \(n+1\), there exists to each divisor \(r\) of \(n+1\) a uniquely determined intermediate lattice \({\mathbb A}_n^r\) with \([{\mathbb A}_n^r:{\mathbb A}_n]=r\). For \(n\) odd, one denotes by \(\text{Cox}_n\) the lattice \({\mathbb A}_n^{(n+1)/2}\).The purpose of the present paper is to study sublattices of finite index in \(\text{Cox}_n\), building upon and extending results by the first author [J. Algebr. Comb. 20, 5–16 (2004; Zbl 1054.05093)].The minimal vectors of \({\mathbb A}_n^*\) are \(\pm v_i\) (\(0\leq i\leq n\)) satisfying the unique dependence relation \(\sum_{i=0}^n v_i=0\). Put \(S=\{ v_i+v_j\mid i<j\}\). Then \(L=\text{Cox}_n\) is in fact generated by \(S\). Let \(M\) be the lattice generated by all the \(v_i\). In section 2, graph-theoretic criteria for the linear independence of subsets \({\mathcal B}\) of \(S\) and for properties of the sublattices generated by \({\mathcal B}\) are derived. This is used in section 3 of the present paper to give a precise description of the abelian groups \(L/N\) and \(M/N\) for sublattices \(N\) generated by subsets of \(S\). In section 4, certain sublattices obtained as cross sections of \(\text{Cox}_n\) are studied, and criteria for when these sublattices are weakly resp. strongly eutactic are given. Reviewer: Detlev Hoffmann (Nottingham) Cited in 1 Document MSC: 11H06 Lattices and convex bodies (number-theoretic aspects) 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 05B30 Other designs, configurations 05C99 Graph theory Keywords:Coxeter lattice; eutactic lattice; spherical design; kissing number; bipartite graph Citations:Zbl 1054.05093 × Cite Format Result Cite Review PDF Full Text: DOI Numdam Numdam EuDML References: [1] A.-M. Bergé, On certain Coxeter lattices without perfect sections. J. Algebraic Combinatorics 20 (2004), 5-16. · Zbl 1054.05093 [2] A.-M. Bergé, J. Martinet, Symmetric Groups and Lattices. Monatsh. Math. 140 (2003), 179-195. · Zbl 1132.11329 [3] B. Bollobás, Modern graph theory. Graduate texts in Mathematics 184, Springer-Verlag, Heidelberg, 1998. · Zbl 0902.05016 [4] H.S.M. Coxeter, Extreme forms. Canad. J. Math. 3 (1951), 391-441. · Zbl 0044.04201 [5] J. Martinet, Perfect Lattices in Euclidean Spaces. Grundlehren 327, Springer-Verlag, Heidelberg, 2003. · Zbl 1017.11031 [6] B. Venkov, Réseaux et designs sphériques (notes by J. Martinet). In Réseaux euclidiens, designs sphériques et groupes, L’Ens. Math., Monographie 37, J. Martinet ed., Genève (2001), 10-86. · Zbl 1139.11320 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.