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Sublattices of certain Coxeter lattices. (English) Zbl 1093.11050

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.

MSC:

11H06 Lattices and convex bodies (number-theoretic aspects)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
05B30 Other designs, configurations
05C99 Graph theory

Citations:

Zbl 1054.05093
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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
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