Sublattices of certain Coxeter lattices
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.
scientific article; zbMATH DE number 5016568