## 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

Zbl 1054.05093
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### References:

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