Spherical 2-designs and lattices from Abelian groups

TL;DR

This paper characterizes when lattices from finite Abelian groups form spherical 2-designs, linking group order properties to lattice optimality and automorphism group structures, and introduces new extreme lattices.

Contribution

It provides a complete characterization of when such lattices are strongly eutactic, leading to new families of extreme lattices and insights into their automorphism groups.

Findings

Lattices are strongly eutactic iff the group is of odd order or a power of order 2.

New infinite family of extreme lattices derived from the characterization.

Enhanced understanding of automorphism groups of these lattices.

Abstract

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction.