Lattices of finite abelian groups

TL;DR

This paper investigates lattices derived from finite abelian groups, proving their eutactic properties, simplifying existing results, and characterizing conditions for strong eutaxy and bases of minimal vectors.

Contribution

It confirms a conjecture that these lattices are eutactic and provides simpler proofs for known properties related to their eutaxy and minimal vectors.

Findings

Lattices from finite abelian groups are eutactic.

Such lattices are strongly eutactic if the group has odd order or is elementary abelian.

The lattice has a basis of minimal vectors except for cyclic groups of order 4.

Abstract

We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B\"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two known results: First, such a lattice is strongly eutactic if and only if the abelian group has odd order or is elementary abelian. Second, such a lattice has a basis of minimal vectors, except for the cyclic group of order 4.