Cyclic and well-rounded lattices

TL;DR

This paper explores the properties and classifications of well-rounded and cyclic lattices, establishing their relationships, counting classes over number fields, and analyzing their cyclic structures in root lattices and algebraic number fields.

Contribution

It demonstrates that all planar well-rounded lattices are similar to cyclic lattices, and classifies cyclic properties of root lattices and lattices from Galois number fields.

Findings

Every planar well-rounded lattice is similar to a cyclic lattice.

Classified simple cyclic root lattices in arbitrary dimensions.

Identified cyclic and well-rounded cyclic lattices from Galois number fields.

Abstract

We focus on two important classes of lattices, the well-rounded and the cyclic. We show that every well-rounded lattice in the plane is similar to a cyclic lattice, and use this cyclic parameterization to count planar well-rounded similarity classes defined over a fixed number field with respect to height. We then investigate cyclic properties of the irreducible root lattices in arbitrary dimensions, in particular classifying those that are simple cyclic, i.e. generated by rotation shifts of a single vector. Finally, we classify cyclic, simple cyclic and well-rounded cyclic lattices coming from rings of integers of Galois algebraic number fields.