On well-rounded sublattices of the hexagonal lattice

TL;DR

This paper explicitly parameterizes well-rounded sublattices of the hexagonal lattice, analyzing their properties and structure, with implications for optimization problems and lattice theory.

Contribution

It provides a new explicit parameterization of well-rounded sublattices of the hexagonal lattice and studies their classification and properties.

Findings

Characterization of similarity classes of well-rounded sublattices

Analysis of minimal norm and signal-to-noise ratio

Existence of a combinatorial structure on similarity classes

Abstract

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.