On well-rounded sublattices of the hexagonal lattice

Details On well-rounded sublattices of the hexagonal lattice On well-rounded sublattices of the hexagonal lattice

2010-07-15

arxiv.org/abs/1007.2667v2

Discrete Mathematics, vol. 310 no. 23 (2010), pg. 3287--3302

Mathematics

Number Theory

21 pages (minor correction to the proof of Lemma 2.1); to appear in Discrete Mathematics

Scientific paper

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.

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