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

On well-rounded sublattices of the hexagonal lattice