On well-rounded sublattices of the hexagonal lattice |
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Authors: | Lenny Fukshansky Daniel Moore |
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Institution: | a Department of Mathematics, 850 Columbia Avenue, Claremont McKenna College, Claremont, CA 91711, United Statesb Department of Mathematics, Loyola Marymount University, 1 LMU Drive, Los Angeles, CA 90045, United Statesc Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, United Statesd Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, United States |
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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. |
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Keywords: | Hexagonal lattice Well-rounded lattices Binary and ternary quadratic forms Epstein zeta function |
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