Vol. 11, No. 1, 2022

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Cyclic and well-rounded lattices

Lenny Fukshansky and David Kogan

Vol. 11 (2022), No. 1, 79–96
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 parametrization 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.

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Keywords
cyclic lattices, well-rounded lattices, root lattices, height functions, circulant matrices
Mathematical Subject Classification
Primary: 11H06, 11H31, 11G50, 11R04
Milestones
Received: 10 October 2021
Revised: 30 January 2022
Accepted: 16 February 2022
Published: 30 March 2022
Authors
Lenny Fukshansky
Department of Mathematical Sciences
Claremont McKenna College
Claremont, CA
United States
David Kogan
Institute of Mathematical Sciences
Claremont Graduate University
Claremont, CA
United States