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Abstract
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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
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Mathematical Subject Classification
Primary: 11H06, 11H31, 11G50, 11R04
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Milestones
Received: 10 October 2021
Revised: 30 January 2022
Accepted: 16 February 2022
Published: 30 March 2022
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