On lattices generated by algebraic conjugates of prime degree

TL;DR

This paper studies lattices generated by algebraic conjugates of prime degree, focusing on their properties, automorphisms, and conditions for being well-rounded, with classifications in low dimensions and a determinant formula for special cases.

Contribution

It provides new classifications of well-rounded lattices from algebraic conjugates, especially for prime degrees, and derives a determinant formula for specific Galois groups.

Findings

Infinite families of well-rounded lattices for large Pisot numbers of prime degree.

Complete classification of 2-dimensional well-rounded lattices from algebraic conjugates.

A determinant formula for lattices with certain Galois group structures.

Abstract

We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially interested in situations when the resulting lattice is well-rounded. We show that this happens for large Pisot numbers of prime degree, demonstrating infinite families of such lattices. We also fully classify well-rounded lattices from algebraic conjugates in the 2-dimensional case and present various examples in the 3-dimensional case. Finally, we derive a determinant formula for the resulting lattice in the case when the minimal polynomial of an algebraic number has its Galois group of a particular type.