On the index system of well-rounded lattices

TL;DR

This paper investigates the structure of well-rounded lattices by examining the possible quotient indices of Minkowskian sublattices within given dimensions and index bounds, contributing to the understanding of lattice index systems.

Contribution

It characterizes the set of possible quotient indices of Minkowskian sublattices in well-rounded lattices for specific dimensions and index bounds.

Findings

Identifies possible quotient indices for dimensions up to 8.

Provides bounds on the index of Minkowskian sublattices.

Classifies index systems in well-rounded lattices.

Abstract

Let $\Lb$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lb'$ be a Minkowskian sublattice of $\Lb$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lb$. We consider the set of possible quotients $\Lb/\Lb'$ which may exists in a given dimension or among not too large values of the index $[\Lb:\Lb']$, indeed $[\Lb:\Lb']\le 4$, or dimension $n\le 8$.