Isodualit\'e des r\'eseaux euclidiens en petite dimension

TL;DR

This paper classifies euclidean isodual lattices of fixed rank using algebraic and geometric methods, revealing their distribution into finite types and parametrization by symmetric spaces, with a complete description up to rank 7.

Contribution

It provides a comprehensive algebraic and geometric classification of low-rank euclidean isodual lattices, including explicit descriptions and parametrizations.

Findings

Finite number of algebraic types of isodual lattices

Parametrization by symmetric spaces associated with classical groups

Complete description of types and Gram matrices up to rank 7

Abstract

We propose an algebraic and a geometric classification of euclidean isodual lattices of fixed rank. First, we prove that these lattices are distribued according to a finite number of algebraic types. Second, we show that they are parametrized by a finite number of symmetric spaces associated to the classical groups ${\bf SO}_0(p,q)$, ${\bf Sp}(2g,{\bf R})$ and ${\bf SU}(p,q)$. We obtain a complete discription of algebraic types and Gram matrices of isodual lattices up to rank 7. The maximal density problem is also discussed.