On the classification of eutactic lattices. (Sur la classification des réseaux eutactiques.) (French) Zbl 0854.11035

In this paper, we enlarge the notion of eutaxy for lattices, so as to make it easier to compute. We extend to these weakly eutactic lattices the Ash finiteness theorem for eutactic lattices. Our methods are however quite different, as they consist in classifying the \(n\)-dimensional lattices by their minimal vectors. We find a finite number of classes, each containing (up to similarity) at most one weakly eutactic lattice. Moreover, we prove that the weakly eutactic lattice (if any) in a given class is the less dense one, and that it is proportional to an algebraic lattice. As an application, we give a complete enumeration of weakly eutactic lattices up to dimension 4, which enables us to check for these dimensions Ash’s mass formula.


11H55 Quadratic forms (reduction theory, extreme forms, etc.)
11H56 Automorphism groups of lattices
11E10 Forms over real fields
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