Automorphisms of modular lattices

TL;DR

This paper extends classification methods for extremal unimodular lattices to modular lattices, introducing the det-type concept to exclude many automorphism types and classifying certain extremal lattices of specific levels and dimensions.

Contribution

It generalizes classification techniques to modular lattices using det-type, enabling the exclusion of many automorphism types and classifying extremal lattices at specific levels and dimensions.

Findings

Excluded many automorphism det-types in modular lattices.

Classified extremal lattices of levels 14, 15, and 6 in specific dimensions.

Extended classification methods beyond unimodular lattices.

Abstract

The methods to classify extremal unimodular lattices with given automorphisms are extended to the situation of modular lattices. A slightly more general notion than the type from the PhD thesis of Michael Juergens is the det-type. The det-type of an automorphism on $L$ determines the one of all partial dual lattices of $L$. This easy observation allows to exclude quite a few det-types of automorphisms left open in the above mentioned thesis. Passing to suitable p-maximal lattices, extremal l-modular lattices of composite level 14 and 15 of dimension 12 and the ones of level 6 and dimension 16 are classified.