Midwest cousins of Barnes-Wall lattices

2009/01/08 by Robert L. Griess, Robert L. Griess Jr, Griess, Robert L.
Mathematics · #11H56 #20C10 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:11H56 #msc:20C10

paper · pdf · doi:10.48550/arxiv.0901.0970

33 pages; submitted

openalex publication_date 2009/01/08 · arxiv created 2009/10/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a rational lattice and suitable set of linear transformations, we construct a cousin lattice. Sufficient conditions are given for integrality, evenness and unimodularity. When the input is a Barnes-Wall lattice, we get multi-parameter series of cousins. There is a subseries consisting of unimodular lattices which have ranks 2^d-1± 2^d-k-1, for odd integers d≥ 3 and integers k=1,2, ..., \frac d-12. Their minimum norms are moderately high: 2^{\lfloor \frac d2 \rfloor -1}.

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