2004/03/28 by Robert L. Griess, Robert L. Griess Jr, Griess, Robert L.
Mathematics · #11H06 #11H56 #20D06 #20G40 #Advanced Algebra and Geometry #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.CO #math.GR #msc:11H06 #msc:11H56 #msc:20D06 #msc:20G40
paper · pdf · doi:10.48550/arxiv.math/0403480
56 pages
arxiv created 2004/03/28 · openalex publication_date 2004/03/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a new existence proof for the rank 2d even lattices usually called the Barnes-Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend. Extending these ideas, we construct in dimensions 2d, for d>>0, the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes-Wall lattices. The number Upsilon (2d) of isometry types here is large: log2 (Upsilon (2d)) has dominant term at least (r/4) d 22d, for any r in [0, 1/2). The Ypsilanti lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate (log2(mass(n)) has dominant term (1/4) log2(n) n2). This work continues our general uniqueness program for lattices, begun in Pieces of Eight. See also our new uniquness proof for the E8-lattice.