Perfect Lattices in Euclidean Spaces Grundlehren Der Mathematischen Wissenschaften Springer

Lattices are discrete subgroups of maximal rank in a Euclidean space. To each such geometrical object, we can attach a canonical sphere packing which, assuming some regularity, has a density. The question of estimating the highest possible density of a sphere packing in a given dimension is a fascinating and difficult problem: the answer is known only up to dimension 3. This book thus discusses a beautiful and central problem in mathematics, which involves geometry, number theory, coding theory and group theory, centering on the study of extreme lattices, i.e. those on which the density attains a local maximum, and on the so-called perfection property. Written by a leader in the field, it is closely related to, though disjoint in content from, the classic book by J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, published in the same series as vol. 290. Every chapter except the first and the last contains numerous exercises. For simplicity those chapters involving heavy computational methods contain only few exercises. It includes appendices on Semi-Simple Algebras and Quaternions and Strongly Perfect Lattices. Front Matter....Pages I-XXI General Properties of Lattices....Pages 1-35 Geometric Inequalities....Pages 37-65 Perfection and Eutaxy....Pages 67-108 Root Lattices....Pages 109-145 Lattices Related to Root Lattices....Pages 147-188 Low-Dimensional Perfect Lattices....Pages 189-225 The Voronoi Algorithm....Pages 227-262 Hermitian Lattices....Pages 263-319 The Configurations of Minimal Vectors....Pages 321-362 Extremal Properties of Families of Lattices....Pages 363-381 Group Actions....Pages 383-426 Cross-Sections....Pages 427-441 Extensions of the Voronoi Algorithm....Pages 443-465 Numerical Data....Pages 467-478 Appendix 1: Semi-Simple Algebras and Quaternions....Pages 479-488 Appendix 2: Strongly Perfect Lattices....Pages 489-495 Back Matter....Pages 497-526