Perfect Lattices in Euclidean Spaces

Jacques Martinet

524 pages, parution le 30/12/2002

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Résumé

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 NJ.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.

Contents

General Properties of Lattices
Geometric Inequalities
Perfection and Eutaxy
Root Lattices
Lattices Related to Root Lattices
Low-Dimensional Perfect Lattices
The Voronoi Algorithm
Hermitian Lattices
The Configurations of Minimal Vectors
Extremal Properties of Families of Lattices
Group Actions
Cross-Sections
Extensions of the Voronoi Algorithm
Numerical Data Appendix 1: Semi-Simple Algebras and Quaternions
Appendix 2: Strongly Perfect Lattices

L'auteur - Jacques Martinet

University of Bordeaux, France

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Jacques Martinet
Parution	30/12/2002
Nb. de pages	524
Format	16 x 24
Couverture	Relié
Poids	935g
Intérieur	Noir et Blanc
EAN13	9783540442363