Dual strongly perfect lattices

• Dual stark perfekte Gitter

Nossek, Elisabeth; Nebe, Gabriele (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2013)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2013

Abstract

The search for the densest sphere packings in every dimension is a classical problem in mathematics. In dimension 3 this problem is know as the Kepler conjecture, and has only been proofed by Thomas Hales in 1998. If we additionally assume, that the centres of the spheres in the packing form a group, a so called lattice, the problem becomes much easier. Already in 1908 Voronoi developed an algorithm, which computes all local densest lattices. But the complexity of this algorithm makes an application in dimensions greater than 8 impossible. Thus the densest sphere packings associated to a lattice are known up to dimension 8 and in dimension 24, only because of the existence of the extremely dense Leech lattice. Venkov united special local densest lattices with the combinatorial concept of spherical designs, which was developed by Delsarte, Goethals and Seidel in 1977. A lattice is called strongly perfect, if its shortest vectors form a spherical 5-design. As Venkov proves, strongly perfect lattices assume local maxima of the density function. Examples for strongly perfect lattices are the Leech lattice, which was mentioned above, the Barnes-Wall lattice and the densest lattices in dimension 2, 4, 6, 7, 8. G. Nebe and B. Venkov completely classified the strongly perfect lattices up to dimension 12. Almost all strongly perfect lattices are dual strongly perfect, which means that their dual lattice is also strongly perfect. In dimension 14 Nebe and Venkov proved that there is exactly one dual strongly perfect lattice. This thesis continues the classification of dual strongly perfect lattices and shows, that in dimension 13 and 15 there are no dual strongly perfect lattices. The methods developed in this thesis allow to give a shorter proof for the classification in dimension 14, but are not sufficient to give a classification in dimension 16. In dimension 17 we impose the additional constraint for the classification and prove that there is no universally perfect lattice, which means that all non empty layers of L are spherical 5-designs. This property can be characterised with theta-series: all theta-series of L with harmonical coefficients up to degree 5 vanish. With the theta-transformation formula we get that the dual lattice of an universally perfect lattice is universally perfect, too. Therefore universally perfect lattices are in particular dual strongly perfect.