Wilhelm Plesken

Computing isometries of lattices

Abstract We present the main ideas for an algorithm to calculate the group of automorphisms of a Euclidean lattice. This algorithm can be applied to related problems, e.g. to compute Bravais groups, to calculate automorphisms of lattices over number fields or, in a slightly modified version, to find isometries between lattices. An implementation of the algorithm by the second author has been successfully applied to lattices up to dimension 40 and allows, for example, obtaining of generators for the automorphism group of the Leech lattice in less than 30 min on a HP 9000/730 workstation.

Linear Pro-p-Groups of Finite Width

Elementary properties of width.- p-adically simple groups .- Periodicity.- Chevalley groups.- Some classical groups.- Some thin groups.- Algorithms on fields.- Fields of small degree.- Algorithm for finding a filtration and the obliquity.- The theory behind the tables.- Tables.- Uncountably many just infinite pro-p-groups of finite width.- Some open problems.

Crystallographic Algorithms and Tables

A survey of definitions, theorems and algorithms for crystallographic groups are given in a dimension-independent fashion. These and some tables (including the Bravais groups up to dimension 6) form the basis of the computer package CARAT, which can handle crystallographic space groups up to dimension 6.

Group rings of finite groups over p-adic integers

Graduated and graduable orders.- The conductor formula for graduated hulls of selfdual orders.- Selfdual orders with decomposition numbers 0 and 1.- Blocks of multiplicity 1.- Examples of group rings.- The principal 2-block of SL2(q) for odd prime powers q.- Blocks with cyclic defect groups.

Constructing integral lattices with prescribed minimum. II

Methods for computing integral laminated lattices with prescribed minimum are developed. Laminating is a process of stacking layers of an (n 1)-dimensional lattice as densely as possible to obtain an n-dimensional lattice. Our side conditions are: All scalar products of lattice vectors are rational integers, and all lattices are generated by vectors of prescribed minimum (square) length et. For m = 3 all such lattices are determined.

On maximal finite irreducible subgroups of (,). I. The five and seven dimensional cases

General methods for the determination of maximal finite absolutely irreducible subgroups of GL(n, Z) are described. For n = 5, 7 all these groups are computed up to Z-equivalence.

Counting Crystallographic Groups in Low Dimensions

We present the results of our computations concerning the space groups of dimension 5 and 6. We find 222 018 and 28927922 isomorph ism types of these groups, respectively. Some overall statistics on the number of Q-classes and Z-classes in dimensions up to six are provided. The computations were done with the package CARAT, which can parametrize, construct and identify all crystallographic groups up to dimension 6.

The lattices of six-dimensional Euclidean space

The lattices of full rank of the six-dimensional Euclidean space are classified according to their automorphism groups (Bravais classification). We find 826 types of such lattices.

Constructing rational representations of finite groups

We present a method to construct irreducible rational matrix representations of finite groups, basedon an efficient construction of fixed points of finite groups acting on complex vector spaces.

On maximal finite irreducible subgroups of (,). V. The eight-dimensional case and a complete description of dimensions less than ten

All maximal finite (absolutely) irreducible subgroups of GL(S, Z) are determined up to Z-equivalence. Moreover, we present a full set of representatives of the Z-classes of the maximal finite irreducible subgroups of GL(n, Z) for n < 9 by listing generators of the groups, the corresponding quadratic forms fixed by these groups, and the shortest vectors of these forms.