Computing Isometries of Lattices

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.