Non-rigidity Degree of a Lattice and Rigid Lattices

Voronoi defines a partition of the cone of positive semidefinite n -ary formsPn into L -type domains. Each L -type domain is an open polyhedral cone of dimensionk , 1 ≤ k ≤ n(n + 1)2, where n is the number of variables and dimension of the corresponding lattice. We define a non-rigidity degree of a lattice as the dimension of the L -type domain containing the lattice. We prove that the non-rigidity degree of a lattice equals the corank of a system of equalities connecting norms of minimal vectors of cosets of 2 L in L. A lattice of non-rigidity degree 1 is called rigid. A lattice is rigid if any of its sufficiently small deformations distinct from a homothety changes its L -type. Using the list of 84 zone-contracted Voronoi polytopes inR5given by Engel [8], we give a complete list of seven five-dimensional rigid lattices.