Generation of Isometries of Certain image-Lattices by Symmetries, Original Research Article

It is well known that every isometry of a quadratic space is generated by symmetries. Although this is not true in general for isometries of lattices, there are certain image-lattices whose isometries are generated by −1 and symmetries. In this paper, we prove that positive image-lattices with determinant an odd prime p (or 2p if p is not possible), which we call primal lattices, have this property if the rank is not too big. More precisely, we prove that every isometry is generated by −1 and symmetries with respect to minimal vectors of length 2 for primal even (or odd) image-lattices of rank less than 16 (or 13, repectively) provided that the minimal length is at least 2, and that the bound of rank in each case is extremal. The most important ingredient in this work is the behavior of the maximal root sublattice which, we found, plays an essential role in shaping the isometry group of a given lattice.