Minimal vectors in the second exterior power of a lattice
A lattice \(L\) in \(n\)-dimensional Euclidean space is said to have property \(S_k\) if the minimal vectors of its \(k\)th exterior power are all split, i.e. \(k\)-products of vectors form \(L\). The author shows that \(S_2\) holds under any of the following three conditions: \(n\leq 11\); \(n\leq 23\) and \(L = K_n\) or \(\Lambda_n\) (laminated lattices); \(L\) is integral and has a 2-dimensional sublattice of determinant at most 8. On the other hand, he shows that for any \(k\geq 2\) there exists an integer \(n_k\) such that \(n\)-dimensional lattices which do not satisfy \(S_k\) exist for any \(n\geq n_k\). For \(k=2\) he obtains explicit counterexamples, the first being the Leech lattice.
scientific article; zbMATH DE number 1066134
scientific article