Minimal vectors in the second exterior power of a lattice
type: Item
instance of: scholarly article
title: Minimal vectors in the second exterior power of a lattice
zbMATH Open document ID: 0885.11041
author: Renaud Coulangeon
published in: Journal of Algebra
publication date: 1998-04-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 52C07
zbMATH DE Number: 1066134
zbMATH Keywords: exterior power of a lattice
zbMATH Keywords: symplectic lattice
zbMATH Keywords: split minimal vectors
reviewed by: Heinz-Georg Quebbemann
MaRDI profile type: Publication
full work available at URL: jabr.1997.7039
OpenAlex ID: W1996773658
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DOI: 10.1006/JABR.1997.7039
description: scientific article; zbMATH DE number 1066134