Abstract
In an earlier paper we studied real laminated lattices (or Z {\mathbf {Z}} -modules) Λ n {\Lambda _n} , where Λ 1 {\Lambda _1} is the lattice of even integers, and Λ n {\Lambda _n} is obtained by stacking layers of a suitable ( n − 1 ) (n - 1) -dimensional lattice Λ n − 1 {\Lambda _{n - 1}} as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing Z {\mathbf {Z}} -module by J J -module, where J J may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which Λ n {\Lambda _n} is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the 6 6 -dimensional integral laminated lattice over Z [ ω ] {\mathbf {Z}}[ \omega ] of minimal norm 2 2 . The paper includes tables of the best real integral lattices in up to 24 24 dimensions.
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call