Complex and integral laminated lattices

Conway, J. H.; Sloane, N. J. A.

doi:10.1090/S0002-9947-1983-0716832-0

Complex and integral laminated lattices
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by J. H. Conway and N. J. A. Sloane PDF

Trans. Amer. Math. Soc. 280 (1983), 463-490 Request permission

Abstract:

In an earlier paper we studied real laminated lattices (or ${\mathbf {Z}}$-modules) ${\Lambda _n}$, where ${\Lambda _1}$ is the lattice of even integers, and ${\Lambda _n}$ is obtained by stacking layers of a suitable $(n - 1)$-dimensional lattice ${\Lambda _{n - 1}}$ as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing ${\mathbf {Z}}$-module by $J$-module, where $J$ may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which ${\Lambda _n}$ 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 $6$-dimensional integral laminated lattice over ${\mathbf {Z}}[ \omega ]$ of minimal norm $2$. The paper includes tables of the best real integral lattices in up to $24$ dimensions.

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