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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 463-490
  • MSC: Primary 11H99; Secondary 52A43
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0716832-0
  • MathSciNet review: 716832