Sphere Packings, Lattices and Groups

  • J. H. Conway
  • N. J. A. Sloane

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 290)

Table of contents

  1. Front Matter
    Pages i-lxxiv
  2. J. H. Conway, N. J. A. Sloane
    Pages 1-30
  3. J. H. Conway, N. J. A. Sloane
    Pages 31-62
  4. J. H. Conway, N. J. A. Sloane
    Pages 63-93
  5. J. H. Conway, N. J. A. Sloane
    Pages 94-135
  6. John Leech, N. J. A. Sloane
    Pages 136-156
  7. J. H. Conway, N. J. A. Sloane
    Pages 157-180
  8. N. J. A. Sloane
    Pages 181-205
  9. J. H. Conway, N. J. A. Sloane
    Pages 206-244
  10. N. J. A. Sloane
    Pages 245-266
  11. J. H. Conway
    Pages 267-298
  12. J. H. Conway
    Pages 299-330
  13. J. H. Conway
    Pages 331-336
  14. A. M. Odlyzko, N. J. A. Sloane
    Pages 337-339
  15. E. Bannai, N. J. A. Sloane
    Pages 340-351
  16. J. H. Conway, N. J. A. Sloane
    Pages 352-405
  17. J. H. Conway, N. J. A. Sloane
    Pages 406-420
  18. R. E. Borcherds
    Pages 421-428
  19. B. B. Venkov
    Pages 429-440
  20. J. H. Conway, A. M. Odlyzko, N. J. A. Sloane
    Pages 441-444

About this book

Introduction

We now apply the algorithm above to find the 121 orbits of norm -2 vectors from the (known) nann 0 vectors, and then apply it again to find the 665 orbits of nann -4 vectors from the vectors of nann 0 and -2. The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16.

Keywords

Graph Group theory Lie algebra classification construction ring theory theory

Authors and affiliations

  • J. H. Conway
    • 1
  • N. J. A. Sloane
    • 2
  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Information Sciences ResearchAT&T Labs — ResearchFlorham ParkUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-6568-7
  • Copyright Information Springer-Verlag New York 1999
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-3134-4
  • Online ISBN 978-1-4757-6568-7
  • Series Print ISSN 0072-7830
  • About this book
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