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The Coxeter–Todd lattice, the Mitchell group, and related sphere packings

Published online by Cambridge University Press:  24 October 2008

J. H. Conway  and
N. J. A. Sloane
J. H. Conway
Affiliation:
University of Cambridge
N. J. A. Sloane
Affiliation:
Mathematics and Statistics Research Center, Bell Laboratories, Murray Hill, New Jersey 07974, USA
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Extract

This paper studies the Coxeter–Todd lattice its automorphism group (which is Mitchell's reflection group 6·PSU(4, 3)·2), and the associated 12-dimensional real lattice K12. We give several constructions for , which is a Z[ω]-lattice where ω = e2πi/3; enumerate the congruence classes of and where θ = ω − ω¯; prove the lattice is unique; determine its covering radius and deep holes; and study its connections with the lattice E6 and the Leech lattice. A number of new dense lattices in dimensions up to about 107 are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 421 - 440
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

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