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

1983/05/01 by John H. Conway, N. J. A. Sloane · 1 citation
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Coding theory and cryptography #Finite Group Theory Research #Coxeter group #Lattice (music) #Combinatorics #Mathematics #Physics

paper · doi:10.1017/s0305004100060746

openalex publication_date 1983/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

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 K 12 . We give several constructions for , which is a Z [ω]-lattice where ω = e 2πi/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 E 6 and the Leech lattice. A number of new dense lattices in dimensions up to about 10 7 are constructed. We also give an explicit basis for the invariants of the Mitchell group. The paper concludes with an extensive bibliography.

Citations

• Definite quadratische formen der dimension 24 und diskriminante 1
• The complex leech lattice and maximal subgroups of the Suzuki group
• The classification of finite simple groups I. Simple groups and local analysis
• The maximal subgroups of Conway's group Co1
• A Group of Order 8,315,553,613,086,720,000
• Invariant Theory
• The covering radius of the Leech lattice
• Laminated Lattices

Cited by

• Coxeter–Todd lattice [wikipedia]