Coxeter–Todd lattice

In mathematics, the Coxeter–Todd lattice K12, discovered by Coxeter and Todd (1953), is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice).

Properties

The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over the Eisenstein integers. The automorphism group of this complex lattice has index 2 in the full automorphism group of the Coxeter–Todd lattice and is a complex reflection group (number 34 on the list) with structure 6.PSU4(F3).2, called the Mitchell group.

The genus of the Coxeter–Todd lattice was described by (Scharlau & Venkov 1995) and has 10 isometry classes: all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.

Construction

Based on Nebe web page we can define K12 using following 6 vectors in 6-dimensional complex coordinates. ω is complex number of order 3 i.e. ω3=1.

(1,0,0,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0),

½(1,ω,ω,1,0,0), ½(ω,1,ω,0,1,0), ½(ω,ω,1,0,0,1),

By adding vectors having scalar product -½ and multiplying by ω we can obtain all lattice vectors. We have 15 combinations of two zeros times 16 possible signs gives 240 vectors; plus 6 unit vectors times 2 for signs gives 240+12=252 vectors. Multiply it by 3 using multiplication by ω we obtain 756 unit vectors in K12 lattice.

Further reading

The Coxeter–Todd lattice is described in detail in (Conway & Sloane 1999, section 4.9) and (Conway & Sloane 1983).

References
  • Conway, J. H.; Sloane, N. J. A. (1983), "The Coxeter–Todd lattice, the Mitchell group, and related sphere packings", Mathematical Proceedings of the Cambridge Philosophical Society, 93 (3): 421–440, Bibcode:1983MPCPS..93..421C, doi:10.1017/S0305004100060746, MR 0698347
  • Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
  • Coxeter, H. S. M.; Todd, J. A. (1953), "An extreme duodenary form", Canadian Journal of Mathematics, 5: 384–392, doi:10.4153/CJM-1953-043-4, MR 0055381
  • Scharlau, Rudolf; Venkov, Boris B. (1995), "The genus of the Coxeter-Todd lattice", Preprint, archived from the original on 2007-06-12
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