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 similar to the Barnes–Wall 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.

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", Math. Proc. Cambridge Philos. Soc. 93 (3): 421–440, doi:10.1017/S0305004100060746, MR 0698347
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR 0920369
• Coxeter, H. S. M.; Todd, J. A. (1953), "An extreme duodenary form", Canadian J. Math. 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

External links

• Coxeter–Todd lattice in Sloane's lattice catalogue