Coxeter–Todd lattice

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In mathematics, the Coxeter–Todd lattice K₁₂, discovered by Coxeter and Todd (1953), is a the 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.

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, and all of them other than the Coxeter-Todd lattice have a root system of maximal rank 12.

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

[edit] 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, MR0698347 
• Conway, John Horton & Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, vol. 290 (3rd ed.), Grundlehren der Mathematischen Wissenschaften, Berlin, New York: Springer-Verlag, MR0920369, ISBN 978-0-387-98585-5 
• Coxeter, H. S. M. & Todd, J. A. (1953), “An extreme duodenary form.”, Canadian J. Math. 5: 384-392, MR0055381 
• Scharlau, Rudolf & Venkov, Boris B. (1995), “The genus of the Coxeter-Todd lattice”, preprint, <http://www.matha.mathematik.uni-dortmund.de/preprints/95-07.html> 

[edit] External links

• Coxeter–Todd lattice in Sloane's lattice catalogue