# Coxeter–Todd lattice

In mathematics, the **Coxeter–Todd lattice** K_{12}, discovered by Coxeter and Todd (1953), is a 12-dimensional even integral lattice of discriminant 3^{6} 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.

## 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.PSU_{4}(**F**_{3}).2, called the Mitchell group.

The genus of the Coxeter–Todd lattice was described by (Scharlau Venkov) 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 K_{12} using following 6 vectors in 6-dimensional complex coordinates. ω is complex number of order 3 i.e. ω^{3}=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 K_{12} lattice.

## Further reading

The Coxeter–Todd lattice is described in detail in (Conway Sloane) and (Conway Sloane).

## 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, doi:10.1017/S0305004100060746 - 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, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, https://archive.org/details/spherepackingsla0000conw_b8u0 - 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 - Scharlau, Rudolf; Venkov, Boris B. (1995), "The genus of the Coxeter-Todd lattice",
*Preprint*, archived from the original on 2007-06-12, https://web.archive.org/web/20070612070900/http://www.matha.mathematik.uni-dortmund.de/preprints/95-07.html

## External links

- Coxeter–Todd lattice in Sloane's lattice catalogue

Original source: https://en.wikipedia.org/wiki/ Coxeter–Todd lattice.
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