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) 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) 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, Bibcode: 1983MPCPS..93..421C
- 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
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