View source for Coxeter–Todd lattice

In mathematics, the '''Coxeter&ndash;Todd lattice''' K<sub>12</sub>, discovered by {{harvs|txt=yes|last=Coxeter|last2=Todd|year=1953}}, is a 12-dimensional even integral [[Lattice (group)|lattice]] of discriminant 3<sup>6</sup>  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&ndash;Wall lattice.  The automorphism group of the Coxeter&ndash;Todd lattice has order 2<sup>10</sup>·3<sup>7</sup>·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|complex reflection group]] (number 34 on the list) with structure 6.PSU<sub>4</sub>('''F'''<sub>3</sub>).2, called the Mitchell group.

The [[Genus of a quadratic form|genus]] of the Coxeter–Todd lattice was described by {{harv|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 [http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/K12.html Nebe] web page we can define K<sub>12</sub> using following 6 vectors in 6-dimensional complex coordinates. ω is complex number of order 3 i.e. ω<sup>3</sup>=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<sub>12</sub> lattice.

==Further reading==
The Coxeter–Todd lattice is described in detail in {{harv|Conway|Sloane|1999|loc=section 4.9}} and {{harv|Conway|Sloane|1983}}.

==References==
* {{citation |last1=Conway |first1=J. H. |last2=Sloane |first2=N. J. A. |year=1983 |title=The Coxeter–Todd lattice, the Mitchell group, and related sphere packings |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=93 |issue=3 |pages=421–440 |doi=10.1017/S0305004100060746 |bibcode=1983MPCPS..93..421C |mr=0698347}}
* {{Citation | last1=Conway | first1=John Horton |  last2=Sloane | first2=Neil J. A. |  year=1999 | title=Sphere Packings, Lattices and Groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=3rd | series=Grundlehren der Mathematischen Wissenschaften | volume=290 | isbn=978-0-387-98585-5 | mr=0920369 | doi=10.1007/978-1-4757-2016-7 | url-access=registration | url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}
* {{citation
|last1=Coxeter |first1=H. S. M. |last2=Todd |first2=J. A.
|title=An extreme duodenary form
|journal=Canadian Journal of Mathematics
|volume=5
|year=1953
|pages=384–392
|mr=0055381
|doi=10.4153/CJM-1953-043-4|doi-access=free
}}
* {{citation
|url=http://www.matha.mathematik.uni-dortmund.de/preprints/95-07.html
|title=The genus of the Coxeter-Todd lattice
|first1=Rudolf
|last1=Scharlau
|first2=Boris B.
|last2=Venkov
|year=1995
|journal=Preprint
|url-status=dead
|archiveurl=https://web.archive.org/web/20070612070900/http://www.matha.mathematik.uni-dortmund.de/preprints/95-07.html
|archivedate=2007-06-12
}}

==External links==
* [http://www.research.att.com/~njas/lattices/K12.html Coxeter–Todd lattice] in Sloane's lattice catalogue

{{DEFAULTSORT:Coxeter-Todd lattice}}
[[Category:Quadratic forms]]

{{Sourceattribution|Coxeter–Todd lattice}}