Coxeter–Todd lattice
In mathematics, the Coxeter–Todd lattice K12, discovered by {{harvs|txt=yes|author1-link=H. S. M. Coxeter|author2-link=J. A. Todd|last=Coxeter|last2=Todd|year=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 {{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 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 {{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 | author1-link=John Horton Conway | last2=Sloane | first2=Neil J. A. | author2-link=Neil Sloane | year=1999 | title=Sphere Packings, Lattices and Groups | publisher=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}}