In mathematics, the Coxeter–Todd lattice K
12, discovered by , 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. The automorphism group of the Coxeter–Todd lattice has order 2
10·3
7·5·7=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
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = ...
. 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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of the Coxeter–Todd lattice was described by and has 10 isometry classes: all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
Construction
Based o
Nebeweb 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 and .
References
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External links
in Sloane's lattice catalogue
{{DEFAULTSORT:Coxeter-Todd lattice
Quadratic forms