Coxeter–Todd lattice


In mathematics, the Coxeter–Todd lattice K12, discovered by, 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.

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 with structure 6.PSU4.2, called the Mitchell group.
The genus 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 on 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.
,,,
½, ½, ½,
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.