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. The automorphism group of the Coxeter–Todd lattice has order = = 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 and has 10 isometry classes: all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
Further reading The Coxeter–Todd lattice is described in detail in and .
References
External links Coxeter–Todd lattice in Sloane's lattice catalogue