Coxeter–Todd lattice
Encyclopedia
In mathematics, the Coxeter–Todd lattice K12, discovered by , is a the 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 similar to the Barnes–Wall lattice.
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,
and all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
The Coxeter–Todd lattice is described in detail in and .
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
fixed by a certain automorphism of order 3, and is similar to the Barnes–Wall lattice.
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
In mathematics, a complex reflection group is a group acting on a finite-dimensional complex vector space, that is generated by complex reflections: non-trivial elements that fix a complex hyperplane in space pointwise...
(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,
and all of them other than the Coxeter–Todd lattice have a root system of maximal rank 12.
The Coxeter–Todd lattice is described in detail in and .
External links
- Coxeter–Todd lattice in Sloane's lattice catalogue