Coxeter–Todd lattice

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In mathematics, the Coxeter–Todd lattice K12, discovered by Coxeter and Todd (1953), 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 (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.

The Coxeter–Todd lattice is described in detail in (Conway & Sloane 1999, section 4.9) and (Conway & Sloane 1983).

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