Niemeier Lattice - Classification

Classification

Niemeier lattices are usually labeled by the Dynkin diagram of their root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number. (The Coxeter number, at least in these cases, is the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams.

The complete list of Niemeier lattices is given in the following table. In the table,

G₀ is the order of the group generated by reflections

G₁ is the order of the group of automorphisms fixing all components of the Dynkin diagram

G₂ is the order of the group of automorphisms of permutations of components of the Dynkin diagram

G_∞ is the index of the root lattice in the Niemeier lattice, in other words the order of the "glue code". It is the square root of the discriminant of the root lattice.

G₀×G₁×G₂ is the order of the automorphism group of the lattice

G_∞×G₁×G₂ is the order of the automorphism group of the corresponding deep hole.