niemeier lattice

In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Niemeier. The best known example is the Leech lattice.

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

The Leech lattice (empty root system), Coxeter number 0.
A₁²⁴, Coxeter number 2.
A₂¹², Coxeter number 3.
A₃⁸, Coxeter number 4.
A₄⁶, Coxeter number 5.
A₅⁴D₄, Coxeter number 6.
D₄⁶, Coxeter number 6.
A₆⁴, Coxeter number 7.
A₇²D₅², Coxeter number 8.
A₈³, Coxeter number 9.
A₉²D₆, Coxeter number 10.
D₆⁴, Coxeter number 10.
E₆⁴, Coxeter number 12.
A₁₁D₇E₆, Coxeter number 12.
A₁₂², Coxeter number 13.
D₈³, Coxeter number 14.
A₁₅D₉, Coxeter number 16.
A₁₇E₇, Coxeter number 18.
D₁₀E₇², Coxeter number 18.
D₁₂², Coxeter number 22.
A₂₄, Coxeter number 25.
D₁₆E₈, Coxeter number 30.
E₈³, Coxeter number 30.
D₂₄, Coxeter number 46.

Properties

Some of the Niemeier lattices are related to sporadic simple groups. The Leech lattice is acted on by a double cover of the Conway group, and the lattices A₁²⁴ and A₂¹² are acted on the the Mathieu groups M₂₄ and M₁₂. The Niemeier lattices, other than the Leech lattice, correspond to the deep holes of the Leech lattice. This implies that the affine Dynkin diagrams of the Niemeier lattices can be seen inside the Leech lattice, when two points of the Leech lattice are joined by no lines when they have distance

\sqrt 4

, by 1 line if the have distance

\sqrt 6

, and by a double line if they have distance

\sqrt 8

Niemeier Lattice

Classification

Properties

Further reading