Niemeier lattice

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

1 Classification
2 The neighborhood graph of the Niemeier lattices
3 Properties
4 References
5 External links

[edit] 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.

[edit] The neighborhood graph of the Niemeier lattices

If L is an odd unimodular lattice of dimension 8n and M its sublattice of even vectors, then M is contained in exactly 3 unimodular lattices, one of which is L and the other two of which are even. (If L has a norm 1 vector then the two even lattices are isomorphic.) The Kneser neighborhood graph in 8n dimensions has a point for each even lattice, and a line joining two points for each odd 8n dimensional lattice with no norm 1 vectors, where the vertices of each line are the two even lattices associated to the odd lattice. There may be several lines between the same pair of vertices, and there may be lines from a vertex to itself. Kneser proved that this graph is always connected. In 8 dimensions it has one point and no lines, in 16 dimensions it has two points joined by one line, and in 24 dimensions it is the following graph:

Each point represents one of the 24 Niemeier lattices, and the lines joining them represent the 24 dimensional odd unimodular lattices with no norm 1 vectors. (Thick lines represent multiple lines.) The number on the right is the Coxeter number of the Niemeier lattice.

In 32 dimensions the neighborhood graph has more than a billion vertices.

[edit] 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 by 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 they have distance $\sqrt 6$ , and by a double line if they have distance $\sqrt 8$ .

[edit] References

Conway, J. H.; Sloane, N. J. A. (1998). Sphere Packings, Lattices, and Groups, 3rd ed., Springer-Verlag. ISBN 0-387-98585-9.

Niemeier, Hans-Volker (1973). "Definite quadratische Formen der Dimension 24 und Diskriminate 1." (In German). Journal of Number Theory 5: 142–178. DOI:10.1016/0022-314X(73)90068-1. MR 0316384.