Niemeier lattice

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 (1973). Venkov (1978) gave a simplified proof of the classification. Witt (1941) mentions that he found more than 10 such lattices, but gives no further details. One example of a Niemeier lattice is the Leech lattice found in 1967.

Classification

Niemeier lattices are usually labelled by the Dynkin diagram of their root lattice. Each Niemeier lattice can be constructed from its root lattice (except for the Leech lattice which has no roots) by adjoining elements known as glue vectors, as detailed in §16.1 of Conway & Sloane (1998). The Dynkin diagrams associated with a Niemeier lattice 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.

Row	Lattice root system	Coxeter number	G₀	G₁	G₂	G_∞
1	Leech lattice (no roots)	0	1	2Co₁	1	Z²⁴
2	A₁²⁴	2	2²⁴	1	M₂₄	2¹²
3	A₂¹²	3	3!¹²	2	M₁₂	3⁶
4	A₃⁸	4	4!⁸	2	1344	4⁴
5	A₄⁶	5	5!⁶	2	120	5³
6	A₅⁴D₄	6	6!⁴(2³4!)	2	24	72
7	D₄⁶	6	(2³4!)⁶	3	720	4³
8	A₆⁴	7	7!⁴	2	12	7²
9	A₇²D₅²	8	8!²(2⁴5!)²	2	4	32
10	A₈³	9	9!³	2	6	27
11	A₉²D₆	10	10!²(2⁵6!)	2	2	20
12	D₆⁴	10	(2⁵6!)⁴	1	24	16
13	E₆⁴	12	(2⁷3⁴5)⁴	2	24	9
14	A₁₁D₇E₆	12	12!(2⁶7!)(2⁷3⁴5)	2	1	12
15	A₁₂²	13	13!²	2	2	13
16	D₈³	14	(2⁷8!)³	1	6	8
17	A₁₅D₉	16	16!(2⁸9!)	2	1	8
18	A₁₇E₇	18	18!(2¹⁰3⁴5.7)	2	1	6
19	D₁₀E₇²	18	(2⁹10!)(2¹⁰3⁴5.7)²	1	2	4
20	D₁₂²	22	(2¹¹12!)²	1	2	4
21	A₂₄	25	25!	2	1	5
22	D₁₆E₈	30	(2¹⁵16!)(2¹⁴3⁵5²7)	1	1	2
23	E₈³	30	(2¹⁴3⁵5²7)³	1	6	1
24	D₂₄	46	2²³24!	1	1	2

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. The number on the left is the Coxeter number of the Niemeier lattice. The red index number in the node indicates the row of the associated table above.

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

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 ${\sqrt {8}}$ .

Niemeier lattices also correspond to the 24 orbits of primitive norm zero vectors w of the even unimodular Lorentzian lattice II_25,1, where the Niemeier lattice corresponding to w is w^⊥/w.

References

Chenevier, Gaëtan; Lannes, Jean (2014), Formes automorphes et voisins de Kneser des réseaux de Niemeier, arXiv:1409.7616, Bibcode:2014arXiv1409.7616C
Conway, J. H.; Sloane, N. J. A. (1998). Sphere Packings, Lattices, and Groups (3rd ed.). Springer-Verlag. ISBN 0-387-98585-9.
Ebeling, Wolfgang (2002) [1994], Lattices and codes, Advanced Lectures in Mathematics (revised ed.), Braunschweig: Friedr. Vieweg & Sohn, doi:10.1007/978-3-322-90014-2, ISBN 978-3-528-16497-3, MR 1938666
Niemeier, Hans-Volker (1973). "Definite quadratische Formen der Dimension 24 und Diskriminate 1" (In German). Journal of Number Theory. 5 (2): 142–178. Bibcode:1973JNT.....5..142N. doi:10.1016/0022-314X(73)90068-1. MR 0316384.
Venkov, B. B. (1978), "On the classification of integral even unimodular 24-dimensional quadratic forms", Akademiya Nauk Soyuza Sovetskikh Sotsialisticheskikh Respublik. Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 148: 65–76, ISSN 0371-9685, MR 0558941 English translation in Conway & Sloane (1998)
Witt, Ernst (1941), "Eine Identität zwischen Modulformen zweiten Grades", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 14: 323–337, doi:10.1007/BF02940750, MR 0005508, S2CID 120849019
Witt, Ernst (1998), Collected papers. Gesammelte Abhandlungen, Springer Collected Works in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-41970-6, ISBN 978-3-540-57061-5, MR 1643949

External links

Aachen University lattice catalogue