Niemeier lattice explained

In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24,which were classified by . gave a simplified proof of the classification. 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 theirroot 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 . 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 Niemeierlattice 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.

Lattice root system	Dynkin diagram	Coxeter number	G0	G1	G2	G∞
Leech lattice (no roots)		0	1	data-sort-value="8315553613086720000"		1	data-sort-value="99999"	Z^24
data-sort-value="A01"	A1^24		2	data-sort-value="16777216"	2^24	1	data-sort-value="244823040"		data-sort-value="4096"	2^12
data-sort-value="A02"	A2^12		3	data-sort-value="2176782336"	3!^12	2	data-sort-value="95040"		data-sort-value="729"	3^6
data-sort-value="A03"	A3^8		4	data-sort-value="110075314176"	4!^8	2	1344	data-sort-value="256"	4^4
data-sort-value="A04"	A4^6		5	data-sort-value="2985984000000"	5!^6	2	120	data-sort-value="125"	5^3
data-sort-value="A05"	A5^4D4		6	data-sort-value="51597803520000"	6!^4(2^34	)	2	24	72
data-sort-value="D04"	D4^6		6	data-sort-value="50096498540544"	(2^34!)^6	3	720	data-sort-value="64"	4^3
data-sort-value="A06"	A6^4		7	data-sort-value="645241282560000"	7!^4	2	12	data-sort-value="49"	7^2
data-sort-value="A07"	A7^2D5^2		8	data-sort-value="5992989327360000"	8!^2(2^45	)^2	2	4	32
data-sort-value="A08"	A8^3		9	data-sort-value="47784725839872000"	9!^3	2	6	27
data-sort-value="A09"	A9^2D6		10	data-sort-value="303395084697600000"	10!^2(2^56	)	2	2	20
data-sort-value="D06"	D6^4		10	data-sort-value="281792804290560000"	(2^56!)^4	1	24	16
data-sort-value="E06"	E6^4		12	data-sort-value="139314069504000"	(2^73^45)^4	2	24	9
data-sort-value="A11"	A11D7E6		12	data-sort-value="8009630236016640000"	12!(2^67	)(2^73^45)	2	1	12
data-sort-value="A12"	A12^2		13	data-sort-value="38775788043632640000"	13!^2	2	2	13
data-sort-value="D08"	D8^3		14	data-sort-value="137464791995252740000"	(2^78!)^3	1	6	8
data-sort-value="A15"	A15D9		16	data-sort-value="1943670270606704600000"	16!(2^89	)	2	1	8
data-sort-value="A17"	A17E7		18	data-sort-value="18586346962676613120000"	18!(2^103^45.7)	2	1	6
data-sort-value="D10"	D10E7^2		18	data-sort-value="15658098963209256960000"	(2^910!)(2^103^45.7)^2	1	2	4
data-sort-value="D12"	D12^2		22	data-sort-value="962351733103908618240000"	(2^1112!)^2	1	2	4
data-sort-value="A24"	A24		25	data-sort-value="15511210043330985984000000"	25!	2	1	5
data-sort-value="D16"	D16E8		30	data-sort-value="477676405704303732326400000"	(2^1516!)(2^143^55^27)	1	1	2
data-sort-value="E08"	E8^3		30	data-sort-value="338214937605319950336000000"	(2^143^55^27)^3	1	6	1
data-sort-value="D24"	D24		46	data-sort-value="5204698426366666226930810880000"	2^2324!	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. (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.

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 A₂¹²are 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.

See also

• Umbral moonshine
• Smith Minkowski Siegel mass formula#Dimension n = 24

References

• • Book: Conway . J. H. . Neil Sloane . Sloane . N. J. A. . Sphere Packings, Lattices, and Groups . registration . 3rd . 1998 . 0-387-98585-9 . Springer-Verlag. John H. Conway .
  • Niemeier. Hans-Volker . Definite quadratische Formen der Dimension 24 und Diskriminate 1. . In German . . 5 . 1973 . 142–178 . 0316384 . 10.1016/0022-314X(73)90068-1 . 2. 1973JNT.....5..142N . free .
• English translation in

External links

• Aachen University lattice catalogue