Leech lattice

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In mathematics, the Leech lattice is a particular lattice Λ in 24-dimensional Euclidean space E²⁴ discovered by John Leech in 1965. (Ernst Witt discovered it in 1940, but did not publish his discovery; see his collected works (Witt 1998, p. 328) for details.)

Contents 1 Characterization 2 Constructions 3 Symmetries 4 Geometry 5 Applications 6 See also 7 References

[edit] Characterization

The Leech lattice Λ is the unique lattice in E²⁴ with the following list of properties:

• It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1.
• It is even; i.e., the square of the length of any vector in Λ is an even integer.
• The shortest length of any non-zero vector in Λ is 2.

The last condition means that unit balls centered at the points of Λ do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball (compare with 6 in dimension 2, as the maximum number of pennies which can touch a central penny; see kissing number).

Cohn and Kumar showed that it gives the densest lattice packing of balls in 24-dimensional space. Their results suggest, but do not prove, that this configuration also gives the densest among all packings of balls in 24-dimensional space.

Conway showed that the Leech lattice is isometric to the Dynkin diagram of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II_25,1.

[edit] Constructions

The Leech lattice can be explicitly constructed as the set of vectors of the form 2^−3/2(a₁, a₂, ..., a₂₄) where the a_i are integers such that

and for each fixed residue class modulo 4, the 24 bit word, whose 1's correspond to the coordinates i such that a_i belongs to this residue class, is a word in the binary Golay code.

The Leech lattice can also be constructed as where w is the norm 0 vector

in the 26-dimensional even Lorentzian unimodular lattice II_25,1. The existence of such an integral vector of norm zero relies on the fact that 1² + 2² + ... + 24² is a perfect square (in fact 70²); the number 24 is the only integer bigger than 1 with this property.

[edit] Symmetries

The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co₁; its order is 8,315,553,613,086,720,000. Many other sporadic simple groups can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.

[edit] Geometry

The covering radius of the Leech lattice is ; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits are the 23 Niemeier lattices other than the Leech lattice.

[edit] Applications

The vertex algebra of the conformal field theory describing bosonic string theory, compactified on the 24-dimensional quotient torus R²⁴/Λ and orbifolded by a two-element reflection group, provides an explicit construction of the Griess algebra that has the monster group as its automorphism group. This monster vertex algebra was also used to prove the monstrous moonshine conjectures. The binary Golay code, independently developed in 1949, is an application in coding theory.

[edit] See also

• Conway group
• Niemeier lattice
• Sphere packing
• E₈ lattice

[edit] References

• Conway, J. H.; Sloane, N. J. A. (1999). Sphere packings, lattices and groups. (3rd ed.) With additional contributions by E. Bannai, R. E. Borcherds, John Leech, Simon P. Norton, A. M. Odlyzko, Richard A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.
• Leech, J, 'Some sphere packings in higher space', Canad. J. Math. 16 (1964), 657-682.
• Leech, J, 'Notes on sphere packings', Canad. J. Math. 19 (1967), 251-267.
• Thompson, Thomas M.: "From Error Correcting Codes through Sphere Packings to Simple Groups", Carus Mathematical Monographs, Mathematical Association of America, 1983.
• Witt, Ernst (1998), Collected papers. Gesammelte Abhandlungen, Berlin, New York: Springer-Verlag, MR1643949, ISBN 978-3-540-57061-5 
• Griess, Robert L.: Twelve Sporadic Groups, Springer-Verlag, 1998.