Leech lattice

From Academic Kids

In mathematics, the Leech lattice is a lattice Λ in R24 discovered by John Leech. It is the unique lattice 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). It seems to be expected that this configuration also gives the densest packing of balls in 24-dimensional space, but this is still open. Cohn and Kumar showed that it is the densest lattice packing in 24-dimensional space.

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

<math>a_1+a_2+\cdots+a_{24}\equiv 4a_1\equiv 4a_2\equiv\cdots\equiv4a_{24}\pmod{8}<math>

and the set of coordinates i such that ai belongs to any fixed residue class (mod 4) is a word in the binary Golay code.

The Leech lattice can also be constructed as <math>w^\perp/w<math> where w is the norm 0 vector

<math>(0,1,2,3,...,22,23,24; 70)<math>

in the 26-dimensional even Lorentzian unimodular lattice II 25,1.

The Leech lattice is highly symmetrical. Its automorphism group is the double cover of the Conway group Co1; its order is approximately 8.3×1018. Many other sporadic simple groups can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.

The covering radius of the Leech lattice is <math>\sqrt 2<math>; 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 <math>\sqrt 2<math> from all lattice points are called the deep holes of the Leech lattice. There are 23 orbits of them, and they correspond to the 23 Niemeier lattices other than the Leech lattice.

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.

See also

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, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.
  • Leech, John, Canad. J. Math. 16 (1964).
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