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Leech Lattice

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A 24-dimensional Euclidean lattice. An automorphism of the Leech lattice modulo a center of two leads to the Conway group Co_1. Stabilization of the one- and two-dimensional sublattices leads to the Conway groups Co_2 and Co_3, the Higman-Sims group HS and the McLaughlin group McL.

Both the Higman-Sims graph and McLaughlin graph can be constructed by picking particular triangles in the Leech lattice, taking as graph vertices lattice points at a certain distance form each triangle vertex, and connecting vertices by an edge if they are a certain distance apart (Conway and Sloane 1993; Gaucher 2013; Brouwer and van Maldeghem 2022, pp. 303 and 338). The Conway graph on 2300 vertices may also be constructed from the Leech lattice (Brouwer and van Maldeghem 2022, pp. 365-366).

The Leech lattice appears to be the densest hypersphere packing in 24 dimensions, and results in each hypersphere touching 196560 others. The number of vectors with norm n in the Leech lattice is given by

theta(n)=(65520)/(691)[sigma_(11)(n)-tau(n)],
(1)

where sigma_(11) is the divisor function giving the sum of the 11th powers of the divisors of n and tau(n) is the tau function (Conway and Sloane 1993, p. 135). The first few values for n=1, 2, ... are 0, 196560, 16773120, 398034000, ... (OEIS A008408). This is an immediate consequence of the theta function for Leech's lattice being a weight 12 modular form and having no vectors of norm two.

theta(n) has the theta series

f(q)=[E_4(q)]^3-720q^2product_(m=1)^(infty)(1-q^(2m))^(24)
(2)
=[E_4(q)]^3-720q^2(q^2,q^2)_infty^(24)
(3)
=[1+240sum_(m=1)^(infty)sigma_3(m)q^(2m)]^3-720q^2product_(m=1)^(infty)(1-q^(2m))^(24)
(4)
=1+196560q^4+16773120q^6+3980034000q^8+...,
(5)

where E_4(q) is the Eisenstein series, which is the theta series of the E_8 lattice (OEIS A004009), (a,q)_infty is a q-Pochhammer symbol, and f(q) can be written in closed form in terms of Jacobi elliptic functions as

f(q)=1/8[theta_2^8(q)+theta_3^8(q)+theta_4^8(q)]-(45)/(16)theta_2^8(q)theta_3^8(q)theta_4^8(q).
(6)

Properties of the Leech lattice are implemented in the Wolfram Language as LatticeData["Leech", prop].

See also

Barnes-Wall Lattice, Conway Graphs, Conway Groups, Coxeter-Todd Lattice, Eisenstein Series, Higman-Sims Graph, Higman-Sims Group, Hypersphere, Hypersphere Packing, Kissing Number, McLaughlin Graph, McLaughlin Group, Tau Function, Theta Series

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References

Brouwer, A. E. and van Maldeghem, H. Strongly Regular Graphs. Cambridge, England: Cambridge University Press, 2022.Conway, J. H. and Sloane, N. J. A. "The 24-Dimensional Leech Lattice Lambda_(24)," "A Characterization of the Leech Lattice," "The Covering Radius of the Leech Lattice," "Twenty-Three Constructions for the Leech Lattice," "The Cellular of the Leech Lattice," and "Lorentzian Forms for the Leech Lattice." §4.11, Ch. 12, and Chs. 23-26 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 131-135, 331-336, and 478-526, 1993.Gaucher, A. P. "Leech Lattice." https://cp4space.hatsya.com/2013/09/12/leech-lattice/. Sep. 12, 2013.Leech, J. "Notes on Sphere Packings." Canad. J. Math. 19, 251-267, 1967.Sloane, N. J. A. Sequences A008408 and A004009/M5416 in "The On-Line Encyclopedia of Integer Sequences."Wilson, R. A. "Vector Stabilizers and Subgroups of Leech Lattice Groups." J. Algebra 127, 387-408, 1989.

Referenced on Wolfram|Alpha

Leech Lattice

Cite this as:

Weisstein, Eric W. "Leech Lattice." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LeechLattice.html

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