Leech Lattice

The Leech lattice is a 24-dimensional Euclidean point lattice. Its automorphism group, modulo its group center of order two, is the Conway group . In the normalization with minimum squared vector norm 4, the stabilizers of individual vectors of squared vector norm 4 and 6 are the Conway groups and , respectively. Stabilizing suitable pairs of vectors gives the Higman-Sims group HS and the McLaughlin group McL (Conway and Sloane 1993).

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 from each triangle vertex, and connecting vertices by a graph 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 gives the densest hypersphere packing in 24 dimensions (Cohn et al. 2017), with each hypersphere touching 196560 others. In this normalization, the number of vectors with squared vector norm in the Leech lattice is given by

(1)

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

has the theta series

					(2)
					(3)
					(4)
					(5)

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

(6)

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