A 24-dimensional Euclidean lattice. An automorphism of the Leech lattice modulo a center of two leads to the Conway
group .
Stabilization of the one- and two-dimensional sublattices leads to the Conway
groups
and
, 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
others. The number of vectors with 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 function for Leech's lattice being a weight 12 modular
form and having no vectors of norm two.
has the theta
series
(2)
| |||
(3)
| |||
(4)
| |||
(5)
|
where
is the Eisenstein series, which is the theta
series of the
lattice (OEIS A004009),
is a q-Pochhammer
symbol, and
can be written in closed form in terms of Jacobi elliptic functions as
(6)
|
Properties of the Leech lattice are implemented in the Wolfram Language as LatticeData["Leech", prop].