Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ₂₄ in 24-dimensional Euclidean space. It was discovered by John Leech (1967). It may also have been discovered (but unpublished) by Ernst Witt in 1940.

History

Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than the Leech lattice. O'Connor & Pall (1944) discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, one of whose two even neighbors is the Leech lattice. The Leech lattice was discovered in 1965 by John Leech (1967, 2.31, p. 262), by improving some earlier sphere packings he found (Leech 1964).

Conway (1968) calculated the order of the automorphism group of the Leech lattice, and, working with John G. Thompson, discovered three new sporadic groups as a by-product: the Conway groups, Co₁, Co₂, Co₃. They also showed that four other (then) recently announced sporadic groups, namely, Higman-Sims, Suzuki, McLaughlin, and the Janko group J₂ could be found inside the Conway groups using the geometry of the Leech lattice. (Ronan, p. 155)