The Leech lattice is the unique even (http://planetmath.org/EvenLattice) unimodular lattice of dimension (http://planetmath.org/Dimension2) 24 having no elements of norm 2. Its automorphism group (http://planetmath.org/EquivalentCode) is the largest Conway group (sometimes denoted by ). The quotient of by its center is called , a sporadic simple group.
1 Construction of the Leech lattice
2 Properties of the Leech lattice
1. The Leech lattice is an unimodular lattice; in other words:
2. Let . Then , , , , .
3. The automorphism group (http://planetmath.org/EquivalentCode) is the largest Conway group with order .
4. The group acts transitively (http://planetmath.org/LeftAction) on the sets , , . For , the imprimitivity blocks of the action of on are the sets where . The imprimitivity blocks of the action of on are sets of 48 vectors called . Any two distinct vectors in a are either or orthogonal, and are congruent (http://planetmath.org/QuotientGroup) modulo .
5. Any vector in is modulo to a vector in for one of . The imprimitivity blocks of the action of on these sets account for all classes (http://planetmath.org/EquivalenceClass) of :
- 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
|Date of creation||2013-03-22 18:43:23|
|Last modified on||2013-03-22 18:43:23|
|Last modified by||monster (22721)|