Leech lattice
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.
The construction of the Leech lattice below depends on the existence of the extended binary Golay code (for a construction of the latter, see miracle octad generator).
1 Construction of the Leech lattice
Let and assume we have constructed the Golay on . The Leech lattice is the set of all points
in where each is an integer, such that
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For some integer , we have for all ;
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For any integer , the set of coordinates is in ;
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.
2 Properties of the Leech lattice
1. The Leech lattice is an unimodular lattice; in other words:
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The set spans all of as an -vector space.
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For any , the scalar product is an integer.
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For any , the norm is an even integer.
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The volume of the fundamental parallelogram of is 1.
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 :
References
- 1 J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices, and Groups. Springer-Verlag, 1999.
Title | Leech lattice |
Canonical name | LeechLattice |
Date of creation | 2013-03-22 18:43:23 |
Last modified on | 2013-03-22 18:43:23 |
Owner | monster (22721) |
Last modified by | monster (22721) |
Numerical id | 7 |
Author | monster (22721) |
Entry type | Definition |
Classification | msc 20D08 |
Classification | msc 20B25 |
Classification | msc 11H56 |
Classification | msc 11H06 |
Classification | msc 51E10 |
Related topic | BinaryGolayCode |
Related topic | MiracleOctadGenerator |