Perfect lattice
Encyclopedia
In mathematics, a perfect lattice (or perfect form) is a lattice
in a positive definite real vector space, that is determined by the set S its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. Perfect lattices were introduced by .
proved that a lattice is extreme if and only if it is both perfect and eutactic.
The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by
1, 1, 1, 2, 3, 7, 33, 10916 . summarize the properties of perfect lattices of dimension up to 7.
verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete.
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in a positive definite real vector space, that is determined by the set S its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. Perfect lattices were introduced by .
proved that a lattice is extreme if and only if it is both perfect and eutactic.
The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by
1, 1, 1, 2, 3, 7, 33, 10916 . summarize the properties of perfect lattices of dimension up to 7.
verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete.