[讨论]偏序集运算和格初步

cjsh

http://new.ssreader.com/ebook/read_10069002.html

cjsh

[这个贴子最后由cjsh在 2011/07/20 02:57pm 第 1 次编辑]

http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html
The root lattices An and their duals for 1 ≤n ≤24
The root lattices Dn and their duals for 1 ≤n ≤24
The root lattices En and their duals for 6 ≤n ≤8
The laminated lattices Λn for 1 ≤n ≤24 including the 16-dimensional Barnes-Wall lattice Λ16 and the Leech lattice Λ24
The Kappa-lattices Kn and their duals for 7 ≤n ≤13 including the Coxeter-Todd lattice K12
The perfect lattices up to dimension 7
The 3-dimensional Bravais lattices
Various interesting lattices in dimensions 20, 24, 28, 32, 40, 80, 105 including, e.g., some of the densest known lattices in dimension 32.

Given a family name X as a string which is one of "A", "B", "C", "D", "E", "F", "G", "Kappa" or "Lambda", together with an integer n, construct a lattice subject to the following specifications:
A:The root lattice An which is the zero-sum lattice in Qn + 1.
B:n ≥2: The root lattice Bn which is the standard lattice of dimension n.
C:n ≥3: The root lattice Cn which is the even sublattice of Zn and is equal to Dn.
D:n ≥3: The root lattice Dn which is the even sublattice of Zn, also called the checkerboard lattice.
E:6 ≤n ≤8: The root lattice En, also called Gosset lattice.
F:n = 4: The root lattice F4 which is equal to D4.
G:n = 2: The root lattice G2 which is equal to A2.
Kappa:1 ≤n ≤13: The Kappa-lattice Kn. For n = 12 this is the Coxeter-Todd lattice.
Lambda:1 ≤n ≤31: The laminated lattice Λn. For n = 16 this is the Barnes-Wall lattice, for n = 24 the Leech lattice.

cjsh

> B := RMatrixSpace(IntegerRing(), 2, 3) ! [1,4,3, 3,2,8];
> B;
L1 := Lattice(B);
> L1;
L3 := LatticeWithBasis(B);
L3;
L4 := LatticeWithBasis(3, [1,2,3, 3,2,1]);
L4;

D := LatticeDatabase();
> NumberOfLattices(D, 6);
L := Lattice(D, 6, 10);
L;
NumberOfLattices(D, 4);
LL := Lattice(D, 4, 10);
LL;

L . 1 ;
L . 5 ;
LL. 2 ;
LL. 4 ;

Zero(L) ;
- 1 ;
4+5;
Norm(1);
Norm(8);

[1 4 3]
[3 2 8]
Lattice of rank 2 and degree 3
Determinant: 777
Factored Determinant: 3 * 7 * 37
Basis:
( 1  4  3)
( 2 -2  5)
Lattice of rank 2 and degree 3
Determinant: 777
Factored Determinant: 3 * 7 * 37
Basis:
(1 4 3)
(3 2 8)
Lattice of rank 2 and degree 3
Determinant: 96
Factored Determinant: 2^5 * 3
Basis:
(1 2 3)
(3 2 1)
19
Standard Lattice of rank 6 and degree 6
Determinant: 351
Factored Determinant: 3^3 * 13
Minimum: 4
Kissing Number: 42
Inner Product Matrix:
[4 1 2 2 2 2]
[1 4 2 2 2 2]
[2 2 4 1 2 2]
[2 2 1 4 2 2]
[2 2 2 2 4 1]
[2 2 2 2 1 4]
20
Standard Lattice of rank 4 and degree 4
Determinant: 1369
Factored Determinant: 37^2
Minimum: 4
Kissing Number: 4
Inner Product Matrix:
[ 4  1  2  1]
[ 1  4  1  0]
[ 2  1  6 -2]
[ 1  0 -2 20]
(1 0 0 0 0 0)
(0 0 0 0 1 0)
(0 1 0 0)
(0 0 0 1)
(0 0 0 0 0 0)
-1
9
1
8