PARTITION of Minimal Classes into Dual-Minimal Classes
in Dimensions 2 to 4
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This file, devoted to numerical data for (well-rounded) minimal classes
in dimensions n=2,3,4, must be read after the file web_minclass.*
(*= dvi, ps or pdf). It is divided into three parts:
PART 1. n=2 or 3.
PART 2. n=4, s>=7.
PART 3. n=4, s<=6.
Results displayed in Parts 1 and 2 are proved results,
those of part 3 (still in construction) are only conjectural.
Before entering into details, we recall general features.
Minimal classes C are denoted by symbols Snxr, where $n$ is the dimension,
r the perfection rank, and x on e of the letters a,b,c,d,
according to the notation of the file Sdim4.gp, which can be downloaded
and read with an editor. (We have s=r except for S4a10=cl(D_4), where s^*=12.) )
This file contains a set of minimal vectors
which characterizes the class , and a matrix depending on k:=n(n+1)/2-r
parameters x_1,...,x_k$ (or x1,...,xk).
NOTATION: Snxr, Mnxr; for short, Snxr=S.
Because we have n<=4, the set S^* on a dual-minimal class
(defined only up to equivalence) is such that the scalar product x.y
(for x in S, y in S^* is equal to 0 or +-1; in particular the components
on the dual basis (e_i^*) of the class are 0 or +-1.
We denote by T_0 the set of pairs of non-zero vectors having components 0,+-1
on the e_i^*; we have |T_0|=(3^n-1)/2 (13 if n-3, 40 if n=4).
We denote by T^* its admissible subsets (those y with |x.y|<=1).
In each part we first list the minimal classes which are at the same time
a dual- minimal class (one part=classes); the list is empty for Part 3.
Otherwise, we first display |T^*| and then the orbit distribution under
the transpose of Aut(C), in the form [i_1,j_1],...,[i_l,j_l],
where i_1,... is the number of x in S orthogonal to any element of the orbit
and j_1,... is the cardinality of the orbit. The orbits are denoted
by o_1,...,o_l, and explicitly written down when they occur in the description
of S^* for some dual-minimal classes.
We write, say, S^*<=o_1\cup o_2 if o_1,o_2 are needed,
and S^*=n) contain a unique dual-eutactic lattice, which is
the eutactic lattice of the class: A_2 in S2a3, Z^2 in S2a2, and A_3 in S3a6.
S3a4. |T^*|=9. [2,6], [0,3]; S^*<=o_1,
o_1={e_1^*,e_2^*,e_3^*,e_1^*-e_2^*,e_1^*-e_3^*,e_2^*-e_3^*}.
S3a4^{(2)}: S^*={e_1^*,e_2^*-e_3^*} or permutations
S3a4^{(4)}: S^*={e_2^*,e_3^*,e_1^*-e_3^*,e_1^*-e_2^*} or perm.
S3a4^{(2)}: S^*=o_1 (similarity class of A_3^*).
S3a4^{(6)}: x_1=x_2=-2/3; S3a4^{(4)}: e.g., x_1=x_2\ne -2/3.
S3a4^{(4)} is weakly isodual and contains a unique isodual lattice,
the ccc lattice, defined over Q(sqrt(2)).
Dual-eutactic: A_3^* in S3a4^{(6)}, the ccc lattice in S3a4^{(4)}.
S3a3. |T^*|=13. [3,3], [1,6], [0,4]; S^*<=o_1={e_1^*,e_2^*,e_3^*}.
S3a3^{(1)}: S^*={e_1^*} or perm.
S3a3^{(2)}: S^*={e_1^*,e_2^*} or perm.
S3a3^{(3)}: S^*=o_1
S3a3^{(3)} is weakly isodual and contains a unique isodual lattice,
the isodual lattice Z^3.
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**************** PART 2 ****************
**************** n=4,s>=7 ****************
***********************************************************************
The one-part minimal classes are S4a10=cl(D_4) (isodual),
S4b10=cl(A_4) (s^*=5), S4a9 (isodual), S4b9 (s^*=2),
S4b8 and S4d7 (s^*=1). The first three (those with s^*>=n) contain a
unique dual-eutactic lattice, which is the eutactic lattice of the class:
D_4 in S4a10, A_4 in S4b10, and A_2 \otimes A_2 in S4a9.
S4a8. |T^*|=17. [5,4], [4,1](S3a4), [4,4](S3b4), [3,4](S3a3), [2,2], [2,2].
o_1={e_1^*,e_2^*,e_3^*,e_4^*}; S^*<=o_1.
S4a8^{(2)}: S^*={e_1^*,e_4^*} or S^*={e_2^*,e_3^*}.
S4a8^{(4)}: S^*=o_1.
S4a7. |T^*|=19. [5,1], [4,4](S3b4), [4,2](S3a4), [3,4], [3,1], [2,4],
[1,2], [1,1] o_1={e_4^*}, o_2=S_1\cup S_2, where
S_1={e_2^*,e_3^*-e_4^*}, S_2={e_3^*,e_2^*+e_4^*};
S^*=7: this belongs to S4a8^{(4)}, is non-eutactic and irrational.
***********************************************************************
**************** PART 3 ****************
**************** n=4,s<=6 ****************
***********************************************************************
FIRST TWO PARTS completed on February 14th, 2017
THIRD PART in CONSTRUCTION
Part 3 will contain a list of dual-minimal classes,
only presumably exhaustive.
Note that by the results of part 2, we have s^*\le 6 except for A_4^*
and for one dual-class contained in S4a4, with s^*=8 -- that of the lattice
L_8^* where L_8 is the dual-eutactic lattice of S4a8^{(4)}.
....................................
The KNOWN dual-eutactic lattices with s=6,5 or 4 are the strongly eutactic
lattices A_2\perp\A_2 in S4d6 (s^*=6), A_4^*\in S4a5 (s^*=10)
and Z^4 in S4a4 (s^*=4), plus L_8^* in S4a4 (s^*=8).
This list is expected to be exhaustive.
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**************** END OF FILE ****************
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