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^*<o_1\cup o_2 if always a strict subset of o_1\cup o_2. 

The dual-minimal classes contained in a class C are written 
C^{(1)}, C^{(2)},..., with superscript is s^*, maybe C^{(1a)},C^{(1b)} 
if there are several dual-classes with s^*=1, etc. 
The existence of the dual-minimal classes that we list is always firmly 
established. Whether they exhaust all dual-minimal classes is not proved 
in Part 3. 
We also list the dual-eutactic lattices in each minimal-class, 
but again in part 3, the list is conjectural (some could have been missed). 

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****************                 PART 1                **************** 
****************                 n=2,3                 **************** 
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The one-part minimal classes are S2a3 (isodual), S2a2 (isodual), 
S3a6=cl(A_3) (s^*=4), S3a5 (s^*=2) and S3b4 (s^*=1). The first three 
(those with 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               **************** 
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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^*<o_1\cup o_2. 
      S4a7^{(1)}: S^*=o_1. 
      S4a7^{(2)}: S^*=S_1 or S_2 
      S4a7^{(3)}: S^*=o_1\cup S_1 or o_1\cup S_2 

S4b7. |T^*|=21. [5,3], [3,8](S3a3), [3,6](S2a3), [1,4]. 
                o_1={e_2^*,e_3^*,e_4^*}; S^*<=o_1. 
      S4b7^{(1)}: {e_2^*} or perm. on 2,3,4. 
      S4b7^{(2)}: {e_2^*,e_3^*} or perm. on 2,3,4. 
      S4b7^{(3)}: o_1. 

S4c7. |T^*|=18. [5,2], [4,4](S3b4), [3,4](S3a3), [2,8]. 
                o_1={e_3^*,e_4^*}; S^*<=o_1. 
      S4c7^{(1)}: {e_3^*} or {e_4^*}. 
      S4c7^{(2)}: o_1. 

Besides the strongly eutactic lattices, D_4\in S4a10, A_4\in S4b10, 
and A_2 tensor A_2\in S4a9, there exists only one more dual-eutactic lattice 
with s>=7: this belongs to S4a8^{(4)}, is non-eutactic and irrational.  


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****************                 PART 3                **************** 
****************                n=4,s<=6               **************** 
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             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)}. 

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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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