Graph

The Program Graph:
==================

This program calculates the "graph of inclusions" for a
geometric class. The following examples shall help you
to understand the output of this program.



Example 1:
==========


File G:
-------
#g2
3       % generator
 1  0  0
 1 -1  0
 0  0 -1
3       % generator
 -1 1 0
 -1 0 0
  0 0 1
2^1  * 3^1   = 6 % order of the group       


Output of "Graph G":
--------------------
3       % graph for the arithmetic classes      (1)
 3 2 1                                          
 0 2 1                                          (2)
 1 0 3
There are 3 Z-Classes with 2 1 2 Space Groups!  (3)
1: 1 (2, 2^1)                                   (4)
1: 1 (4, 2^2)
1: 1 (3, 3^1)  2 (6, 3^1)                       (5)
2: 2 (2, 2^1)
2: 2 (4, 2^2)
1: 3 (18, 3^1)
1: 4 (9, 3^1)
2: 5 (9, 3^1)
3: 3 (2, 2^1)
3: 3 (4, 2^2)
3: 4 (3, 3^1)  5 (6, 3^1)
4: 1 (1, 3^1)
5: 2 (1, 3^1)
4: 4 (2, 2^1)
4: 4 (4, 2^2)
4: 4 (3, 3^1)  5 (6, 3^1)
5: 5 (2, 2^1)
5: 5 (4, 2^2)
5       % inclusions for all spacegroups
 3 1 1 1 0                                      (6)
 0 2 0 0 1
 0 0 2 1 1
 1 0 0 3 1
 0 1 0 0 2  

(1) The number in this line is the number of Z-classes.
(2) This matrix yields information about the graph
    of the arithmetic classes. There are 3 + 2 + 1 = 6
    maximal sublattices of a representative of the first
    Z-class which are invariant under this group. If you
    conjugate this representative with the sublattices, 
    you get 3 groups in the first Z-class, 2 groups in 
    the second Z-class and 1 group in the third Z-class.
(3) This line gives you the numbers of the affine classes
    in the various Z-classes.
    The affine classes are numbered in ascending order
    with respect to the Z-classes. So the first affine class
    of the third Z-class gets the number 4.
    The first affine class in each Z-class contains the
    symmorphic space groups.
(4) The first space group has 2 maximal k-subgroups of index
    2^1 which are conjugated under the affine normalizer
    of the spacegroup. These subgroups are in the first
    affine class.
(5) The first space group has 3 maximal k-subgroups of index
    3^1 which are conjugated under the affine normalizer
    of the spacegroup. These subgroups are in the first
    affine class. There are 2 maximal k-subgroups of index
    3^1 which are conjugated under the affine normalizer
    of the spacegroup. The translation lattices for all
    these subgroups are in one orbit under the stabilizer of
    the cocycle of the spacegroup, so we print the orbits
    in one line.
(6) This matrix gives you the numbers of orbits under the
    affine normalizer of a spacegroup on the maximal
    k-subgroups. There are 3 + 1 + 1 + 1 orbits for a
    representative of the first affine class. The groups
    in 3 of these orbits are in the first affine class.
    The groups in one orbit are in the second affine
    class, etc.



Example 2:
==========


File G:
-------
g2
3       % generator
 -1  0  0
  0 -1  0
  0  0 -1
3       % generator
  0  1 0
 -1 -1 0
  0  0 1
2^1  * 3^1   = 6 % order of the group  


Output of "Graph G":
--------------------
2       % graph for the arithmetic classes
 4 2
 1 2
There are 2 Z-Classes with 1 1 Space Groups!
1: 1 (2, 2^1)
1: 1 (4, 2^2)
1: 1 (3, 3^1)                                   (1)
1: 1 (3, 3^1)                                   (2)
1: 2 (6, 3^1)
2: 1 (3, 3^1)
2: 2 (2, 2^1)
2: 2 (4, 2^2)
2       % inclusions for all spacegroups
 4 1
 1 2            

In this example, there are two orbits each with 3 maximal
k-subgroups of a representative of the first affine class.
They are printed in separate lines ((1) and (2)) because
the translation lattices for these groups are NOT conjugated
under the stabilizer of the cocycle of a representative for
the first affine class.