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| Original file line number | Diff line number | Diff line change |
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| The Program Graph: | ||
| ================== | ||
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| This program calculates the "graph of inclusions" for a | ||
| geometric class. The following examples shall help you | ||
| to understand the output of this program. | ||
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| Example 1: | ||
| ========== | ||
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| 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 | ||
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| 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 | ||
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| (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. | ||
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| Example 2: | ||
| ========== | ||
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| 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 | ||
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| 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 | ||
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| 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. | ||
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