CARAT Introduction / Programs / Vector_systems

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## Program Vector_systems

### Usage

    Vector_systems ['file1'] 'file2' ['file3'] [-n] [-i] [-t=n] [-v] [-C] [-H] [-F] [-S]

file1: matrix_TYP containing a presentation of the group (cf. Presentation)
file2: bravais_TYP containing the group (and its normalizer).
file3: (OPTIONAL) containing cocycles for identification.

### Description

Calculates representatives of the isomorphism classes of extension of the group in file2 with presentation in file1 with the natural lattice. Output lists representative vectorsystems. (Output can be used as input for Extract). Also the isomorphism type of the extension group is given.

### Options

     -n:      only calculates the number of extensions, without computing
              representatives. This option is much faster for big cohomology
              groups. WARNING: Try this option first for 2-groups in dimension
              greater than 4.
     -v:      verbose mode. Give some echoing to stderr to
              indicate a little what the program is doing.
     -i:      Identify the cozycles given in file3, ie. give
              the described space groups a name. CAUTION: The
              name will depend on the generating set of the group
              in file2 and the presentation in file1.
              Can be used to test isomorphism of space groups with equal
              point groups. The name is 0 iff the extension splits.
     -t=n:    Has an effect only if given with -i. Outputs the
              isomophism needed to transform the space group
              By default, only the linear part is calculated. To
              get a full transformation matrix, use -t=2.
     -C:      Ignore the operation of the normalizer, just work
              on the level of extensions.
     -H:      echo the isomorphism type of the cohomology group
              H^1(G,Q^n/Z^n) to stderr.
     -F:      Only construct those extensions which gie rise to
              torsion free space groups. Does not work in conjunction
              with -n.
     -S:      write corresponding space groups in files

### Remarks

THE PROGRAM RELIES HEAVILY ON THE FOLLOWING TWO FACTS:

- THE PRESENTATION GIVEN IN file1 IS INDEED A PRESENTATION. CHECKED IS ONLY THAT THE GIVEN GENERATORS FULFILL THE RELATIONS.
- THE NORMALIZER IN GLn(Z) FOR THE GROUP IN file2 IS CORRECT. THE PROGRAM MIGHT NOT FINISH OR GIVE WRONG ANSWERS OTHERWISE.

See also for Extract, Presentation, Same_generators and Zass_main.

'file1' can be left out, but the program is faster, if the presentation is given.

Vector_systems is a synonym for Extensions.

### Examples

1. Find all space groups with a given point group and decide for which superlattices each extension splits.
2. Find all Z-classes, affine classes and torsion free space groups in a given Q-class.

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