Program Extensions

Usage

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

Description

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 cocycles 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 GL_n(Z) FOR THE GROUP IN file2 IS CORRECT. THE PROGRAM MIGHT NOT FINISH OR GIVE WRONG ANSWERS OTHERWISE.

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

See also for Extract, Presentation, Same_generators and Zass_main.

Extensions is a synonym for Vector_systems.

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.