Example 10
Find the stabilizer of the sublattice generated by the columns of the matrix M below in the Bravais
group of the unit form F=I6.
   6
   1 3 1 0 0 0
   0 1 3 1 0 0
   0 0 1 3 1 0
   0 0 0 1 3 1
   1 0 0 0 1 3
   3 1 0 0 0 1
Used Programs
Aut_grp,
Bravais_catalog/Datei,
Elt,
Orbit,
Order
Solution
First Solution
   - 
      Call
      
         Bravais_catalog
      and input 6-1 as family symbol. Write the Bravais group 6-1_1_1
      onto file 'B' (i. e. calculate the bravais-groups, input filename 'B',
      choose "a selection of bravais-groups", and enter 1 for i and j).
      Now 'B' contains the Bravais group of I6.
- 
      Write the above matrix M on a file 'm' and call
      
         Elt m
      to make sure that the lattice spanned by the columns of M has full rank.
      We get:
         6       % ELT of matrix
          1 0 0 0 0  0
          0 1 0 0 0  0
          0 0 1 0 0  0
          0 0 0 1 0  0
          0 0 0 0 8  0
          0 0 0 0 0 40
      
- 
      Call
      
         Orbit -g -S m B > mB
      to write the orbit and a generating set for the stabilizer on
      the file 'mB'. The length of the orbit runs out to be 1920, and
      hence the order of the stabilizer is 24. Edit the file 'mB' by
      omitting the orbit and leaving the generators of the stabilizer (i.e. the last 23 matrices).
      By calling
         Order mB
      one sees that the order is 24 indeed. Now 'mB' contains a set of
      generators of the stabilizer.
Second Solution
   - 
      Prepare a file 'f' with the following content
      
         #3
         12 d0
         1
         12 d1
         1 1 1 1 1 1 0 0 0 0 0 0
         12x0
         0
         0 0
         0 0 0
         0 0 0 0
         0 0 0 0 0
         0 0 0 0 0 0
         1 3 1 0 0 0 0
         0 1 3 1 0 0 0 0
         0 0 1 3 1 0 0 0 0
         0 0 0 1 3 1 0 0 0 0
         1 0 0 0 1 3 0 0 0 0 0
         3 1 0 0 0 1 0 0 0 0 0 0
      and note that the Bravais group of the space of forms generated
      by these three matrices consists of the matrices
      diag(g,m-1 g m) with g in the stabilizer of the lattice
      generated by the columns of m in the Bravais group of I6.
- 
      Call
      
         Aut_grp f > Bf
      to get generators of the group described above in the file
      'Bf'.
 Note the stabilizer of the lattice generated by the colums of M can be
      extracted from the file 'Bf' in its action on the standard
      lattice and also in its action on the sublattice.
   
   last change: 11.09.2000