CARAT Introduction / Examples / Find the stabilizer of a sublattice in the Bravais group of the unit form F=I_6.
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 10 - Find the stabilizer of a sublattice in the Bravais group of the unit form I<sub>6</sub>.
Keywords: CARAT, crystallography, mathematics, algorithms, tables, algorithm, table, examples, example, Aut_grp, Bravais_catalog, Elt, Order, stabilizer, sublattice

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### 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

1. 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.
2. 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

3. 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

1. 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.
2. 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.

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last change: 11.09.2000