Example 6

Used Programs

Solution

1. Call

            Bravais_catalog
         

   and input 1;1;1;1 as family symbol.
   It turns out that there are nine Z-classes of Bravais groups in the family only one of which is homogeneously decomposable. Though we might not need all of these groups explicitly, we write all of them in the file 'b'. Split 'b' up into nine files 'b1', 'b2', 'b3', 'b4', 'b5', 'b6', 'b7', 'b8' and 'b9', so that 'bi' contains 1;1;1;1_1_i .
2. Call

            Bravais_catalog
         

   and input 4-1 as family symbol. Write all Bravais groups in this family to a file 'B'. There are two groups in file 'B' now. We want the Bravais group of the form represented by the unit matrix, which is the first group. The idea is that any of the subgroups of this group is Z-equivalent (actually equal) to its transposed.
3. Call

            Bravais_inclusions B > Bin
            grep 1\;1\;1\;1 Bin
         

   We get the file 'Bin' and the output

            Symbol: 1;1;1;1  homogeneously d.: 1 zclass: 2
            Symbol: 1;1;1;1  homogeneously d.: 1 zclass: 1
            Symbol: 1;1;1;1  homogeneously d.: 1 zclass: 5
            Symbol: 1;1;1;1  homogeneously d.: 1 zclass: 9
         

   We find that the groups 1;1;1;1_1_i with i = 1, 2, 5, or 9 contain the unit matrix in their form space (at least up to Z-equivalence).
   Hence these groups are Z-equivalent to their transposed groups. Note that an odd number of groups still have to be paired. Hence one group must be Z-equivalent to itself, without fixing a form Z-equivalent to the identity form.
4. Call

            Tr_bravais b3 > b3t
            Bravais_type b3t > b3type
         

   to find that 1;1;1;1_1_3 and 1;1;1;1_1_6 are paired. Similarly one finds that 1;1;1;1_1_4 and 1;1;1;1_1_8 are paired and that 1;1;1;1_1_7 is paired to itself (without fixing I_4).
   Doing this we get the files b3t, b3type, b4t, and b4type.