Example 4

4d1
-1 1 1 1

4
0 0 1 0
0 0 0 1
0 1 0 0
1 0 0 0

Used Programs

Solution

1. Write the two matrices in a file 'g' and add

            #g2
         

   as top line. Now 'g' has bravais_TYP as format.
2. Call

            Normalizer g > gn
         

   to produce a file 'gn' containing G, a basis of the space of invariant quadratic forms of G, generators for the normalizer of G in GL₄(Z), and the order of G. Note 'gn' has bravais_TYP as format.
3. Call

            Vector_systems gn > gout
         

   to write a set of representatives of the vector systems under the normalizer action into the file 'out'. It turs out that the order of the extension group is 8 and that there are 6 orbits, i. e. one has 6 affine classes going with the Z-class of G.
4. For each of the 6 cocycles write it into a file 'coz' and call

            Extract -r gn coz > g.i
         

   to construct the affine classes in the Z-class of G in standard form (i.e. the translation lattice is Zⁿ):

   g.1

   g.2

   g.3

   g.4

   g.5

   g.6

5. Call

            Tr_bravais gn > gn_tr
         

   to get the transposed group 'gn_tr of 'g'.
6. Call

            Sublattices -b gn_tr > gl
         

   to write the matrices representing the bases of the G^tr-sublattices of Z⁴ of 2-power index into file 'gl'. (Note G and G^tr have order 2⁶.) These sublattices are ordered by inclusion and form a chain:

   L1 = Z⁴ > L2 > L3 > L4 > 2 L1

7. For each inverse write it into a file 'inv' call

            Tr inv > l.i
         

   to transpose it. So we get the G-invariant superlattices of Zⁿ:

   l.1 = Zⁿ < l.2 < l.3 < l.4 < 1/2 Zⁿ

8. The following is trivial:
   • The first extension splits for every superlattice.
   • The first extension splits and the other extensions don't split for 1/2 Zⁿ.
   So we only have to consider 'g.2' to 'g.6' with the lattices 'l.2' to 'l.4'.
9. Enlarge the matrices in the files 'l.2', 'l.3' and 'l.4', i.e. change

            4       /2              % tranposed of 1-th matrix of inv
             2  0  0  1
             0 -2  0 -1
             0  0 -2 -1
             0  0  0 -1
         

   to

             5       /2              % tranposed of 1-th matrix of inv
              2  0  0  1 0
              0 -2  0 -1 0
              0  0 -2 -1 0
              0  0  0 -1 0
              0  0  0  0 2
         

10. To decide for which superlattice each extension splits, one has to add the translations of a superlattice 'l.j' to a spacegroup 'g.i' and transform this group such that the translation lattice is Zⁿ (Cf. Standard_affine_form). This can be done in one step by calling:

             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Conj_bravais -i g.i l.j > g.i.j ;
                done ;
             done
          

    We get the following spacegroups:

    g.2.2	g.2.3	g.2.4
    g.3.2	g.3.4	g.3.4
    g.4.2	g.4.3	g.4.4
    g.5.2	g.5.3	g.5.4
    g.6.2	g.6.3	g.6.4

11. For each of these groups, extract the cocycle and the point group:

             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Extract -p g.i.j > pg.i.j ;
                   Extract -c g.i.j > cg.i.j ;
                done ;
             done
          

    We get the following files:

    pg.2.2 pg.2.3 pg.2.4 pg.3.2 pg.3.4 pg.3.4 pg.4.2 pg.4.3 pg.4.4 pg.5.2 pg.5.3 pg.5.4 pg.6.2 pg.6.3 pg.6.4

    cg.2.2 cg.2.3 cg.2.4 cg.3.2 cg.3.4 cg.3.4 cg.4.2 cg.4.3 cg.4.4 cg.5.2 cg.5.3 cg.5.4 cg.6.2 cg.6.3 cg.6.4

12. Call

             for i in 2 3 4 5 6 ; do
                for j in 2 3 4 ; do
                   Vector_systems -i pg.$i.$j cg.$i.$j ;
                done ;
             done
          

    to identify the spacegroups in the files 'g.i.j'. The name is 0 iff the extension splits. We get the following output:

             Name for the 1-th extension in pg.2.2: 1
             Name for the 1-th extension in pg.2.3: 1
             Name for the 1-th extension in pg.2.4: 0
             Name for the 1-th extension in pg.3.2: 2
             Name for the 1-th extension in pg.3.3: 0
             Name for the 1-th extension in pg.3.4: 0
             Name for the 1-th extension in pg.4.2: 2
             Name for the 1-th extension in pg.4.3: 0
             Name for the 1-th extension in pg.4.4: 0
             Name for the 1-th extension in pg.5.2: 3
             Name for the 1-th extension in pg.5.3: 1
             Name for the 1-th extension in pg.5.4: 0
             Name for the 1-th extension in pg.6.2: 0
             Name for the 1-th extension in pg.6.3: 0
             Name for the 1-th extension in pg.6.4: 0
          

Remarks

• Step 3 and 4 can be done in one step with Vector_systems/Extensions -S.
• Minimal klassengleich supergroups of spacegroups can also be calculated with KSupergroups.