Get the irreducible representations for a space-group represented on non-standard basis.

[hongyi-zhao]

See my following example:

gap> SGGenSet227me;
[ [ [ 0, -1, 0, 1/2 ], [ 0, 0, -1, 1/2 ], [ -1, 0, 0, 1/2 ], [ 0, 0, 0, 1 ] ], [ [ -15/4, 29/4, -15/4, -15/16 ], [ -33/8, 55/8, -25/8, -25/32 ], 
      [ -25/8, 55/8, -33/8, -41/32 ], [ 0, 0, 0, 1 ] ] ]
gap> S1:=SpaceGroupOnLeftIT(3,227);
SpaceGroupOnLeftIT(3,227,'2')
gap> S2:=AffineCrystGroupOnLeft(SGGenSet227me);
<matrix group with 2 generators>
gap> conj:=AffineIsomorphismSpaceGroups(S2, S1);
[ [ 5/2, 3, 5/2, -5 ], [ 5/2, 5/2, 3, -21/4 ], [ 3, 5/2, 5/2, -5 ], [ 0, 0, 0, 1 ] ]
gap> S2^(conj^-1)=S1;
true

Here, S1 is the 3d space group 227 given in ITA, then how can I use your package to compute the irreducible representations of S2?

Regards,
Zhao

[lan496]

When you have symmetry operations in a non-standard basis, I think spgrep.get_spacegroup_irreps_from_primitive_symmetry can compute irreps for the space group.

[hongyi-zhao]

If I understand correctly, the above method first converts the given representation onto a primitive cell and then computes all irreducible representations of it up to an arbitrary unitary transformation. If so, the resulting irrep is corresponding to the primitive cell, instead of the initial space group representation. Any more hints/comments will be helpful.

By the way, could you give an example using the generators I provided above to do the corresponding computation?

[lan496]

The above function does not rely on a primitive cell.
For generators, it will be easy to tabulate symmetry operations from given generators. Then, the tabulated symmetry operations can be passed to the function.

[hongyi-zhao]

The above function does not rely on a primitive cell.

Then, what's the implication of primitive_symmetry used in the function name?

For generators, it will be easy to tabulate symmetry operations from given generators. Then, the tabulated symmetry operations can be passed to the function.

It is still difficult for me to figure out the concrete computation steps with your package. Any more hints will be grateful.

[lan496]

Then, what's the implication of primitive_symmetry used in the function name?

"primitive" indicates rotations and translations are based on a primitive basis.

It is still difficult for me to figure out the concrete computation steps with your package. Any more hints will be grateful.

Sorry, I do not have a concrete snippet for this purpose.
The following code will be shown how to generate group from generators.

from queue import Queue

def traverse(
    generators: list[Operation],
) -> list[Operation]:
    """
    Generate all coset operations
    """
    coset = set()
    que = Queue()  # type: ignore
    identity = generators[0].identity(ndim=3)
    que.put(identity)

    while not que.empty():
        g = que.get()
        if g in coset:
            continue
        coset.add(g)

        for h in generators:
            gh = remainder1_symmetry_operation(g * h)
            que.put(gh)

    return ret