Current implementation may return wrong number of symmetrized tensors for higher rank...

[hongyi-zhao]

I noticed the following note in the source code:

spgrep/src/spgrep/tensor.py

Lines 88 to 91 in 0ecee48

	Note
	----
	Current implementation may return wrong number of symmetrized tensors for higher rank...
	"""

Can you give an example to illustrate this situation? How large a rank will trigger this problem, and is there a way to deal with it, such as performing some sort of decomposition to lower the rank?

Regards,
Zhao

[lan496]

I observe this issue for the fourth-order symmetric tensor for hexagonal (6/mmm).

spgrep/tests/test_tensor.py

Lines 113 to 122 in a928f36

	@pytest.mark.skip()
	def test_symmetric_tensor_hexagonal_rank4():
	symmetry = get_symmetry_from_database(hall_number=485)
	rotations = symmetry["rotations"]
	rep = get_representation_on_symmetric_matrix(rotations)
	tensors = get_symmetry_adapted_tensors(rep, rotations, rank=4, real=True)
	sym_tensors = apply_intrinsic_symmetry(tensors)
	assert len(sym_tensors) == 18

This issue seems to happen when input representation matrices have noise. The noise is amplified for higher ranks. Currently, I have no idea to properly handle it.

[hongyi-zhao]

I tested the above example and found that len(sym_tensors) is 19, so the test fails.

But to be honest, I am still not very clear about the physical (mathematical) meaning of the following test steps performed in this test:

    rep = get_representation_on_symmetric_matrix(rotations)
    tensors = get_symmetry_adapted_tensors(rep, rotations, rank=4, real=True)
    sym_tensors = apply_intrinsic_symmetry(tensors)

    assert len(sym_tensors) == 18

As a user of GAP, I try to check the representation of these rotations, aka, the point group, as follows:

gap> rotations:=[[[ 1,  0,  0],
>         [ 0,  1,  0],
>         [ 0,  0,  1]],
> 
>        [[-1,  0,  0],
>         [ 0, -1,  0],
>         [ 0,  0, -1]],
> 
>        [[ 1, -1,  0],
>         [ 1,  0,  0],
>         [ 0,  0,  1]],
> 
>        [[-1,  1,  0],
>         [-1,  0,  0],
>         [ 0,  0, -1]],
> 
>        [[ 0, -1,  0],
>         [ 1, -1,  0],
>         [ 0,  0,  1]],
> 
>        [[ 0,  1,  0],
>         [-1,  1,  0],
>         [ 0,  0, -1]],
> 
>        [[-1,  0,  0],
>         [ 0, -1,  0],
>         [ 0,  0,  1]],
> 
>        [[ 1,  0,  0],
>         [ 0,  1,  0],
>         [ 0,  0, -1]],
> 
>        [[-1,  1,  0],
>         [-1,  0,  0],
>         [ 0,  0,  1]],
> 
>        [[ 1, -1,  0],
>         [ 1,  0,  0],
>         [ 0,  0, -1]],
> 
>        [[ 0,  1,  0],
>         [-1,  1,  0],
>         [ 0,  0,  1]],
> 
>        [[ 0, -1,  0],
>         [ 1, -1,  0],
>         [ 0,  0, -1]],
> 
>        [[ 0, -1,  0],
>         [-1,  0,  0],
>         [ 0,  0, -1]],
> 
>        [[ 0,  1,  0],
>         [ 1,  0,  0],
>         [ 0,  0,  1]],
> 
>        [[-1,  0,  0],
>         [-1,  1,  0],
>         [ 0,  0, -1]],
> 
>        [[ 1,  0,  0],
>         [ 1, -1,  0],
>         [ 0,  0,  1]],
> 
>        [[-1,  1,  0],
>         [ 0,  1,  0],
>         [ 0,  0, -1]],
> 
>        [[ 1, -1,  0],
>         [ 0, -1,  0],
>         [ 0,  0,  1]],
> 
>        [[ 0,  1,  0],
>         [ 1,  0,  0],
>         [ 0,  0, -1]],
> 
>        [[ 0, -1,  0],
>         [-1,  0,  0],
>         [ 0,  0,  1]],
> 
>        [[ 1,  0,  0],
>         [ 1, -1,  0],
>         [ 0,  0, -1]],
> 
>        [[-1,  0,  0],
>         [-1,  1,  0],
>         [ 0,  0,  1]],
> 
>        [[ 1, -1,  0],
>         [ 0, -1,  0],
>         [ 0,  0, -1]],
> 
>        [[-1,  1,  0],
>         [ 0,  1,  0],
>         [ 0,  0,  1]]];;
gap> 
gap> P:=Group(rotations);;
gap> char:=Irr(P);;
gap> rep:=IrreducibleRepresentationsDixon(P, char:unitary);;
gap> List(rep, r -> List(P, g -> g^r));
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      [ [ -1/2, -1/2*E(12)^7+1/2*E(12)^11 ], [ -1/2*E(12)^7+1/2*E(12)^11, 1/2 ] ], [ [ -1/2, 1/2*E(12)^7-1/2*E(12)^11 ], [ 1/2*E(12)^7-1/2*E(12)^11, 1/2 ] ], 
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      [ [ -1/2, 1/2*E(12)^7-1/2*E(12)^11 ], [ 1/2*E(12)^7-1/2*E(12)^11, 1/2 ] ], [ [ -1/2, -1/2*E(12)^7+1/2*E(12)^11 ], [ -1/2*E(12)^7+1/2*E(12)^11, 1/2 ] ], 
      [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1/2, 1/2*E(12)^7-1/2*E(12)^11 ], [ 1/2*E(12)^7-1/2*E(12)^11, 1/2 ] ], 
      [ [ -1/2, -1/2*E(12)^7+1/2*E(12)^11 ], [ -1/2*E(12)^7+1/2*E(12)^11, 1/2 ] ], [ [ 1, 0 ], [ 0, -1 ] ], 
      [ [ -1/2, 1/2*E(12)^7-1/2*E(12)^11 ], [ 1/2*E(12)^7-1/2*E(12)^11, 1/2 ] ], [ [ -1/2, -1/2*E(12)^7+1/2*E(12)^11 ], [ -1/2*E(12)^7+1/2*E(12)^11, 1/2 ] ], 
      [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1/2, 1/2*E(12)^7-1/2*E(12)^11 ], [ 1/2*E(12)^7-1/2*E(12)^11, 1/2 ] ] ] ]

But this is very different to the restults given by the rep variable of your code, as shown below:

In [31]: rep
Out[31]: 
array([[[ 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.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 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.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 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.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 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.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
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        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          1.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , -1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  1.        ]],

       [[ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  0.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  0.        ,
         -1.        ,  0.        ],
        [ 1.41421356,  0.        ,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]],

       [[ 1.        ,  1.        ,  0.        ,  0.        ,
          0.        , -1.41421356],
        [ 0.        ,  1.        ,  0.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  1.        ,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        ,  1.        ,
         -1.        ,  0.        ],
        [ 0.        ,  1.41421356,  0.        ,  0.        ,
          0.        , -1.        ]]])

If you are willing to provide a more intuitive explanation of these results given by your code, I would be willing to conduct more GAP based comparative studies in order to pinpoint the cause of the problem you are referring to.

Anyway, the point group here is very simple, as shown below:

gap> IdGroup(P);
[ 24, 14 ]
gap> StructureDescription(P);
"C2 x C2 x S3"

So, I don't understand why such a simple finite unimodular matrtix group can trigger such a strange problem.

Regards,
Zhao

[lan496]

In this test, I consider the action of rotations on 3x3 symmetric matrices (for example, elastic constants). The procedure to generate representation matrices is described in an example page: https://spglib.github.io/spgrep/examples/symmetry_adapted_tensor.html.

[hongyi-zhao]

#100 (comment)
This issue seems to happen when input representation matrices have noise. The noise is amplified for higher ranks. Currently, I have no idea to properly handle it.

If I understand you correctly, you are referring to the cumulative error caused by decimal number approximations. As far as I know, it's a difficult thing to do float point computation in group theory. Strictly speaking, it is impossible to conduct exact computational group theory calculation over the real number field, so GAP only has extensive supports on the complete Cyclotomic field.

See here for the related discussion.

#100 (comment)
In this test, I consider the action of rotations on 3x3 symmetric matrices (for example, elastic constants). The procedure to generate representation matrices is described in an example page: https://spglib.github.io/spgrep/examples/symmetry_adapted_tensor.html.

I'll try to see if I can understand and reproduce it in GAP.

One thing that puzzles me is that your example page doesn't use any imprecise numbers, such as approximate floating point representations of irrational numbers, so this example cannot reflect the noise problem you've mentioned above.

[hongyi-zhao]

#100 (comment)
This issue seems to happen when input representation matrices have noise. The noise is amplified for higher ranks. Currently, I have no idea to properly handle it.

I'm not sure if using higher precision floating point numbers would help avoid this problem.