How to Creat a Symbolic Rank 4 Symmetric tensor

Question

I would like to create a rank 4 symbolic tensor with this symmetries

(1) C_ijkl = C_jikl

(2) C_ijkl = C_ijlk

(3) C_ijkl = C_klij

is there any way to apply symmetry (3)?

symmetry (1) and (2) can be applied as:

cAr = SymmetrizedArray[pos_ -> c[pos], {3, 3, 3, 3}, {Symmetric[{1, 2}], 
Symmetric[{3, 4}]}]; cAr // MatrixForm

Answer 1

You can use the general language of symmetries (of which Symmetric and Antisymmetric are particular cases). Your symmetries (1), (2) and (3) correspond respectively to these generators:

sym = {
    {Cycles[{{1, 2}}], 1},
    {Cycles[{{3, 4}}], 1},
    {Cycles[{{1, 3}, {2, 4}}], 1}
};

The 1 means that there is no sign change. For example, the permutation symmetry of the Riemann tensor has -1 in the first two generators, corresponding to antisymmetries.

Then the symmetrized array would be, for dimension 3:

cAr = SymmetrizedArray[pos_ -> c[pos], {3, 3, 3, 3}, sym];
cAr // MatrixForm