Tensor symmetric in a pair of indices and under the chage of pairs

[Elias Leite Mendonça]

Dear colleagues,

1. I am trying to define a rank-4 tensor with the following symmetry properties:
   Pα,β,μ,ν​

Symmetric in (αβ), symmetric in (μν), and symmetric under the exchange of the pairs (αβ)↔(μν).

I was able to impose the symmetry within each pair separately, but I could not manage to implement the symmetry under the exchange of the two pairs.

2. My second question is the following:
   I have managed to symmetrize an explicit tensor expression so that it satisfies the symmetries described above. However, is there a way to automate this process? That is, without having to symmetrize term by term and manually impose the pair exchange symmetry by brute force?

I have attached my attempt in the notebook.

I would appreciate any help or suggestions.

[Thomas Bäckdahl]

Hi!

You don't have to use just symmetric, or antisymmetric groups. You can specify any list of generators. xAct will automatically compute the corresponding strong generating set.
In your case you can write:
DefTensor[\[ScriptCapitalP][-\[Mu], -\[Nu], -\[Rho], -\[Sigma]], M, GenSet[Cycles[{1, 2}], Cycles[{3, 4}], Cycles[{1, 3}, {2, 4}]]]

Cycles[{1, 2}] corresponds to the first symmetric pair.
Cycles[{3, 4}] corresponds to the second symmetric pair.
Cycles[{1, 3}, {2, 4}] corresponds to the exchange symmetry.

All of the group theoretical aspects are handled by the xPerm package. If you need to you can look at the documentation for that package.

Regards
Thomas

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