# Independent permutation symmetry of a tensor [Using TensorSymmetry command]

Suppose I have this tensor $$A_{ijkl} = \epsilon_{ik} \epsilon_{jl}+\epsilon_{il} \epsilon_{jk}$$. Now I want to find all the independent permutation symmetries of the indices of this tensor. The answer is $$(ij)$$,$$(kl)$$,$$(ik)(jl)$$, where by $$(ij)$$ I mean the tensor is symmetric under the permutation of $$i,j$$. Similarly $$(ik)(jl)$$ represents the multiplication of two disjoint cycles.

I am very new to computation and I want to implement this in Mathematica. Here's what I tried:

ss = TensorProduct[LeviCivitaTensor[2], LeviCivitaTensor[2]]; A = ss + TensorTranspose[ss, {Cycles[{{2, 4}}], 1}]; TensorSymmetry[A]


This yields the result:

{{Cycles[{{2, 4}}], 1}, {Cycles[{{1, 2}, {3, 4}}], 1}, {Cycles[{{1, 2, 3, 4}}], 1}}

Which when translated back into my notation reads $$(kl)$$, $$(ik) (jl)$$ and $$(ikjl)$$, but we can see that the results are not simplified. For example we can use $$(ik) (jl)$$ to simplify $$(ikjl)$$ as follows:

$$(ikjl) = (ikj) (jl) = (jik) (jl) = (ji) (ik) (jl)$$

So $$(ijkl)$$ when used with $$(ik)(jl)$$ reduces to $$(ij) \equiv (ji)$$ as wanted. I want to do this simplification using Mathematica, in other words I want Mathematica to use all the cycles in the list to reduce it to a simplified and independent one. Could someone please help me to implement this? Thanks in advance. On a side note I find manipulating symbolic tensors in Mathematica very difficult. Suggestion for packages that helps this kind of calculation is much appreciated. I know there exists packages like Ricci which are dedicated for GR related calculations, I want something more simple and accessible for tensor analysis.

I don't think your TensorProduct expression agrees with your notation. If you correct the definition of A you get your desired symmetries:
A = TensorTranspose[TensorProduct[LeviCivitaTensor[2],LeviCivitaTensor[2]],{1,3,2,4}] +