Antisymmetrize a rank 4 tensor

Question

Suppose I have a rank 4 tensor $T_{abcd}$, I can build a SparseArray to input some initial values like $T_{1435}=1$ (a huge amounts of elements are equal to 1) etc. Now I want to antisymmetrize this tensor, then automatically diagonal terms are all 0, and $T_{4135}=T_{1453}=T_{1345}=T_{3415}=T_{5431}=T_{1534}=-1$ and $T_{4153}=1$ etc. How can I achieve this?

PS: I have tried SymmetrizedArray, but it can not do something like {{1,2,3,4},{2,1,4,2}} -> 1.

Answer 1

You're almost there. You just need to add Antisymmetric[{1, 2, 3, 4}]:

sa = SymmetrizedArray[
  {{1, 4, 3, 5} -> 1},
  {5, 5, 5, 5}, 
  Antisymmetric[{1, 2, 3, 4}]
]

Then you can simply visually check with

Normal[sa] // MatrixForm

that by specifying just $T_{1435} = 1$ above, you automatically get $T_{4135}=T_{1453}=T_{1345}=T_{3415}=T_{5431}=T_{1534}=-1$ and $T_{4153}=1$ etc.