Pair-wise antisymmetric tensor

Question

I am trying to define a tensor $T$ which is antisymmetric under pair-wise exchange of its indices:

$T = \delta_{a],[b}\,\delta_{c],[d}\,\delta_{e],[f}\,\delta_{g],[h}$

where $T$ is antisymmetric under the exchanges $b \leftrightarrow c, d \leftrightarrow e, f \leftrightarrow g ,h \leftrightarrow a$.

I guess that defining $T$ as a Table will take too much memory. Maybe defining it as a function is better. In both cases I have no clue on how to do this in a clever, short way without writing down all possible exchanges by hand. Can you help?

Answer 1

Expanding on a comment by @kglr above, if you want a general form of the tensor with only the independent components, you can use SymmetrizedArray as follows:

antiSymmetricTensor[dim_, symbol_ : A] := 
 SymmetrizedArray[pos_ :> symbol @@ pos, ConstantArray[dim, 8],
  {Cycles[{#}], -1} & /@ Partition[RotateLeft@Range[8], 2]]

as a demonstration, here's the $i=1,j=2,k=3,l=2$ for $d=3$:

This is indeed in the subdomain of arrays with dimension dim and the specified symmetries:

With[{
  dim = 3,
  symbol = A,
  domain = Arrays[ConstantArray[dim, 8], Reals, 
    Antisymmetric /@ Partition[RotateLeft@Range[8], 2]]
  },
 Element[antiSymmetricTensor[dim, symbol], domain];
 Simplify[%, symbol[__] ∈ Reals]
 ]
(* True *)