Independent components of a rank-3 antisymmetric array in dimension 4:

This is an array with symmetry:

These are its independent components:

Extract the values of the independent components:

In an array with no symmetry, all components are independent:

Independent components of a rank-4 array whose symmetry is generated by a cycle:

Same situation, but having the generator with negative phase:

Use a nontrivial symmetry with complex generators:

Independent components of a general 3-dimensional stiffness tensor in elasticity theory:

Independent components of a general 4-dimensional Riemann curvature tensor, without taking into account its cyclic symmetry:

A skew-symmetric array in dimension is zero if its depth is larger than the dimension, and hence there are no independent components:

Independent components under rank- antisymmetry in dimension dim are equivalent to the list of -subsets of Range[dim]:

Independent components under rank- symmetry in dimension dim are equivalent to the list of -multisets of Range[dim]:

Given an array with symmetry, the values of its independent components can be extracted in several forms:

The computation of independent components can be simulated using representatives of the orbits of components under Permute action: