  Using the SU($N$) representation of the group theory, we derive the general
form of the spin swapping operator for the quantum Heisenberg spin-$s$ systems.
We further prove that such a spin swapping operator is equal to the spin
singlet pairing operator under the partial transposition. For SU(2) invariant
states, it is shown that the expectation value of the spin swapping operator
and its generalizations, the permutations, can be used as an entanglement
witness, especially, for the formulation of observable conditions of
entanglement.
