Yang-Baxter R-operators and parameter permutations

We present an uniform construction of the solution to the Yang-Baxter equation with the symmetry algebra sℓ(2) and its deformations: the q-deformation and the elliptic deformation or Sklyanin algebra. The R-operator acting in the tensor product of two representations of the symmetry algebra with arbitrary spins ℓ1 and ℓ2 is built in terms of products of three basic operators S1, S2, S3 which are constructed explicitly. They have the simple meaning of representing elementary permutations of the symmetric group S4, the permutation group of the four parameters entering the RLL-relation.