On a Complete Basis in the Space of Rotationally Invariant Operators of $N$ Quantum Spins $1/2$

Systems of quantum spins $1/2$ with isotropic Heisenberg interaction play an important role in physics. In studying such systems, it may be useful to have a complete, yet non-overcomplete, basis of operators each of which has the symmetry of the Hamiltonian, i.e., is invariant with respect to rotations (global $\mathrm {SU}(2)$ transformations of the Pauli matrices). This paper presents an algorithm for constructing such a basis. The algorithm is implemented in Wolfram Mathematica.