Coupling Algorithm for $\mathrm{Sp}(3,R)$ Irreducible Representations

The nuclear symplectic model based on $\mathrm{Sp}(3,R)$ -- the smallest algebra that contains both the shell model Hamiltonian and the rotor algebra -- connects the microscopic shell model to collective rotational behavior and naturally extends the Elliot $\mathrm{SU(3)}$ model to multiple shells. However, $\mathrm{Sp}(3,R)$ is only an approximate symmetry of the nucleus which can be broken by spin-orbit interactions, tensor force interactions, and pairing. The Hamiltonians in most physical situations will break $\mathrm{Sp}(3,R)$ symmetry, causing their eigenstates to become linear combinations of symplectic irreducible representations (irreps). Calculations with those eigenstates will then involve multiple irreps. We report a computer algorithm for enumerating the irreps that arise from the coupling of two symplectice irreps and evaluating their multiplicities in the product. This should assist in performing such multi-irrep calculations and facilitate computing symplectic coupling coefficients.