Abstract. In nuclei with valence nucleons that are identical nucleons and occupy r number of j-orbits, there will be 2^r-1 number of multiple pairing (quasi-spin) SU(2) algebras with the generalized pair creation operator S₊ being a sum of single-j pair creation operators with arbitrary phases. Also, for each set of phases there will be a corresponding Sp(2Ω) algebra in U(2Ω) ⊃ Sp(2Ω); Ω = ∑ (2j+1)/2. Using this correspondence, derived is the condition for a general one-body operator of angular momentum rank k to be a quasi-spin scalar or a vector vis-a-vis the phases in S₊. These will give special seniority selection rules for electromagnetic transitions. We found that the phase choice advocated by Arvieu and Moszkowski gives pairing Hamiltonians having maximum correlation with well known effective interactions. All the results derived for identical fermion systems are shown to extend to identical boson systems such as sd, sp, sdg and sdpf interacting boson models (IBM's) with SU(2) → SU(1,1) and Sp(2\Omega) → SO(2Ω). Going beyond pairing, for a given set of oscillator orbits, there are multiple rotational SU(3) algebras both in shell model and IBM's. Different SU(3) algebras in IBM's are shown, using sdg IBM as an example, to give different geometric shapes.

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