Shell-model representations of the microscopic version of the Bohr-Mottelson collective model

The structure of the irreducible collective spaces of the group $Sp(12,R)$, which many-particle nuclear states are classified according to the chain $Sp(12,R) \supset U(6) \supset SO(6) \supset SU_{pn}(3) \otimes SO(2) \supset SO(3)$ of the proton-neutron symplectic model (PNSM), is considered in detail. This chain of the PNSM was shown to correspond to a microscopic shell-model version of the Bohr-Mottelson collective model. The construction of the relevant shell-model representations of the $Sp(12,R)$ group along this chain is considered for three nuclei with varying collective properties and from different mass regions. It is shown that the $SU_{pn}(3)$ basis states of the $Sp(12,R)$ representations are always Pauli allowed for $\upsilon \geq \upsilon_{0}$, but organized in a different way into different $SO(6)$ shells. This is in contrast to the case of filling the levels of the standard three-dimensional harmonic oscillator and using the plethysm operation. Although the $SU_{pn}(3)$ multiplets with $\upsilon < \upsilon_{0}$ are not all Pauli forbidden, it is safe to discard them, as it was actually done in the practical applications.