Multiple multi-orbit pairing algebras in shell model and interacting boson models

In nuclei with valence nucleons are say identical nucleons and say these nucleons occupy several-$j$ orbits, then it is possible to consider pair creation operator $S_+$ to be a sum of the single-$j$ shell pair creation operators $S_+(j)$ with arbitrary phases, $S_+=\sum_j \alpha_j S_+(j); \alpha_j=\pm 1$. In this situation, it is possible to define multi-orbit or generalized seniority that corresponds to the quasi-spin $SU(2)$ algebra generated by $S_+$, $S_-=(S_+)^\dagger$ and $S_0=(\hat{n} -\Omega)/2$ operators; $\hat{n}$ is number operator and $\Omega=[\sum_j (2j+1)]/2$. There are now multiple pairing quasi-spin $SU(2)$ algebras. Also, the $\alpha_j$'s and the generators of the corresponding generalized seniority generating sympletic algebras $Sp(2\Omega)$ in $U(2\Omega) \supset Sp(2\Omega)$ have one-to-one correspondence. Using these, derived are the special seniority selection rules for electromagnetic transitions. A particular choice for $\alpha_j$'s as advocated by Arvieu and Moszkowski (AM) in the past gives pairing Hamiltonians having maximum correlation with well known effective interactions. The various results derived for identical fermion systems are shown to extend to identical boson systems with the bosons occupying several-$\ell$ orbits as for example in $sd$, $sp$, $sdg$ and $sdpf$ IBM's. The quasi-spin algebra here is $SU(1,1)$ and the generalized seniority quantum number is generated by $SO(2\Omega)$ in $U(2\Omega) \supset SO(2\Omega)$. The different $SO(2\Omega)$ algebras here will be important in the study of quantum phase transitions and order-chaos transitions in nuclei.