Coupling coefficients of SUn⊃SOnfor On-scalar microscopic theory of collective states

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Abstract
Boson realisations of all On-scalar states which span a subspace of the Hilbert space for the collective degrees of freedom of the nucleus and are labelled by irreducible representations of the chain of subgroups U3 contains/implies U2 contains/implies U1 of the complementary group Sp(6,R), are given. Two general expressions for isoscalar factors (reduced Wigner coefficients) of SUn contains/implies SOn needed for constructing On-scalar basis functions depending on the microscopic collective variables in the oscillator basis are derived. These isoscalar factors couple three symmetric representations of SUn(Un) (p10)*(p20)*(p30) to the representation (h1h2h3) of SUn subduced to the SOn(On)-scalar representation and, more generally, to the SOn irreducible representation (fff), which appears in the case of the closed shells.