Dual bases and some coupling coefficients for SU₄⊃ SU₂× SU₂, SU_n⊃ SO_nand Sp₄⊃ U₂

Abstract

General analytical expressions for isoscalar factors (reduced Wigner coefficients) for the couplings (p0) × (0q) to (λ 0 μ) and (p10) × (p20) to (λ ν 0) in the chains SU4 ⊃ SU2 × SU2 and SUn ⊃ SOn (n ≥ 3) are derived by using the algebra of complementary groups and methods of analytical continuation. A review of alternative approaches to the problem grounded on the concept of biorthogonal systems of non-canonical bases is given. It is demonstrated that the construction of dual bases (polynomial and stretched or related ones) is associated with the use of certain bilinear combinations of isoscalar factors and solutions of the boundary value problem for isoscalar factors. Overlaps are given for different non-canonical basis states of the two-parametric irreducible representations for the chains SU4 ⊃ SU2 × SU2, SUn ⊃ SOn and Sp4 ⊃ U2, as well as the expansions of the projected basis states for Sp4 ⊃ U2 in terms of canonical basis states. A new expression for some special SU2 Clebsch-Gordan coefficients is also given.