The Biedenharn-Louck-Hecht Resolution of the Outer Multiplicity Problem for the U(3) and Uq(3) Groups
Abstract
The solution of the outer multiplicity problem in the tensor product of U(3) irreducible representations (irreps) developed by Biedenharn et al.(1-7) and realized through the well-known Draayer-Akiyama (DA) computer code(8) is extended to the quantum algebra Uq(3). An analytic formula for special stretched Uq(3) Wigner coefficients, (formula presented) is derived using a projection operator method.(9-10) In this expression Hi denotes the highest weight vector of the (λiλi) irrep; the subscript "max" means coefficients corresponding to a unit tensor operator with a maximal characteristic null space; and q is the usual quantum label so the standard U(3) Wigner coefficient, which is required in the DA code, can be obtained in the q → 1 limit of the theory. To illustrate the theory, some Uq(3) Wigner coefficients for the tensor product (22) × (22) are calculated. The procedure for evaluating nonhighest weight Wigner Uq(3) coefficients follow the DA prescription.