# 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.

*This paper has been withdrawn.*