Solving the multiplicity problem in the direct product of irreducible representations of su(3) and su(q)(3) algebras by diagonalizing Gram matrices

A method for solving the multiplicity problem in the direct product of irre ducible representations of the su(3) and su(q)(3) algebras is described. Th is method is based on diagonalizing the Gram matrix for a set of originally nonorthonormalized vectors. By considering the example of the direct produ ct (Il) x (ll), it is shown that this method provides a natural solution to the problem of splitting the degenerate irreducible representations of the su(q)(3) quantum algebra, yielding elegant analytic expressions for Wigner coefficients at small degeneracy multiplicities.