SU(3) Clebsch-Gordan coefficients and some of their symmetries

We discuss the construction and symmetries of Clebsch-Gordan coefficients arising from basis states constructed as triple tensor products of two-dimensional harmonic oscillator states. Because of the symmetry of the basis states, matrix elements and recursion relations are easily expressed in terms of technology. As the Weyl group has a particularly simple action on these states, Weyl symmetries of the coupling coefficients generalizing the well known symmetry of coupling can be obtained, so that any coefficient can be obtained as a sum of Weyl-reflected coefficients lying in the dominant Weyl sector. Some important cases of multiplicity-free decompositions are discussed as examples of applications.