The problem which motivated the writing of this paper is that of finding structure behind the decomposition of the sl
3 representation spaces V* ⊗ W = Hom(V, W) for finite dimensional irreducible sl
3-modules V and W. For sl
2 this extends the classical Clebsch-Gordon problem. The question has been considered for sl
3 in a computational way in [5]. In this paper we build a conceptual algebraic framework going beyond the enveloping algebra of sl
3.
For each dominant integral weight α let V
α
be an irreducible representation of sl
3 of highest weight α. It is well known that, for weights α, μ, λ, the multiplicity of Vλ
in Hom(Vα, Vα+μ
) is bounded by the multiplicity of μ in Vλ
, with equality for generic α. This suggests the possibility of a single construction of highest weight vectors of weight X in Hom(Vα, Vα+μ
) which is valid for all a.