This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the algebra of sl(2) intertwiners of d copies of the defining rep of sl(2) (and then asks about a supersymmetric version).
My question: in what ways does Sammy's story extend to other simple Lie groups, e.g. sl(3)? For example, I think that the Schur-Weyl duality still holds, so that HomUsl(3)(V⊗d) is a quotient of the group algebra of the symmetric group Sd (here V is the defining representation of sl(3)). Is it something like the Young Tableaux with three rows? And is there a diagrammatic description of this algebra?