Let V be a 3-dimensional complex vector space and V' its dual. For some positive integers a and b, the ring of SL_3-invariant complex-valued functions on a copies of V and b copies of V' is a classical object in invariant theory. In their 2016 paper, Fomin and Pylyavskyy use tensor diagrams (and webs) to construct cluster structures in these rings, and find that looking at the product of vector spaces under different cyclic orderings provides different cluster structures for the same ring. In this talk, we explore their construction and conjectures and discuss further developments of the theory.