Tensor decompositions for SL(2) and outerplanar graphs

The main result of this article is the decomposition of tensor products of representations of SL(2) in the sum of irreducible representations parametrized by outerplanar graphs. An outerplanar graph is a graph with the vertices 0, 1, 2, ..., m, edges of which can be drawn in the upper half-plane without intersections. I allow for a graph to have multiple edges, but don't allow loops.