A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence

We define the hive ring, which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.