A positive proof of the Littlewood-Richardson rule using the octahedron recurrence

Abstract

We define the hive ring, which has a basis indexed by dominant weights for GLn(ℂ), and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of GLn tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel′fand-Tsetlin schemes. . ."]). We use the octahedron rule from [Robbins-Rumsey, "Determinants. . ."] 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 GLn(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. . ."], whose results we use to give a closed form for the associativity bijection.