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A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. (English) Zbl 1053.05119

Summary: We define the hive ring, which has a basis indexed by dominant weights for \(\text{GL}_n(\mathbb C)\), and structure constants given by counting hives [A. Knutson and T. Tao, J. Am. Math. Soc. 12, 1055–1090 (1999; Zbl 0944.05097) and ibid. 17, 19–48 (2004; Zbl 1043.05111)] (or equivalently honeycombs, or BZ patterms [A. D. Berenshtein and A. V. Zelevinskii, Sov. Math., Dokl. 37, 799–802 (1988); translation from Dokl. Akad. Nauk SSSR 300, 1291–1294 (1988; Zbl 0674.20024)]).
We use the octahedron rule from [D. P. Robbins and H. Rumsey jun., Adv. Math. 62, 169–184 (1986; Zbl 0611.15008)] 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 \(\text{GL}_n(\mathbb C)\).
In the honeycomb interpretation, the octahedron rule becomes “scattering” of the honeycombs. This recovers some of the “crosses and wrenches” diagrams from D. Speyer’s very recent preprint [Perfect matchings and the octahedron recurrence, preprint May 2003], whose results we use to give a closed form for the associativity bijection.

MSC:

05E05 Symmetric functions and generalizations
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