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

Allen Knutson ; Terence Tao ; Christopher Woodward

The electronic journal of combinatorics, Tome 11 (2004) no. 1

Résumé

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.

Détail

DOI : 10.37236/1814

Classification : 05E05
Mots-clés : Littlewood-Richardson rule, hive ring, Pieri rule, representation ring, honeycomb, octahedron rule