Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. | A positive proof of the Littlewood-Richardson rule using the octahedron recurrence Allen Knutson Mathematics Department UC Berkeley Berkeley California allenk@ Terence Tao Mathematics Department UCLA Los Angeles California tao@ Christopher Woodward Mathematics Department Rutgers University New Brunswick New Jersey ctw@ Submitted Jun 18 2003 Accepted Jul 4 2004 Published Sep 13 2004 Mathematics Subject Classifications 52B20 05C05 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 Gelfand-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. AK was supported by NSF grant 0072667 and a Sloan Fellowship. TT was supported by the Clay Mathematics Institute and the Packard Foundation. CW was supported by NSF grant 9971357. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R61 1 Contents 1 Introduction 2 Acknowledgements. 3 2 Hives 3 3 Recognizing the representation ring Rep GLn C 5 4 The hive ring satisfies the det-1 and Pieri rules 6 5 The hive ring is associative 8 6 The honeycomb interpretation scattering 11 Honeycomb scattering vs. hive excavation. 14 The scattering rule in GP . 14 7 A closed form for the associativity bijection 14 1 Introduction Let Rep GLn C denote the ring of formal differences of .
