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Báo cáo toán học: "Promotion operator on rigged configurations of type A"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Promotion operator on rigged configurations of type A. | Promotion operator on rigged configurations of type A Anne Schilling Department of Mathematics University of California One Shields Avenue Davis CA 95616-8633 U.S.A. anne@math.ucdavi s.edu Qiang Wang Department of Mathematics University of California One Shields Avenue Davis CA 95616-8633 U.S.A. xqwang@math.ucdavis.edu Submitted Aug 17 2009 Accepted Jan 30 2010 Published Feb 8 2010 Mathematics Subject Classifications 05E15 Abstract In 14 the analogue of the promotion operator on crystals of type A under a generalization of the bijection of Kerov Kirillov and Reshetikhin between crystals or Littlewood-Richardson tableaux and rigged configurations was proposed. In this paper we give a proof of this conjecture. This shows in particular that the bijection between tensor products of type An crystals and unrestricted rigged configurations is an affine crystal isomorphism. 1 Introduction Rigged configurations appear in the Bethe Ansatz study of exactly solvable lattice models as combinatorial objects to index the solutions of the Bethe equations 5 6 . Based on work by Kerov Kirillov and Reshetikhin 5 6 it was shown in 7 that there is a statistic preserving bijection between Littlewood-Richardson tableaux and rigged configurations. The description of the bijection is based on a quite technical recursive algorithm. Littlewood-Richardson tableaux can be viewed as highest weight crystal elements in a tensor product of Kirillov-Reshetikhin KR crystals of type An1 . KR crystals are affine finitedimensional crystals corresponding to affine Kac-Moody algebras in the setting of 7 of type An1 . The highest weight condition is with respect to the finite subalgebra An. The bijection can be generalized by dropping the highest weight requirement on the elements in the KR crystals 1 yielding the set of crystal paths P. On the corresponding set of unrestricted rigged configurations RC the An crystal structure is known explicitly 14 . One of the remaining open questions is to define the .