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 . anne@ Qiang Wang Department of Mathematics University of California One Shields Avenue Davis CA 95616-8633 . xqwang@ 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 .
