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 and evacuation on standard Young tableaux of rectangle and staircase shape. | Promotion and evacuation on standard Young tableaux of rectangle and staircase shape Steven Pon Department of Mathematics University of Connecticut CT USA Qiang Wang Department of Psychiatry and Behavioral Sciences UC Davis Medical Center CA USA Submitted Mar 12 2010 Accepted Jan 7 2011 Published Jan 12 2011 Mathematics Subject Classification 05E18 Abstract Dual- promotion and dual- evacuation are bijections on SYT A for any partition A. Let cr denote the rectangular partition c . c of height r and let sck k 2 denote the staircase partition k k 1 . 1 . We demonstrate a promotion-and evacuation-preserving embedding of SYT sck into SYT kk 1 . We hope that this result together with results by Rhoades on rectangular tableaux can help to demonstrate the cyclic sieving phenomenon of promotion action on SYT sck . 1 Introduction Promotion and evacuation denoted here by d and e respectively - see Definitions and are closely related permutations on the set of standard Young tableaux SYT A for any given shape A. Schutzenberger studied them in 14 15 16 as bijections on SYT A and later as permutations on the linear extensions of any finite poset. Edelman and Greene 3 and Haiman 6 described some of their important properties in particular they showed that the order of promotion on SYT sck is k k 1 where sck k k 1 1 Both authors were partially supported by NSF grants DMS-0636297 DMS-0652641 and DMS-0652652 for this work while studying in the Mathematics Department at the University of California Davis. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P18 1 is the staircase tableau. In 2008 Stanley gave a terrific survey 20 of previous results on promotion and evacuation. In the paper we report the construction of an embedding I SYT sck - SYT k k 1 that preserves promotion and evacuation I o d d o I and I o e e o I. This result arises from our on-going project aimed at understanding the promotion cycle structure on .
