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: Quantized dual graded graphs. | Quantized dual graded graphs Thomas Lam Department of Mathematics University of Michigan Ann Arbor MI 48109 USA tfylam@ Submitted Mar 10 2009 Accepted Jun 1 2010 Published Jun 14 2010 Mathematics Subject Classifications 05C30 Abstract We study quantized dual graded graphs which are graphs equipped with linear operators satisfying the relation DU qUD rl. We construct examples based upon the Fibonacci differential poset permutations standard Young tableau and plane binary trees. 1 Introduction Fomin s dual graded graphs Fom and Stanley s differential posets Sta are constructions developed to understand and generalize the enumerative consequences of the Robinson-Schensted algorithm. The key relation in these constructions is DU UD rl where U D are up and down operators acting on the graphs or posets1. In this article we develop some of the basic theory of quantized dual graded graph which are equipped with up-down operators U D satisfying the q-Weyl relation DU qUD rl. Here q can be considered a parameter which the graph depends upon and which can be specialized. One of the motivations for the current work were the signed differential posets developed in Lam which correspond to the relation DU UD rl. Thus quantized dual graded graphs specialize to usual dual graded graphs at q 1 and to signed differential posets or their dual graded graph equivalent at q 1. The motivating enumerative identity in the subject developed by Fomin and Stanley is D A 2 n AHn -The author was supported by NSF grants DMS-0600677 DMS-0652641 and DMS-0901111 and by a Sloan Fellowship. Tomin also considered more general relations of the form DU f UD THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R88 1 where the sum is over partitions of n and fx is the number of standard Young tableau of shape A. The corresponding analogue Theorem 4 for a quantized dual graded graph r r reads Y fr q fí r n 1 V where the sum is over vertices of height n the polynomials fv q and fV q are weighted .
