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Báo cáo toán học: "hains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order"

CTuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: hains, Subwords, and Fillings: Strong Equivalence of Three Definitions of the Bruhat Order. | Chains Subwords and Fillings Strong Equivalence of Three Dehnitions of the Bruhat Order Catalin Zara Department Mathematics and Statistics Penn State Altoona Altoona PA czara@psu.edu Submitted Jan 5 2006 Accepted Mar 1 2006 Published Mar 7 2006 Mathematics Subject Classification 05E15 05C38 Abstract Let Sn be the group of permutations of n 1 . n . The Bruhat order on Sn is a partial order relation for which there are several equivalent definitions. Three well-known conditions are based on ascending chains subwords and comparison of matrices respectively. We express the last using fillings of tableaux and prove that the three equivalent conditions are satisfied in the same number of ways. 1 Preliminaries Let Sn be the group of permutations of n 1 . n . The Bruhat order on Sn is a partial order relation that appears frequently in various contexts and for which there are several equivalent definitions. In this section we recall three of them and introduce some reformulations of these definitions. For more about the Bruhat order including details and proofs of the equivalence of Definitions 1 2 and 3 see BB Fu or Hu . 1.1 Chains For 1 6 i j 6 n let i j 2 Sn be the transposition i j. We say that v A i j v if and only if the values i and j are not inverted in v. Definition 1. The Bruhat order on Sn is the transitive closure of A. In other words v 4 w if and only if there exists a chain . 1bl . 2 02 im 3m T v V0 V1 V2 --- vm w 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N5 1 such that for all k 1 . m we have vk-1 4 ik jk vk-1 vk. To allow reflexivity v 4 v we allow chains with no edges . Then w0 n n 1 . 1 is the unique maximum in the Bruhat order and v 4 w if and only if ww0 4 vw0. Definition. We say that the ascending chain 1 is a relevant chain if i1 6 i2 6 6 im. Example 1. There are twenty-two ascending chains from 2134 to 4231 but only two of them are relevant 2134 2431 4231 2134 2314 2341 3241 4231 Notation. Let C v w be the set of relevant chains from v to w.