tailieunhanh - Báo cáo toán học: "The combinatorics of orbital varieties closures of nilpotent order 2 in sl n"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The combinatorics of orbital varieties closures of nilpotent order 2 in sl n . | The combinatorics of orbital varieties closures of nilpotent order 2 in sln Anna Melnikov Department of Mathematics University of Haifa 31905 Haifa Israel and Department of Mathematics the Weizmann Institute of Science 76100 Rehovot Israel melnikov@ Submitted Sep 12 2002 Accepted Apr 28 2005 Published May 6 2005 Mathematics Subject Classifications 05E10 17B10 Abstract. We consider two partial orders on the set of standard Young tableaux. The first one is induced to this set from the weak right order on symmetric group by Robinson-Schensted algorithm. The second one is induced to it from the dominance order on Young diagrams by considering a Young tableau as a chain of Young diagrams. We prove that these two orders of completely different nature coincide on the subset of Young tableaux with 2 columns or with 2 rows. This fact has very interesting geometric implications for orbital varieties of nilpotent order 2 in special linear algebra sln. 1. Introduction Let Sn be a symmetric group that is a group of permutations of 1 2 . ng. Respectively let Sn be a group of permutations of n positive integers mi m2 . mng where mi i. It is obvious that there is a bijection from Sn onto Sn obtained by mi i so we will use the notation Sn in all the cases where the results apply to both Sn and Sn. In this paper we write a permutation in a word form w ax a2 . an where ai w mi . . All the words considered in this paper are permutations . with distinct letters only. Set pw mi j if aj mi in other words pw mi is the place index of mi in the word form of w. If w 2 Sn then pw i w-1 i . Supported in part by the Minerva Foundation Germany Grant No. 8466 THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R21 1 We consider the right weak Bruhat order on Sn that is we put w y if for all i j 1 i j n the condition pw mj pw mi implies py mj py mi . Note that m1 m2 . mn is the minimal word and mn mn-1 . m1 is the maximal word in this order. Let A Al A2 Ak 0 be a partition .