Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Deformation of Chains via a Local Symmetric Group Action. | Deformation of Chains via a Local Symmetric Group Action Patricia Hersh Department of Mathematics Room 2-588 Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge MA 02139 hersh@ Submitted May 13 1998 Accepted March 3 1999. AMS Subject Classification 05E25 06A07. Abstract A symmetric group action on the maximal chains in a finite ranked poset is local if the adjacent transpositions act in such a way that i i 1 sends each maximal chain either to itself or to one differing only at rank i. We prove that when Sn acts locally on a lattice each orbit considered as a subposet is a product of chains. We also show that all posets with local actions induced by labellings known as R S-labellings have symmetric chain decompositions and provide R S-labellings for the type B and D noncrossing partition lattices answering a question of Stanley. 1 Introduction A symmetric group action on the maximal chains in a finite ranked poset was defined by Stanley in St2 to be local if for each i the adjacent transposition si i i 1 sends each maximal chain either to itself or to one differing from it only at rank i. This work was supported by a Hertz Foundation Graduate Fellowship. THE ELECTRONIC .JOURNAL OF COMBINATORICS 6 1999 R27 2 There is a correspondence between rhombic tilings of a planar region and equivalence classes of reduced expressions for a permutation up to commutation. This naturally translates symmetric group structure to poset structure when Sn acts locally on the maximal chains in a poset. We begin by reviewing this correspondence which is thoroughly examined in El because we it will allow us to explain why orbits of local symmetric group actions on lattices are always products of chains. When a permutation w is written as a product of adjacent transpositions w sa1 sa2 . sai with l as small as possible such a product is called a reduced expression for w. To obtain a rhombic tiling from this begin with a vertical path consisting of n 1 nodes as
