tailieunhanh - Báo cáo toán học: "Promotion and Evacuation"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Promotion and Evacuation. | Promotion and Evacuation Richard P. Stanley Department of Mathematics . Cambridge Massachusetts USA rstan@ Submitted Jul 22 2008 Accepted Apr 21 2009 Published Apr 27 2009 Mathematics Subject Classifications 06A07 Dedicated to Anders Bjorner on the occasion of his sixtieth birthday. Abstract Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schutzenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset P may be regarded as maximal chains in the lattice J P of order ideals of P. The generalizations concern permutations of the maximal chains of a wider class of posets or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of F then the results can be stated in terms of the expansion of certain Hecke algebra products. 1 Introduction. Promotion and evacuation are bijections on the set of linear extensions of a finite poset. Evacuation first arose in the theory of the RSK algorithm which associates a permutation in the symmetric group n with a pair of standard Young tableaux of the same shape 31 pp. 320-321 . Evacuation was described by . Schtitzenberger 25 in a direct way not involving the RSK algorithm. In two follow-up papers 26 27 Schutzenberger extended the definition of evacuation to linear extensions of any finite poset. Evacuation is described in terms of a simpler operation called promotion. Schuitzenberger established many fundamental properties of promotion and evacuation including the result that evacuation is an involution. Schutzenberger s work was simplified by Haiman 15 and This material is based upon work supported by the National Science Foundation under Grant No. 0604423. Any opinions findings and conclusions or recommendations expressed in this material are those of the .