Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A Macdonald Vertex Operator and Standard Tableaux Statistics for the Two-Column (q, t)-Kostka Coefficients. | A Macdonald Vertex Operator and Standard Tableaux Statistics for the Two-Column q t -Kostka Coefficients Mike Zabrocki Centre de Recherche Mathématiques Université de Montreal LaCIM Université de Québec à Montreal email zabrocki@ Submitted September 30 1998 Accepted November 2 1998 MR Subject Number 05E10 Keywords Macdonald polynomials tableaux symmetric functions q t-Kostka coefficients Abstract The two parameter family of coefficients KARq t introduced by Macdonald are conjectured to q t count the standard tableaux of shape A. If this conjecture is correct then there exist statistics a T and such that the family of symmetric functions Hp X q t MA K q t sx X are generating functions for the standard tableaux of size ụ in the sense that HM X q t MrQ M r t M r SA r X where the sum is over standard tableau of of size p . We give a formula for a symmetric function operator H with the property that HqiH 2a1b X q t H 2a i1b X q t . This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that H 2a 1b X q t is the generating function for standard tableaux of size 2a b and hence that KA 2a1b q t is a polynomial with non-negative integer coefficients . The inductive proof gives an algorithm for building the standard tableaux of size n 2 from the standard tableaux of size n and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics a T and when p is of the form 2a 1b and that these statistics prove conjectures about the relationship between adjacent rows of the q t -Kostka matrix that were suggested by Lynne Butler. 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 5 1998 R45 2 1 Introduction The Macdonald basis for the symmetric functions generalizes many other bases by specializing the values of t and q. The symmetric function basis P. X q t is dehned 14 p. 321 as being self-orthogonal and .
