Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A Combinatorial Formula for the Hilbert Series of bigraded Sn-modules. | A Combinatorial Formula for the Hilbert Series of bigraded Sn-modules Meesue Yoo Department of Mathematics University of California San Diego CA meyoo@ Submitted Oct 20 2009 Accepted Jun 15 2010 Published Jun 29 2010 Mathematics Subject Classification 05C88 Abstract We prove a combinatorial formula for the Hilbert series of the Garsia-Haiman bigraded Sn-modules as weighted sums over standard Young tableaux in the hook shape case. This method is based on the combinatorial formula of Haglund Haiman and Loehr for the Macdonald polynomials and extends the result of A. Garsia and C. Procesi for the Hilbert series when q 0. Moreover we construct an association of the fillings giving the monomial terms of Macdonald polynomials with the standard Young tableaux. 1 Introduction In 1988 Mac88 Macdonald introduced a family of symmetric functions with two variables that are known as the Macdonald polynomials which form a basis for the space of symmetric functions. Upon introducing these polynomials Macdonald conjectured that the coefficients of the plethystic Schur expansion of Macdonald polynomials are polynomials in the parameters q and t with nonnegative integer coefficients. To prove this positivity conjecture of Macdonald polynomials Garsia and Haiman GH93 introduced certain bigraded Sn modules and Haiman proved Hai01 that the bigraded Frobenius characteristic F Mm which by definition is simply the image of the bigraded character of Mm under the Frobenius map is given by Yi. x q t Ỵíx q t where A x q t are the modified Macdonald polynomials HHL05 and X xi x2 . For the Garsia-Haiman module Mm if we define Hh k MM to be the subspace of Mm THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R93 1 spanned by its bihomogeneous elements of degree h in X and degree k in Y we can write a bivariate Hilbert series such as n k n y hm m EE thqk dim Hh k MJ . h 0 k 0 Noting that the degree of the Sn character Xx is given by Pn Sx where is the usual inner product on symmetric .
