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Row-strict Quasisymmetric Schur Functions, Characterizations of Demazure Atoms, and Permuted Basement Nonsymmetric Macdonald Polynomials

出版	University of California, Davis, 2011
ISBN	12672386669781267238665
URL	http://books.google.com.hk/books?id=rqRqAQAACAAJ&hl=&source=gbs_api

註釋We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. This expansion follows from several new properties of an insertion algorithm defined by Mason and Remmel (2011) which inserts a positive integer into a row-strict composition tableau. We then apply this Littlewood-Richardson type rule to give a basis for the quotient of quasisymmetric functions by the ideal generated by symmetric functions with zero constant term. We then discuss a family of polynomials called Demazure atoms. We review the known characterizations of these polynomials and then present two new characterizations. The first new characterization is a bijection between semi-standard augmented fillings and triangular arrays of nonnegative integers, which we call composition array patterns. We also provide a bijection between composition array patterns with first row [gamma] and Gelfand-Tsetlin patterns whose first row is the partition [lambda] whose parts are the parts of [gamma] in weakly decreasing order. The second new characterization shows that Demazure atoms are the polynomials obtained by summing the weights of all Lakshmibai-Seshadri paths which begin in a given direction. Finally, we consider a family of polynomials called permuted basement nonsymmetric Macdonald polynomials which are obtained by permuting the basement of the combinatorial formula of Haglund, Haiman, and Loehr for nonsymmetric Macdonald polynomials. We show that these permuted basement nonsymmetric Macdonald polynomials are the simultaneous eigenfunctions of a family of commuting operators in the double affine Hecke algebra.