From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix

Abstract

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ x λ s λ , and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α y α Q α . For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients y α and wishes to compute the Schur coefficients x λ . This paper gives a general combinatorial formula expressing each x λ as a linear combination of the y α ’s, where each coefficient in this linear combination is + 1 , − 1 , or 0. This formula arises by suitably modifying Eğecioğlu and Remmel’s combinatorial interpretation of the inverse Kostka matrix involving special rim-hook tableaux.