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

Every symmetric function f can be written uniquely as a linear combination of Schur functions, say f = ∑ λ xλsλ, and also as a linear combination of fundamental quasisymmetric functions, say f = ∑ α yαQα. For many choices of f arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients yα and wishes to compute the Schur coefficients xλ. This paper gives a general combinatorial formula expressing each xλ as a linear combination of the yα’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 rimhook tableaux.