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

Every symmetric function ff can be written uniquely as a linear combination of Schur functions, say f=∑λxλsλf=∑λxλsλ, and also as a linear combination of fundamental quasisymmetric functions, say f=∑αyαQαf=∑αyαQα. For many choices of ff arising in the theory of Macdonald polynomials and related areas, one knows the quasisymmetric coefficients yαyα and wishes to compute the Schur coefficients xλxλ. This paper gives a general combinatorial formula expressing each xλxλ as a linear combination of the yαyα’s, where each coefficient in this linear combination is +1+1, −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.