Schur Coefficients of the Integral Form Macdonald Polynomials

Abstract

In this paper, we consider the combinatorial formula for the Schur coefficients of the integral form of the Macdonald polynomials. As an attempt to prove Haglund's conjecture that $\Biggl<\frac{J_{\mu}(X;q,q^k)}{(1-q)^{|\mu|}},s_{\lambda}(X)\Biggr> \in \mathbb{N}[q]$, we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases [Yoo12]. A result of Egge-Loehr-Warrington [ELW] gives a combinatorial way of getting Schur expansion of symmetric functions when the expansion of the function in terms of Gessel's fundamental quasi symmetric functions is known. We apply this result to the combinatorial formula for the integral form Macdonald polynomials of Haglund [Hag] in quasi symmetric functions to prove the Haglund's conjecture in more general cases.