Meesue Yoo (유미수), Schur expansion of the integral form of Macdonald polynomials

In this talk, 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 ⟨J_λ[X;q,q^k]/(1-q)ⁿ,s_μ(X)⟩∈ℕ[q], we have found explicit combinatorial formulas for the Schur coefficients in one row case, two column case and certain hook shape cases. Egge-Loehr-Warrington constructed 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 given. We apply this method to the combinatorial formula for the integral form Macdonlad polynomials of Haglund-Haiman-Loehr in quasi symmetric functions to get the Schur coefficients and prove the Haglund’s conjecture in more general cases.