Combinatorial formulas connected to diagonal harmonics and Macdonald polynomials

We study bigraded Sn-modules introduced by Garsia and Haiman as an approach to prove the Macdonald positivity conjecture. We construct a combinatorial formula for the Hilbert series of Garsia-Haiman modules as a sum over standard Young tableaux, and provide a bijection between a group of fillings and the corresponding standard Young tableau in the hook shape case. This result extends the known property of Hall-Littlewood polynomials by Garsia and Procesi to Macdonald polynomials. We also study the integral form of Macdonald polynomials and construct a combinatorial formula for the coefficients in the Schur expansion in the one-row case and the hook shape case