A polynomial expression for the character of diagonal harmonics

Based on his study of the Hilbert scheme from algebraic geometry, Haiman [Invent. Math. 149 (2002), pp. 371–407] obtained a formula for the character of the space of diagonal harmonics under the diagonal action of the symmetric group, as a sum of Macdonald polynomials with rational coefficients. In this paper we show how Haiman’s formula, combined with identities involving plethystic symmetric function operators, yields a new formula for this character. Our formula doesn’t involve any mention of Macdonald polynomials, and the coefficients are visibly polynomials (not rational functions), although they are not manifestly positive. Our formula can be expressed as either a sum of weighted Tesler matrices, or as the constant term in a multivariate Laurent series.