A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants

Abstract

A special case of Haimanʼs identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q , t . In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haimanʼs formula for the Hilbert series into an explicit polynomial in q , t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.