A basis for the Y1 subspace of diagonal harmonic polynomials

Abstract

The space DHn of Sn diagonal harmonics is the collection of polynomials P(x, y) = P(x1,…,xn,y1,…,yn) which satisfy the differential equations ∑i=1n ∂xir∂yis, P(x, y) = 0 for all r,s ⩾ 0 (with r + s > 0). Computer explorations by Haiman have revealed that DHn has a number of remarkable combinatorial properties. In particular DHn is an Sn module whose conjectured representation, graded by degree in y, is a sign twisted version of the action of Sn on the parking function module. This conjecture predicts the character of each of the y-homogeneous subspaces Yj of DHn. The space Y0 of diagonal harmonics with no y dependence is known in the classical theory. In this article we construct a basis for the subspace Y1 of diagonal harmonics linear in y. Using this basis we prove that the Y1 specialization of the Parking function conjecture is correct, and we provide a formula for the character of Y1 graded by degree in x. This last formula matches the Y1 specialization of a master conjecture of Garsia and Haiman which expresses the bigraded character of DHn.