A conjecture of a basis for the diagonal harmonic alternants

Abstract

Let Q[X, Y] denote the ring of polynomials with rational coefficients in the variables X = {x1, x2,…, xn} and Y = {y1, y2,…, yn}. Garsia and Haiman in A remarkableq, tCatalan and q-Lagrange inversion (see [4]) define the diagonal harmonics as the solution space DHn = {P(X,Y) ∈ Q[X,Y] : ∑i=1n ∂xih∂yikP = 0 , they define the diagonal harmonic alternantsDHAn as DHAn = {P(X, Y) ∈ DHn : σP = sign(σ)P} and conjecture that the dimension of DHAn is the Catalan number 1(n+1)2nn, they conjecture that the Hilbert series of DHAn is a q, t polynomial generalization of the Catalan numbers. In this paper, I conjecture that a collection of polynomials closely related to a collection of Schur functions is a basis for DHAn. This conjecture has been verified by computer up to n = 7. Furthermore, a ring EVn analogous to DHAn is studied and it is shown that in EVn the analogous conjecture is true. The Hilbert series of EVn gives a different q, t polynomial generalization of the Catalan numbers than that given by Garsia and Haiman.