Preface

Abstract

Abstract Let Q [ X, Y ] denote the ring of polynomials with rational coefficients in the variables X = { x 1 , x 2 ,…, x n } and Y = { y 1 , y 2 ,…, y n }. Garsia and Haiman in A remarkable q , t Catalan and q-Lagrange inversion (see [4]) define the diagonal harmonics as the solution space DH n = {P(X,Y) ∈ Q [X,Y] : ∑ i=1 n ∂ x i h ∂ y i k P = 0 , they define the diagonal harmonic alternants DHA n as DHA n = {P(X, Y) ∈ DH n : σP = sign (σ)P} and conjecture that the dimension of DHA n is the Catalan number 1 (n+1) 2n n , they conjecture that the Hilbert series of DHA n 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 DHA n . This conjecture has been verified by computer up to n = 7. Furthermore, a ring EV n analogous to DHA n is studied and it is shown that in EV n the analogous conjecture is true. The Hilbert series of EV n gives a different q , t polynomial generalization of the Catalan numbers than that given by Garsia and Haiman.