Identities and Positivity Conjectures for some remarkable Operators in the Theory of Symmetric Functions

Abstract. Let Jμ[X; q, t] be the integral form of the Macdonald polynomial and set H̃μ[X; q, t] = tJμ[X/(1− 1/t); q, 1/t ], where n(μ) = ∑ i(i− 1)μi. This paper focusses on the linear operator ∇ defined by setting ∇H̃μ = tq ′)H̃μ. This operator occurs naturally in the study of the Garsia-Haiman modules Mμ. It was originally introduced by the first two authors to give elegant expressions to Frobenius characteristics of intersections of these modules (see [3]). However, it was soon discovered that it plays a powerful and ubiquitous role throughout the theory of Macdonald polynomials. Our main result here is a proof that ∇ acts integrally on symmetric functions. An important corollary of this result is the Schur integrality of the conjectured Frobenius characteristic of the Diagonal Harmonic polynomials [11]. Another curious aspect of ∇ is that it appears to encode a q, t-analogue of Lagrange inversion. In particular, its specialization at t = 1 (or q = 1) reduces to the q-analogue of Lagrange inversion studied by Andrews [1], Garsia [7] and Gessel [17]. We present here a number of positivity conjectures that have emerged in the few years since ∇ has been discovered. We also prove a number of identities in support of these conjectures and state some of the results that illustrate the power of ∇ within the Theory of Macdonald polynomials.