Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions

Abstract

Let J^ [X; q, t] be the integral form of the Macdonald polynomial and set Hp [X; q, t] _ t^^J^X/O.-1/t); g, l/t], where 71(11) = X!;(* -l)/^-This paper focusses on the linear operator V defined by setting VH^ = t n ^qn ^ ^H^.This operator occurs naturally in the study of the Garsia-Haiman modules M^.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 V 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 V is that it appears to encode a q, ^-analogue of Lagrange inversion.In particular, its specialization at t = 1 (or q = 1) reduces to the g-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 V 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 V within the Theory of Macdonald polynomials.