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

• DOI: 10.4310/maa.1999.v6.n3.a7
• Publication Date: 2016-02-25T15:53:34Z
• Authors: F. Bergeron; A. M. Garsia; M. Haiman; G. Tesler
• Keywords: Symmetric function; Corollary; Operator (biology); Schur polynomial

Let J^ [X; q, t] be the integral form of Macdonald polynomial and set Hp _ t^^J^X/O.-1/t); g, l/t], where 71(11) = X!;(* -l)/^-This paper focusses on linear operator V defined by setting VH^ t n ^qn ^ ^H^.This occurs naturally in study Garsia-Haiman modules M^.It was originally introduced first two authors to give elegant expressions Frobenius characteristics intersections these (see [3]).However, it soon discovered that plays a powerful ubiquitous role throughout theory polynomials.Our main result here is proof acts integrally symmetric functions.An important corollary this Schur integrality conjectured characteristic Diagonal Harmonic polynomials [11].Another curious aspect appears encode ^-analogue Lagrange inversion.In particular, its specialization at 1 (or q 1) reduces g-analogue inversion studied Andrews [1], Garsia [7] Gessel [17].We present number positivity conjectures have emerged few years since has been discovered.We also prove identities support state some results illustrate power within Theory polynomials.