Adriano M. Garsia, University of California, San Diego, U.S.A.
e-mail:  garsia@schur.ucsd.edu
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Some New Methods in the Theory of Symmetric Functions

Abstract

Let $J_\mu[X;q,t]$ be the  {\it  integral form} of the Macdonald polynomial and set $\TH_\mu[X;q,t]=t^{n(\mu)}J_\mu[X/(1-1/t);q,1/t\,]$, where $\ssp n(\mu)=\sum_i(i-1)\mu_i$. The talk focusses on the linear operator $\nabla$ defined by setting $\nabla\TH_\mu=t^{n(\mu)}q^{n(\mu')}\TH_\mu$. This operator  occurs  naturally in the study of the Garsia-Haiman modules $\BM_\mu$. It was originally introduced to give elegant expressions to Frobenius characteristics of intersections of these modules. However, it was soon discovered that it plays a powerful and ubiquitous role throughout the theory of Theory of Symmetric Functions. Our  main result is a proof that $\nabla$ 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. Another curious aspect of $\nabla$ 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, Garsia and Gessel. We will present a number of positivity conjectures that have emerged in the few years since $\nabla$ 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 $\nabla$ within the Theory of Symmetric functions.