\def \TH {{\tilde H}}
\def \BM {{\bf M}}

\def \ssp {\, \,}
\centerline  {\bf Some new methods in the Theory of Symmetric Functions}
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\centerline  {A. M. Garsia, University of California, San Diego}
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\centerline {{\bf Abstract }}
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%\noindent
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.
\end