# Applications of Macdonald Polynomials (07w5048)

Arriving in Banff, Alberta Sunday, September 9 and departing Friday September 14, 2007

## Organizers

(Université du Québec à Montréal)

(University of Pennsylvania)

(University of California, San Diego)

## Objectives

The study of Macdonald polynomials is one of the most active current areas of
research in Algebraic Combinatorics. It exhibits natural ties with many
area of mathematics: Algebraic Geometry,
Representation Theory, Special Function Theory, etc.,
and raises new exciting questions in all of these subjects. The techniques
involved in this study are also wide ranging, going from the
uses of Double Affine Hecke Algebras to the study of Hilbert Schemes, passing
through the study of deep combinatorial statistics on tableaux. For example,
in the mid 90's Cherednik showed that nonsymmetric Macdonald polynomials are
intimately tied up with the representation theory of Double Affine Hecke Algebras, and
resolved the Macdonald constant term-conjectures" for arbitrary root systems.
These conjectures were a focal point of research in Algebraic Combinatorics throughout the 1980's. Another example is the work of Haiman, who showed that there are deep connections between algebraic geometry,
the representation theory of the space of diagonal harmonics,
and the the theory of Macdonald polynomials. He was awarded
the 2004 Moore AMS prize for this work.
Haiman subsequently extended his methods to prove a long-standing conjecture for the character of diagonal harmonics as an analytic expression involving a sum of
rational functions in two parameters q,t. A related result was obtained by
Iain Gordon, who using Cherednik's approach proved an analogue of Haiman's
result on the dimension of diagonal harmonics for other root systems.
In addition Lapointe and Morse have introduced a
generalization of Schur functions called k-Schur functions which have
many unexpected connections to geometry and Macdonald polynomial theory.

There has also been a lot of progress understanding the combinatorics
underlying the representation theory of the space of diagonal harmonics.
Garsia and Haglund proved a purely combinatorial formula for the Hilbert series of the subspace of alternants, which turns out to be a q,t
analogue of the Catalan numbers.
The proof uses plethystic symmetric function identities developed over a
period of several years by F. Bergeron, Garsia, Haiman, Tesler and others.
Haglund, Haiman, Loehr, Remmel and Ulyanov have introduced a conjecture,
which is still open,
giving a combinatorial conjecture for the character of diagonal harmonics.
Recently this conjecture led Haglund to the discovery of the first conjectured
combinatorial formula for Macdonald polynomials, which was subsequently proved by Haglund, Haiman and Loehr. This promises to open up an entirely new approach
to the many important open conjectures that still need to be
the Bergeron-Garsia-Haiman-Tesler conjecture regarding the so-called Nabla
operator, and the Bergeron-Garsia conjectures
known as the Science-Fiction conjectures.

Another parallel line of research has been carried out by N. Bergeron and Zabrocki
involving q,t analogs of noncommutative symmetric functions and quasi-symmetric
functions.
All of these and many other significant results we haven't mentioned here
have resulted in a coherent body of theorems and conjectures which form
a beautiful blend of combinatorial enumeration, geometry, algebra, and analysis.
The group that we intend to bring together contains representatives of each of
the main areas involved in this large picture.
It will certainly be the first time that such a group is brought together, and it is a great time to do this in view of the very recent
developments that have occurred in the area. Our list is a good equilibrium
of the forefront established researchers in the field, as well as bright rising stars. Many of them have already confirmed their intention to come, namely
F. Bergeron, N. Bergeron, M. Haiman, R. Biagioli, A. Garsia, J. Haglund, F.
Hivert, L. Lapointe, J. Morse, J. Remmel,
N. Loehr, M. Zabrocki ...