# Applications of Macdonald Polynomials (07w5048)

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

## Organizers

Francois Bergeron (Université du Quebec a Montréal)

James Haglund (University of Pennsylvania)

Jeffrey Remmel (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

addressed, examples of which include

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 ...

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

addressed, examples of which include

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 ...