[Todos] Recordatorio: Coloquios del Departamento de Matemática (jueves 20/8/09)

[Daniel Carando]

Hoy Jueves 20 de agosto a las 16:00, aula E24

Erdal Emsiz
Universidad de Talca (Chile)

Macdonald polynomials and explicit commuting  operators diagonalized by them.


Resumen:
In the 1980's Ian Macdonald formulated a series of conjectures
concerning the value of  the constant term of certain power series
indexed by parameters related to a semisimple Lie algebra (more
concretely to its
root system $R$). The conjectures when they first appeared seemed to
be isolated curiosities and it was not clear what lay behind them. That became
clear a few years later with the introduction of (nowadays called)
Macdonald polynomials. These are polynomials $P^{q,t}_\lambda$ in several
variables, depending on parameters $q$ and $t$, indexed by the
dominant weights $\lambda$ for the above root system $R$.  Their various
$q,t$-specialization yield for example the monomial symmetric
functions,
Jack's symmetric functions, zonal spherical functions on certain
symmetric spaces for $p$-adic groups and other classical families of
functions.
Macdonald constant term conjectures in their more general form also
predict the specialization of $P^{q,t}_\lambda$ and a certain duality
between them. These conjectures and related problems concerning Macdonald
polynomials generated huge activity in the last twenty years in
representation theory, combinatorics and theory of quantum integrable
systems, amongst others.

In this talk I  will give an overview of the Macdonald polynomials and
 talk about the above mentioned conjectures of Macdonald. These
conjecture are actually all theorems now, although I will not say much about the
proofs.  If time permits I will talk about recent joint work with Jan
Felipe van Diejen concerning explicit commuting difference operators
diagonalized
by the Macdonald polynomials.


Están todos cordialmente invitados,

Daniel Carando