MA 792J  Introduction to Macdonald polynomials
Spring 2008

Instructor: N. Jing, Office: 234 HA, 3-3584,
Tentative Schedule, TBA (We will meet twice a week)

Course Description:
In this course we will present the following topics:
A) Hall-Littlewood polynomials (if time permits connection with p-adic groups)
B) Macdonald polynomials (combinatorial method)
C) Selected topics for their connections to crstal graphs, representation theory, and vertex representations

Representation theory of symmetric groups and general linear groups
 are widely used in many mathematical researches. The importance can be partly explained
by the fact that the subject has been associated with famous names such as Euler, Gauss, Jacobi, Cauchy, Schur,
Weyl, etc. This classical subject has become a central part of algebraic combinatorics during the last several
decades. Moreover the study of symmetric polynomials has turned out to be a multi-disciplinary field that has
connections with algebraic geometry, invariant theory, representation theory, Lie groups and algebras,
PDE (KdV and KP hierarchy equations), statistical mechanics, random matrix models, to name a few.
For example, Hall-Littlewood polynnomials are actually spherical functions for the general linear
groups over p-adic fields.

Recently conjectured by Haglund and proved by Haglund-Haiman-Loehr, a combinatorial formula for Macdonald polynomials
similar to Lascous-Schutzenberger's formula to Hall-Littlewood polynomials
has been finally known. It is interesting that this formula is proved by some combinatorial method similar
to crystal graph theory in quantum affine algebras. We plan to go over the proof and
then discuss the connection of symmetric polynomials and infinite dimensional Lie algebras and quantum
groups.

Students with maturity in linear algebra will have sufficient background for the
course. No previous knowledge about crystal bases is needed, though it is certainly helpful.
Group theory and related algebraic notions will be reviewed when needed. I will try to build the
materials from scratch and present the course in a down-to-earth fashion.

References:
There will be no formal textbooks and the materials are drawn
from the following:
1. I. G. Macdonald, Symmetric functions and Hall polynomials, 1995
2. M. Geck and G. Pfeiffer: Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford, 2000.
3. I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the Monstor, 1988
and research papers of myself and others.

Grading Policy:
The student is required to give a short presentation at the end of the semester.