MATH 280: Macdonald polynomials and crystal bases
Winter 2014, UC Davis

Lectures: WF 4:30-5:50pm, MSB 3106
CRN 70001
Instructor: Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: after class or by appointment
Text: The course will not strictly follow a particular text. Some useful references include:
  • I.G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, second edition, 1995.
  • I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press 157, 2003.
  • W. Fulton, Young Tableaux, London Mathematical Society, Student Texts 35, 1997.
  • A. Bjoerner, F. Brenti, Combinatorics of Coxeter groups, Springer, Graduate Texts in Mathematics, 2005.
  • J. Hong, S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics 42, AMS, 2002.
We will also use recent papers which I will announce or hand out in class.
Computing: During class, I will illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using Sage Online Notebook, or you can make an account on the LaCIM server. You can also run Sage on by typing the command `sage` to launch a Sage session in the terminal.
Grading: Most of all, this class is intended to expose graduate students to this active research area! We will discuss many open research problems. Every registered student is required to present a paper in class (see list below for examples) and write a brief module on this paper on the MathWiki or as a latex file. It would be great if you can team up in groups of two. Also, each group should peer-review at least one other module on the MathWiki or latex file. Suitable topics will be discussed in class.

Course description

Macdonald polynomials form a two parameter basis for the ring of symmetric functions and have a very rich structure. For example, (nonsymmetric) Macdonald polynomials can be understood as eigenvectors of certain operators. In special limits they relate to Demazure characters. Recently, their structure has also been related to crystal bases, which originally came from the representation theory of quantum groups. This course will investigate these exciting new connections!

Topics include:

Suggested papers for presenations:

Crystals Macdonald polynomials Demazure crystals