MATH 280: Macdonald polynomials and crystal bases
Winter 2014, UC Davis
||WF 4:30-5:50pm, MSB 3106
||Anne Schilling, MSB 3222, phone: 554-2326,
Office hours: after class or by appointment
||The course will not strictly follow a particular text. Some useful references include:
We will also use recent papers which I will announce or hand out in class.
- 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.
||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 fuzzy.math.ucdavis.edu by typing the command `sage` to launch a Sage session in the terminal.
||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.
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!
- Symmetric functions, in particular Macdonald polynomials
- Combinatorics of Coxeter groups, weak and strong Bruhat order, quantum Bruhat graph
- Crystal graphs, Demazure crystals
- Models for crystals
Suggested papers for presenations:
- T. Nakashima, Crystal base and a generalization of the Littlewood-Richardson rule
for the classical Lie algebras, Comm. Math. Phys. 154 (1993) 215-243
- A. Lascoux, B. Leclerc, J.-Y. Thibon, Crystal graphs and q-analogues of weight
multiplicities for the root system A_n, Lett. Math. Phys. 35 (1995) 359-374
- A. Lascoux, B. Leclerc, J.-Y. Thibon, The plactic monoid
- C. Lenart, A. Postnikov, A combinatorial model for crystals of Kac-Moody algebras, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4349-4381
- L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture,
Duke Math. J. 116 (2003), no. 1, 103-146
- J. Haglund, M. Haiman, N. Loehr, A combinatorial formula for Macdonald polynomials,
J. Amer. Math. Soc. 18 (2005), no. 3, 735-761
- J. Haglund, M. Haiman, N. Loehr, A combinatorial formula for nonsymmetric Macdonald polynomials,
Amer. J. Math. 130 (2008), no. 2, 359-383
- A.M. Garsia, M. Zabrocki, Polynomiality of the q,t-Kostka revisited,
Algebraic combinatorics and computer science, 473-491, Springer Italia, Milan, 2001.
( math/0008199 )
- S. Mason, An explicit construction of type A Demazure atoms,
J. Algebraic Combin. 29 (2009), no. 3, 295-313
- Y.B. Sanderson, On the connection between Macdonald polynomials and Demazure characters,
J. Algebraic Combin. 11 (2000), no. 3, 269-275
- B. Ion, Nonsymmetric Macdonald polynomials and Demazure characters, Duke Math. J. 116 (2003), no. 2, 299-318