Prof. Jim Haglund , jhaglund@math.upenn.edu
Course webpage: http://www.math.upenn.edu/~jhaglund/581/

Office hours: M 10:00am-11:00am, R 11:45am-12:45pm.

Lecture: TR 10:15-11:45pm in DRL 4C8.

Course : This is a self contained course on the combinatorics of symmetric functions, specifically on LLT polynomials and the chromatic symmetric function. Several outstanding open problems in this area (such as the Stanley-Stembridge Conejcture and its q-analog by Shareshian and Wachs) will be discussed in some detail. Specific theorems to be proved include the fact that LLT polynomials are symmetric in the X variables, the chromatic function corresponding to a Dyck path is Schur positive, basic results about plethysm, superization, and expansions in the Gessel fundamental quasisymmetric function basis, and applications to Garsia and Remmel's theory of q-rook polynomials. A few lectures covering the basics of the representation theory of the symmetric group will be included early in the course. The course is primarily designed for graduate students or faculty who are interested in doing research in algebraic combinatorics or combinatorial representation theory, or possibly undergraduate students looking for a research experience in this area. It would be nice to have had a course like Math 5800 on enumerative combinatorics as background, but is not essential. Homework assignments which will be posted at the bottom of this webpage. Auditors are welcome.

One background source for the lectures is Chapters 1, 2, 6 and Appendix A of my book The q,t-Catalan Numbers and the Space of Diagonal Harmonics. The official bound version of this book can be ordered from the AMS. It is part of the AMS University Lecture Series. Other useful reference texts include "Algebraic Combinatorics and Coinvariant Spaces" by Francois Bergeron, "Enumerative Combinatorics", Volume 2 by Richard Stanley and "Symmetric Functions and Hall Polynomials", 2nd Edition, by I.G. Macdonald.

Homework 1 Due in class on Tuesday, Feb. 27.

Homework 2 Due May 13, as a pdf attachment to an email or under my office door.