**Math 581, Spring 2022**

**Prof. Jim Haglund **, jhaglund@math.upenn.edu

Course webpage:
http://www.math.upenn.edu/~jhaglund/581/

**Office hours: **W 10:00am-10:50am, F 11:00-11:50am, via zoom. See canvas site or email me for zoom link.

**Lecture:** TR 12:00-1:30pm. The first three lectures (Jan. 13, 18 and 20) will be online using zoom
(see canvas site for link, or if you are not a Penn student send me an email to
obtain an invitation to the zoom lectures). Starting Jan. 25, the lectures will be given in person in DRLB 2C6.

**Course :** This is a self contained course on the
combinatorics of symmetric functions, specializing on the problem of finding a combinatorial interpretation for the
coefficients of a given symmetric function when expanded in the Schur basis. This includes some of the more
important open problems in algebraic combinatorics, which are fairly easy to describe (but very challenging to solve).
Examples we will cover
include Macdonald and LLT polynomials, and chromatic symmetric functions corresponding to colourings of graphs.
In these cases and other situations we typically have a combinatorial interpretation for the coefficients in the
monomial basis, and a geometric interpretation for the coefficients in the Schur
basis which guarentees they are positive integers, but no combinatorial interpretation for these positive integers.
Some of the techniques we will show
give insight into this problem include the RSK algorithm, crystal graphs, quasisymmetric expansions, and sign-reversing involutions.
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 580 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.

**Lecture Notes (created using Ipad app Notability)
**

Lecture 5 was taken directly from Appendix A, pp. 123, 124, and 126 of my book "The q,t-Catalan Numbers ..."

(2) Do exercise A.10 from the q,t-Catalan book.

(3) Prove that for any symmetric function G, the Hall scalar product of G with h_d e_{n-d} equals the Hall scalar product of \Delta _{e_{n-d}} G with h_n. Here the \Delta operator is defined in equation (4.48) on p. 68 of the q,t-Catalan book.