Math 581, Spring 2020

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

Office hours:W 10:00am-11:00am, in DRL 4E2B.

Office phone: (215) 573-9093

Lecture: TR 1:30-3:00 in DRL 2C2

Course : This course is a self contained course on the combinatorics of Macdonald polynomials and coinvariant rings. Macdonald polynomials are symmetric functions in a set of variables X which also depend on two parameters q,t. They have become important in algebraic combinatorics as well as mathematical physics and knot homology. We will be focusing on the combinatorial side of the picture. One primary source for the lectures is my book The q,t-Catalan Numbers and the Space of Diagonal Harmonics. The official bound version of this book can now be ordered from the AMS. It is part of the AMS University Lecture Series. A good companion text to what we will be covering is the book "Algebraic Combinatorics and Coinvariant Spaces" by Francois Bergeron. Other useful reference texts on the theory of symmetric functions are "Enumerative Combinatorics", Volume 2 by Richard Stanley and also "Symmetric Functions and Hall Polynomials", 2nd Edition, by I.G. Macdonald.

The first part of the course will focus on the basics of the theory of symmetric functions, including plethystic identities. We will then discuss the combinatorial description of Macdonald polynomials and its applications to enumeration. The second part of the course will cover the combinatorics associated with the study of certain modules of the symmetric group, namely coinvariant and diagonal coinvariant rings. Formulas for the bigraded Hilbert sereies of these rings will involve q,t-versions of familiar objects such as Catalan numbers and parking functions. Taking into account the symmetric group action will involve Macdonald polynomials.

Time permitting, in the third part we discuss Carlsson and Mellit's second proof of the Shuffle Conjecture, which gives a combinatorial formula, for the expansion of the character of the diagonal coinvariant ring into monomials, in terms of weighted parking functions.

A number of open research problems will be discussed throughout the course. It is hoped the course will be useful both for graduate students interested in learning about a topic of current interest in algebraic combinatorics as well as researchers in this area.

Each student will be encouraged to give a 30-60 minute presentation on some topic connected to the course.