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 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5 was taken directly from Appendix A, pp. 123, 124, and 126 of my book "The q,t-Catalan Numbers ..."
Lecture 6 (Showing how the Lascoux-Schutzenberger Cochargre Theorem follows from the Macdonald formula)
Lecture 7 (Superization)
Lecture 8 (Consequences of an Expansion into Fundamentals)
Lecture 9 (A Combinatorial Formula for Jmu)
Lecture 9 continued (More on turning a Fundamental expansion into a Schur expansion via the Egge-Loehr-Warrington-Garsia Theorem)
Lecture 10 (The formula for the nonsymmetric Macdonald polynomial E_{alpha})
Lectures 11 and 12 (Proof of the two column rule involving Crystals and Yamanouchi words)
Lecture 13 (Proof of the two column rule, continued)
Lectures 14, 15 (Notes on Boolean Schur Positivity Problem)
Lecture 16 (The 3 Column Rule and Butler's Conjecture)

Homework 1: (Due Thursday, March 3). (1) Do exercise 1.5 from the q,t-Catalan book. Then find a combinatorial formula for the Schur coefficients in the modified Macdonald polynomial {\tilde H}_{\mu}(X;q,t) when the shape mu is a hook.
(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.

Homework 2: Due Thursday, March 31. Do exercises A.5, and A.18 from the book (Jennifer has noted that in the statement of Exercise A.18 the q^{\alpha _n -1} should be q^{1-\alpha _n}). Also, as defined in the Lectures 11 and 12 notes above, let Yam(lambda), for lambda a partition of n, be the set of Yamanouchi words w_1\cdots w_n of content lambda, where in any final segment w_j \cdots w_n there are at least as many i's as (i+1)'s, for all i>=1. Prove that Yam(lambda) is in bijection with SYT(lambda).

Homework 3: Due Tuesday, May 10 by 5pm (either under the door of my office or via email as a pdf attachement). Do exercises 1.11, 1.18, 1.21, 2.12, and 2.18 from the q,t-Catalan book.