Math 581, Spring 2021

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

Office hours:W 10:00am-11:00am, see canvas site for zoom link.

Lecture: TR 12:00-1:20 on zoom (see canvas site for link, or if you are not registered for the course send me an email to obtain an invitation to the zoom lecture)



Course : This is a self contained course on the combinatorics of symmetric functions and the diagonal coinvariant ring. In particular we will (eventually) target presenting a proof of the Delta Conjecture. This conjecture has two sides, one involving an analytic expression containing Macdonald symmetric functions, and the other weighted combinatorial objects known as parking functions. Special cases of the conjecture contain a large number of intriguing combinatorial indentities. There is also an algebraic side to the story, involving invariant theory and the action of the symmetric group on a certain quotient ring. This algebraic side remains largely conjectural and undeveloped, but two proofs of the Delta Conjecture have recently been found.

The first three weeks of the course will cover the basics of symmetric function theory and the combinatorics of Dyck paths (lattice paths consisting of unit North and East steps going from (0,0) to (n,n) and never going below the diagonal x=y). This will mainly consist of material in Chapters 1 through 3 of my book "The q,t-Catalan Numbers ..." (see link below). After covering this base and some other material from my book we will move on to describing the two sides of the Delta Conjecture, and discuss some of its special cases. Then we will work through the new proof by D'Adderio and Mellit of the Compositional Delta Conjecture (which implies the Delta Conjecure). The basic tools used in the proof are plethystic symmetric function identities and the Dyck path algebra. We should (time permitting) also be able to cover the new proof by Blasiak, Haiman, Morse, Pun and Seelinger of the "Extended Delta Conjecture" (which also implies the Delta Conjecture. Neither the Compositional Delta Conjecture or the Extended Delta Conjecture implies the other). A number of open research problems will be discussed throughout the course. The course will hopefully prove interesting to anyone who likes combinatorics, or who works with symmetric functions and representations of the symmetric group. There will be intermittent hw assignments which will be posted at the bottom of this webpage.

The 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 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.

Lecture Notes (created using Ipad app Notability)

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17

Lecture 18

Lecture 19

Lecture 20

Lecture 21

Lecture 22

Lecture 23

Lecture 24

Lecture 25

Lectures 26 and 27




Homework 1 (Due Thursday, February 25).

Homework 2 (Due Tuesday, April 13).