Illinois State University: Diagonal harmonics and shuffle theorems

Abstract: The algebra of multivariate symmetric polynomials over the rational numbers has been used by mathematicians for hundreds of years and contains rich combinatorial structures. In the 1980’s, a basis of symmetric polynomials called Macdonald polynomials was introduced with extra parameters that, when specialized, recover various classical bases. The theory of Macdonald polynomials has given rise to many curious identities relating algebraic quantities arising from representation theory with the combinatorics of Dyck paths. The first such identity, relating the bigraded character of the so-called “module of diagonal harmonics” to the combinatorics of Dyck paths lying below a line of slope -1, was conjectured in 2005 and proven in 2018. Recently, many generalizations of this identity have been proven by relating the action of an Elliptic Hall algebra with various infinite sums of symmetric Laurent polynomials. I will start with an overview of symmetric polynomials and how to get symmetric function expressions from spaces of harmonic polynomials. Then I will move on to discussing various shuffle theorems and the framework that has led to their proofs. A good part of this talk should be accessible to undergraduate Math majors and is based on joint work with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.

Date published: Thursday, October 27, 2022