The Combinatorics of Macdonald Polynomials and Symmetric Function Operators

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Symmetric functions, long important in algebra, are playing an increasing role in recent years in many other areas of mathematics including algebraic geometry and mathematical physics, algebraic combinatorics, statistical mechanics and representation theory. There are many examples in mathematics and science of polynomials which depend on several variables, and which have important applications. In this project the combinatorics associated with these polynomials will be investigated. Macdonald polynomials, which are multi-variate symmetric functions which satisfy an orthogonality relation and play a central role in algebraic combinatorics with applications to special functions, algebraic geometry, and statistical mechanics. Recently, Macdonald polynomials have arisen in mathematical physics and the study of knot invariants connected to string theory. There has been a lot of recent progress on the combinatorics of these objects, especially for special knots known as torus knots, but many important questions remain open. These questions involve both Macdonald polynomials and various operators which act on them, and this project will further develop the theory of Macdonald polynomials and their operators. It seems likely that Macdonald polynomials will play an increasing role in mathematical physics, representation theory, and combinatorics over the coming years, so results on Macdonald polynomials and their operators arising from this project will likely have applications to many areas of mathematics. Macdonald polynomials, which depend on a set of variables, a partition, and two parameters, are closely connected to the study of the Hilbert scheme, and arise in the study of knot invariants and character formulas. This project seeks to build on previous results on the combinatorics of Macdonald polynomials and character formulas described via operators applied to a Macdonald polynomial. A particularly important example is the character of diagonal harmonics. The recently proved ''shuffle conjecture'' gives a nice combinatorial expression for this character. A central focus of this project is to study a recent generalization of the shuffle conjecture called the Delta conjecture. A number of interesting applications of this conjecture to combinatorics have already been found, and a proof would significantly expand our understanding of Macdonald polynomials, and character formulas connected to the Hilbert scheme and knot invariants. Two other projects involve combinatorial objects connected to diagonal harmonics such as Tesler matrices and LLT polynomials (a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon), as well as the study of Demazure characters and atoms. These last three projects will involve more direct combinatorial analysis and classical symmetric function theory.