Quasisymmetric Macdonald Polynomials

Abstract

My main research focus right now is symmetric and quasisymmetric functions. In particular, I am interested in specializations of Macdonald polynomials and generalizations of Schur functions. A symmetric function is a polynomial which remains unchanged when the variables are permuted. The Schur function basis for symmetric functions is related to many different areas of mathematics and can be generated in many ways, but I am most interested in its combinatorial aspects and the properties of semi-standard Young tableaux (boxes containing numbers placed according to certain rules) which can be used to construct Schur functions. Most of my recent work in this area has been on a related collection of polynomials, called "quasisymmetric Schur functions" which can also be constructed using tableaux and which decompose Schur functions in a natural way. These objects come directly from Haglund's recent combinatorial formula for Macdonald polynomials and are a treasure-trove of interesting problems which provide a new perspective for some of the classical problems in this area as well. For example, if a function is symmetric and quasisymmetric Schur-positive, then it is automatically Schur positive, meaning quasisymmetric Schur functions provide a new avenue for investigating Schur positivity. One interesting open problem is to understand the multiplication of quasisymmetric schur functions; there is no known rule for the product of two arbitrary quasisymmetric Schur functions, although a number of special cases are known. Another open problem involves exploring Macdonald polynomial connections. Specifically, since quasisymmetric Schur functions are situated in between Demazure atoms (specializations of nonsymmetric Macdonald polynomials) and Schur functions (specializations of Macdonald polynomials), it is natural to study the quasisymmetric object that sits between nonsymmetric Macdonald polynomials and Macdonald polynomials. This object should generalize quasisymmetric Schur functions in a straightforward manner.