A symmetric function is a multi-variate polynomial F(x1,x2,...,xn) which is invariant under any permutation of the variables. In recent years such functions have become quite important to several fields of mathematics, and the representation theoretic, analytic, algebraic, geometric, and combinatorial properties of different families of symmetric functions form active areas of current research. Under this proposal the PI will investigate the combinatorial properties of various families of symmetric functions and other objects of relevance to representation theory and algebra. One particularly important family of symmetric functions central to the PI's research program is the family of Macdonald polynomials. These are symmetric functions in a set of variables {x1,...,xn} which depend on two additional parameters q,t and also on an integer partition. Macdonald polynomials have found many significant applications to date, but there are also several captivating open problems associated with them, particularly with their combinatorial properties. They are closely related to the space of diagonal harmonics, an important topic in algebraic combinatorics and algebraic geometry.
Symmetric functions are becoming increasingly important as tools for researchers in mathematics and mathematical physics. Through the work of Haiman, Cherednik and many others, Macdonald polynomials and diagonal harmonics are linked to mainstream topics such as the Hilbert Scheme from algebraic geometry and the double affine Hecke algebra. New developments in the combinatorics of Macdonald polynomials and other symmetric functions, and related objects such as the Hilbert series of diagonal harmonics, are potentially applicable to many areas of mathematics and science. The explicit and concrete nature of recent combinatorial advances in Macdonald polynomials and the space of diagonal harmonics makes these subjects accessible to a broad range of researchers and students. Thus problems with connections to advanced theoretical mathematics can be attacked using techniques from more concrete and accessible areas such as bijective combinatorics. The PI currently has three Ph. D. students, and maintains collaborations with faculty at primarily undergraduate institutions.
