The PI plans to work on three related projects involving the combinatorics of Macdonald polynomials and the space of diagonal harmonics. The first is an attempt to build on recent joint work between Haiman, Loehr and the PI which gives a combinatorial formula for type A nonsymmetric Macdonald polynomials. The existence of this formula in the type A setting suggests that similar formulas exist for other root systems; the discovery of such formulas would open up the subject significantly, as very few explicit identities of any kind have been discovered beyond the type A case. The second project attempts to build on the combinatorial formula for (type A) symmetric Macdonald polynomials, which was discovered empirically by the PI and proved in subsequent joint work with Haiman and Loehr. The main open problem in this regard is to find a combinatorial formula for the coefficients in the Schur expansion; as a step in this direction the PI describes a new conjectured formula for the special case of augmented hook shapes. The third project involves trying to prove a conjectured formula for the monomial expansion of the character of the space of diagonal harmonics, due to Haiman, Loehr, Remmel, Ulyanov and the PI. This conjecture, which led the PI to the empirical formula for symmetric Macdonald polynomials mentioned above, has a number of implications to enumerative combinatorics and representation theory.
Orthogonal polynomials are families of polynomials which satisfy an orthogonality condition, typically meaning that the integral of any two different elements of the family, against some given weight function, is zero. The study of orthogonal polynomials is several centuries old, and they have quite a number of applications throughout mathematics and science. Symmetric functions are polynomials in several variables which are invariant under any permutation of the variables. They have a large number of applications to several branches of mathematics such as representation theory and the roots of polynomial equations. In 1988 Macdonald introduced a new family of orthogonal polynomials, which depend on a set of variables X and two extra parameters q,t. They are symmetric functions in the variables X, and contain most useful symmetric functions as special cases. They were immediately recognized as being important to several area of mathematics including combinatorics and special functions. In 2000 Haiman proved they have a complicated interpretation in terms of an advanced, abstract and theoretical branch of mathematics known as algebraic geometry. Macdonald's construction is indirect however, and neither it nor Haiman's interpretation give a way of easily describing these polynomials. Recently the PI discovered, and with Haiman and Loehr proved, a simple combinatorial formula for Macdonald's polynomials. The PI is trying to apply this result to answer some of the open questions surrounding them. Macdonald and Cherednik now have a much more general construction which contains almost all previously studied symmetric functions and orthogonal polynomials as special cases. The PI, in collaboration with Haiman, is trying to extend the simple combinatorial formula to this more general construction.
