A major result of Professor Haiman's earlier work was the discovery, starting in 2004, of combinatorial formulas in the theory of Macdonald polynomials, something that had been sought ever since Macdonald introduced his polynomials in 1988 (this aspect of Haiman's research was carried out in collaboration with Jim Haglund and Nick Loehr). The formulas connect Macdonald polynomials with other special q-symmetric functions recently studied by combinatorialists, namely the LLT polynomials of Lascoux, Leclerc and Thibon, and the k-Schur functions of Lapointe, Lascoux and Morse. From the point of view of Lie theory, all these developments are connected with general linear groups and therefore with the root systems of type A. The guiding themes of the proposed research will be to unify these recent combinatorial discoveries, to connect them with underlying algebraic, geometric and representation theoretic phenomena, and to extend them to Lie groups and root systems of other types.
In a broader optic, combinatorics is the part of mathematics that deals with the passage from the abstract to the concrete. Thus Lie theory in the abstract is the theory of continuous symmetries. However, by one of the great theorems in mathematics, concrete combinatorial data--the root systems--govern the structure of the most important Lie groups. While the link between Lie groups and root systems is classical, there are also other, more subtle, combinatorial structures associated with Lie theory, which mathematicians are still striving to understand. One way to seek such understanding is to begin by exploring the combinatorial side, which by nature lends itself to explicit computation and the search for patterns, and afterwards to try to explain the observed combinatorial phenomena by reference to more abstract underlying concepts from group theory, geometry and representation theory. This is the mode of understanding which Haiman seeks to pursue in the proposed research.
The project sought to improve our understanding of phenomena involving certain special families of symmetric functions: Macdonald polynomials, LLT (Lascoux-Leclerc-Thibon) polynomials, and the k-atom polynomials of Lascoux, Lapointe and Morse. These polynomials are the subject of a number of conjectures as to their expected properties and methods of computing them. Because the conjectures are combinatorial and explicitly computational, we can test them in individual examples, and this provides evidence for the belief that they are true in all cases. However, it is not yet understood how to establish the conjectures in general, nor what the underlying reasons are for the observed phenomena. The aim of the project was to make progress on these problems by connecting the combinatorial questions with constructions involving the same polynomials in geometry and representation theory. A summary of the main results achieved is as follows. 1. In collaboration with I. Grojnowski, we improved results of Leclerc and Thibon relating LLT polynomials to matrix coefficients in Hecke algebras. In this way we were able to establish conjectured positivity properties and also to define and obtain similar properties for new polynomials of LLT type associated with simple Lie groups. In this collaboration we also laid the foundations for a new theory which we expect will eventually lead to a proof of the principal investigator's unsolved conjectures on positivity of Hecke algebra characters. 2. In 2003 I had conjectured the existence and basic properties of generalized Macdonald polynomials associated to Nakajima quiver varieties. Bezrukavnikov and Kaledin later proved these conjectures by geometric means, but this left open the problem of understanding the polynomials algebraically. Under this award I succeeded in solving the latter problem by obtaining formulas for the new Macdonald polynomials in terms of non-symmetric Macdonald polynomials. 3. With a doctoral student, L.-C. Chen, we constructed modules whose characters we expect to be equal to k-atom polynomials. In fact, Chen managed to carry my initial idea on how to do this much further and constructed a larger family of modules for which we can conjecture algebraic and combinatorial character formulas, as well as a geometric interpretation. We can also reduce positivity conjectures for k-atorms to expected properties of Chen's modules. 4. In collaboration with F. Bergeron, we gained a new understanding of Pieri formulas for Macdonald polynomials. Building on this work and recent results of Schiffman, Vassserot, Feigin and Tsymbauliak, I have been able to find the first formulas for 3-parameter versions of some of the intriguing 2-parameter combinatorial quantities associated with Macdonald polynomials.