Combinatorics of Special Functions in Geometry and Representation Theory

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ABSTRACTPrincipal Investigator: Haiman, Mark Proposal Number: DMS - 0801262Institution: University of California-BerkeleyTitle: Combinatorics of Special Functions in Geometry and Representation TheoryA 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.