Mathematical Sciences: Representation Theoretical Methods in the Theory of Special Functions

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9532049 Garsia This award supports an investigation that straddles the boundary between Representation Theory and the Theory of Special Functions. The bridge between Representation Theory and Special Functions is provided by the Theory of Symmetric Functions. In 1988, Macdonald introduced a symmetric function basis containing two free parameters q and t. This basis promises to be a central element in the connnection between Representation Theory and the Theory of Special Functions. The investigation is a continuation of a project that the investigator and his collaborators have pursued for more than four years. The main object is to understand the Representation Theoretical and Special Function Theoretical properties of this new basis. Earlier joint work with M. Haiman has lead to the construction of some natural bigraded modules whose bivariate Frobenius characteristic appear to be very closely related to Macdonald polynomials. An extensive computer calculation carried out by the investigator and his collaborators produced a variety of conjectures. Some of these conjectures have been proved in full generality; others only in special cases. The central one, the n-factorial conjecture, has lead to mathematical problems of the first magnitude in various areas of Mathematics that range from Combinatorics to Algebraic Geometry. The primary object of this investigation is to explore the various Representation Theoretical, Special Function Theoretical, and Combinatorial implications of the recent discoveries with a view to resolving some of the open Garsia- Haiman and Macdonald conjectures. This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research.