The investigator and his associates study the connections between Representation Theory and the Theory of Special functions. He bridge for this connection is provided by the Frobenius map which relates group characters to symmetric functions. In the ten years since its discovery, the Macdonald symmetric function basis has progressively emerged has a central element in this connection. Efforts at proving a variety of conjectures surrounding the Macdonald basis, have led to some truly remarkable discoveries in the Theory of Symmetric functions as well as in Representation Theory. In particular this led the principal investigator and his associates to the discovery of new and efficient tools to carry out calculations (both at the theoretical as well as practical levels) within the theory of symmetric functions. The surprising development is that there is a family of plethystic operators which play a remarkable role within the theory of symmetric functions and unravel the complexity of the Macdonald basis. Investigations, made possible by these discoveries reveal that this basis encodes some truly surprising properties of the diagonal action of the symmetric group Sn on polynomials on two sets of variables x1,...,xn and y1,...,yn. The implications of these developments in several areas which range from Algebraic Combinatorics to Algebraic Geometry and Theoretical Physics are presently under intensive investigation.
To this date very few researchers and educators in the pure as well as the applied sciences truly appreciate the vastness of new horizons offered by the combination of symbolic manipulation software and processing power of present day computers. Several areas of mathematics have truly become experimental sciences. In computer explorations, investigators that have mastered this art, are being daily amazed to see conjectures and theorems literately jump out of the screen. In the early days of the computer era, in the late sixties and early seventies involvement with the space exploration efforts carried out at the Jet Propulsion Laboratories, the principal investigator was made keenly aware of the needs to develop tools that enabled the translation of theoretical discoveries into practical computational methods. This led to a frantic effort towards the replacement of pure "existence" proofs by "algorithmic" arguments. Nevertheless, the ability to carry extensive computations by hand for days without committing a single mistake is a quality that has been reserved to only a handful of humans over the ages. It has become clear that the power of such giants as Gauss and Euler was substantially based on this rare ability. However, it is not irreverent to say that, nowadays, any talented mathematics graduate student in possession of a 400MHz PC with MAPLE or MATHEMATICA can easily computationally outdo weeks of both Gauss and Euler ombined in a matter of minutes. Yet a great deal remains to be done. The few utilities that are added constantly to these symbolic manipulation packages, nowhere near exhaust the possibilities that have become available. Surprisingly, very few investigators realize the power of the theory of symmetric functions as a symbolic manipulation tool. The latter stems from the fact that non linear problems may be linearized by the introduction of an infinite number of variables. It develops that the change of bases matrices of symmetric function theory may be used to "mock" the presence of infinities within a finite device, thereby permitting the linearization of many a computational problem. This given it is easily seen how important it is to pursue investigations in the theory of symmetric functions that extend and deepen the computational power of the theory. This is the foremost goal of the present project.
