The proposed research is in the field of Algebraic Combinatorics. This is a growing field whose beginnings were prompted by the dawn of the computer age. Its practitioners have the same goals as the invariant theorists of the 19th century. The emphasis is on constructions and algorithms. At the start of the 20th century Hilbert showed how much easier it is to prove the existence of a mathematical object than to construct it. At the time when all constructions had to be done by hand it was very alluring to abandon constructions. Abstract mathematics flourished with powerful new results and methods to the present date. Computers combined with emergence of powerful symbolic software brought back interest and ability to construct. The power of combinatorial constructs emerged at the same time to create a field of growing promise. What is a combinatorial construct? The answer is simple: it is a visual realization of a mathematical construct. The old saying "A Picture is Worth a Thousand Words" cannot be more appropriate in this case. When we translate a mathematical construct into a single visual image a variety of properties of the construct emerge that were not as evident in the original purely mathematical formulation. We must see to believe how these properties predict identities relating some of the most abstruse mathematical constructs. The interplay between algebra and combinatorics is at the heart of the research activities supported by this award.
Research of the last two decades has shown in a unequivocal way that symmetric function theory is a powerful computational tool not only for theoretical investigations also but for obtaining computer data in various branches of mathematics. Therefore the most significant by-products of the planed research are new symmetric function tools and identities. Early computer explorations by the principal investigator and Haiman yielded data which revealed a surprisingly intimate connection between the space of Diagonal Harmonics, Parking Functions, and the Theory of Macdonald polynomials. This development produced a variety of problems and conjectures some of which are still open. Parallel to this development the researchers in the Theory of Torus Knots obtained symmetric function constructs identical to those derived in our field using Parking Functions. The proposed work aims to exploit this connection. The resulting discoveries can positively affect areas that have been connected with the present field: Representation Theory, Symmetric Function Theory, Combinatorics, Algebraic Geometry, and Computational Algebra. Algebraic Combinatorics is particularly suitable for computer experimentation. This activity is highly effective for training young researchers and allowing them to discover the manner in which research can be carried in our Computer Age. Under this setting, even students with limited background can be brought to experience the joy of discovery. The variety of discoveries that research in the proposed areas has already created, and has the potential of creating, is an enrichment of the Mathematical Magics that can inspire future generations of young researchers.
