Representation Theoretical Methods in the Theory of Special Functions

University Of California-San Diego, La Jolla CA

Investigators

Abstract

This research is on the Theory of Symmetric Functions and its applications to Representation Theory and Combinatorics. The connection between Representation Theory and the Theory of Symmetric Functions is provided by the Frobenius map which relates group characters to symmetric functions. Combinatorics plays a role in that multiplicities of irreducibles are often obtained by counting tableaux, paths, trees and a growing variety of newly emerging discrete structures. In the more than ten years since its discovery, the Macdonald basis has progressively emerged as a central element in these connections. For more than a decade of research in the Theory of Macdonald Polynomials, the investigator and M. Haiman have been led to a variety of conjectures in Representation Theory, Algebraic Geometry, Combinatorics and Symmetric Function Theory. Efforts in proving these conjectures have yielded fundamental facts and methods in each of these areas. More recently the investigator discovered a variety of summation formulas (PNAS V. 98 (April 2001) 4313-4316) which permitted the proof of the first significant positivity result in the Theory of Macdonald Polynomials. In joint work with J. Haglund the investigator proved a beautiful combinatorial formula (conjectured by J. Haglund) for a rational function which had come to be known as the $q,t$-Catalan. The investigator in collaboration with students and associates plans to use his recently discovered symmetric function identities for a direct attack of some of the conjectures that are still unresolved. The Theory of Symmetric Functions is a powerful symbolic manipulation tool. The reason for this is that non linear problems may often be linearized by the introduction of an infinite number of variables. Now it develops that the change of bases matrices of Symmetric Function Theory may be used to "mimic" the presence of infinities within a finite device, thereby permitting the linearization and solution of many a computational problem. Discoveries in the theory and applications of symmetric functions, should also turn out to be of significant impact in the various areas of mathematics in which symmetric function methods have been shown to be effective. This given, we can see 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.