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

9532049 Garsia该奖项支持跨越表征理论和特殊函数理论之间界限的研究。对称函数论是表示论与特殊函数之间的桥梁。1988年，Macdonald引入了一个包含两个自由参数q和t的对称函数基。这一基础有望成为表征论和特殊函数论之间联系的中心要素。这项调查是调查人员及其合作者四年多来一直在进行的一个项目的继续。主要目的是了解这个新的基础的表示理论和特殊功能理论的属性。与M的早期合作。海曼导致建设一些自然的双阶模块的二元Frobenius特征似乎是非常密切相关的麦克唐纳多项式。研究人员和他的合作者进行了广泛的计算机计算，得出了各种各样的结果。这些命题中有一些已被证明在充分的一般性;其他人只在特殊情况下。中心的一个，n阶乘猜想，导致了数学问题的第一个规模在各个领域的数学，范围从组合数学代数几何。本研究的主要目的是探讨各种表示理论，特殊功能理论和组合的影响，最近的发现，以解决一些开放的Garsia-海曼和麦克唐纳的理论。 这项研究是在组合数学的一般领域。组合数学的目标之一是找到有效的方法来研究如何安排对象的离散集合。 离散系统的行为对现代通信极为重要。例如，大型网络的设计，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。

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.