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

9532049加西亚这个奖项支持一项跨越表象理论和特殊功能理论之间的调查。对称函数理论为表示论和特殊函数之间架起了一座桥梁。1988年，Macdonald引入了一种含有两个自由参数q和t的对称函数基，这个基有望成为表示论和特殊函数论之间联系的中心元素。这项调查是调查人员和他的合作者四年多来一直在进行的一个项目的延续。主要目的是了解这一新基础的表象理论和特殊函数的理论性质。早期与M.Haiman的合作导致了一些自然双格子模的构造，其二元Frobenius特征似乎与Macdonald多项式密切相关。研究人员和他的合作者进行了大量的计算机计算，得出了各种猜测。这些猜想中的一些已经在一般情况下得到了充分的证明，另一些则只在特殊情况下得到了证明。核心的n阶乘猜想在从组合学到代数几何的各个数学领域都产生了一流的数学问题。这项研究的主要目的是探索最近发现的各种表示理论、特殊函数理论和组合理论的含义，以期解决一些公开的Garsia-Haiman和Macdonald猜想。这项研究属于组合学的一般领域。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。

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.