Representation Theoretical Methods in the Theory of Special Functions

本研究方向为代数组合学。这是一个正在发展的领域，它的开端是由计算机时代的曙光推动的。它的实践者与19世纪的不变量理论家有着相同的目标。重点是结构和算法。20世纪初，希尔伯特向我们展示了证明一个数学对象的存在比构造它要容易得多。在所有的建筑都必须手工完成的时候，放弃建筑是非常诱人的。抽象数学以强有力的新结果和新方法蓬勃发展至今。计算机与强大的符号软件的出现相结合，重新唤起了人们对建筑的兴趣和能力。组合结构的力量同时出现，创造了一个不断增长的前景领域。什么是组合结构？答案很简单：它是数学结构的可视化实现。俗话说“一图胜千言”，在这种情况下再合适不过了。当我们把一个数学结构转换成一个单一的视觉图像时，这个结构的各种特性就会出现，而这些特性在最初的纯数学公式中并不明显。我们必须看到并相信这些属性是如何预测一些最深奥的数学结构的恒等式的。代数和组合之间的相互作用是该奖项支持的研究活动的核心。过去二十年的研究已经明确地表明，对称函数理论不仅是理论研究的一个强大的计算工具，也是在数学的各个分支中获取计算机数据的一个强大的计算工具。因此，规划研究最重要的副产品是新的对称函数工具和恒等式。首席研究员和海曼在早期的计算机探索中获得的数据揭示了对角谐波空间、停车函数和麦克唐纳多项式理论之间惊人的密切联系。这种发展产生了各种各样的问题和猜测，其中一些仍然是开放的。与此同时，环面结理论的研究人员也得到了对称的函数结构，与我们领域中使用停车函数得到的相同。拟议的工作旨在利用这种联系。由此产生的发现可以积极地影响与当前领域相关的领域：表示理论、对称函数理论、组合学、代数几何和计算代数。代数组合学特别适合于计算机实验。这项活动对培训年轻的研究人员非常有效，并使他们能够发现在计算机时代进行研究的方式。在这样的环境下，即使是背景有限的学生也能体验到发现的乐趣。这些领域的研究已经创造了各种各样的发现，并且有潜力创造，这丰富了数学的魔力，可以激励未来几代年轻的研究人员。

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.