Representation Theoretical Methods in the Theory of Special Functions

该研究属于代数组合学领域。这是一个不断发展的领域，其起源是由计算机时代的曙光所推动的。 它的实践者与19世纪的不变理论家有着相同的目标。重点是结构和算法。在20世纪初的世纪希尔伯特表明，这是多么容易证明存在的一个数学对象比建设它。在当时的所有建设必须做的手是非常诱人的放弃建设。抽象数学蓬勃发展与强大的新成果和方法到目前为止。 计算机与强大的符号软件的出现相结合，重新唤起了人们对构建的兴趣和能力。组合结构的力量同时出现，创造了一个越来越有前途的领域。什么是组合结构？答案很简单：它是一个数学构造的可视化实现。“一图胜千言”这句老话用在这里再合适不过了。当我们把一个数学结构转换成一个单一的视觉图像时，这个结构的各种属性就会出现，而这些属性在最初的纯数学公式中并不明显。我们必须相信这些性质是如何预测与某些最深奥的数学结构有关的恒等式的。代数和组合学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表明，对称函数理论是一个强大的计算工具，不仅用于理论研究，而且用于在数学的各个分支中获得计算机数据。因此，计划研究的最重要的副产品是新的对称函数工具和恒等式。早期的计算机探索的主要研究者和海曼产生的数据，揭示了一个令人惊讶的密切联系之间的空间的对角谐波，停车功能，和理论的麦克唐纳多项式。这种发展产生了各种各样的问题和挑战，其中一些仍然是开放的。与此同时，环面结理论的研究人员获得了与我们使用停车函数导出的对称函数相同的对称函数构造。这项工作的目的是利用这种联系。由此产生的发现可以积极影响与本领域相关的领域：表示论，对称函数论，组合数学，代数几何和计算代数。 代数组合学特别适合于计算机实验。 这项活动是非常有效的培训年轻的研究人员，让他们发现在我们的计算机时代进行研究的方式。在这种设置下，即使是背景有限的学生也可以体验发现的乐趣。在所提出的领域的研究已经创造了各种各样的发现，并有创造的潜力，是数学魔术的丰富，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.