Representation Theoretical Methods in the Theory of Special Functions

这位调查者和他的同事研究了表象理论和特殊函数理论之间的联系。这种连接的桥梁是由Frobenius映射提供的，该映射将群字符与对称函数联系起来。在发现麦克唐纳对称函数基的十年中，逐步出现了麦克唐纳对称函数基在这方面的核心要素。为了证明围绕麦克唐纳基础的各种猜想，人们在对称函数理论和表示理论中取得了一些真正了不起的发现。特别是，这导致首席研究人员和他的同事发现了在对称函数理论范围内(在理论和实践层面上)进行计算的新的有效工具。令人惊讶的发展是，有一族体积算符在对称函数理论中扮演着显着的角色，并揭开了麦克唐纳基础的复杂性。由这些发现而来的研究表明，这个基编码了对称群锡对两组变量x1，…，xn和y1，…，yn上多项式的对角作用的一些真正令人惊讶的性质。这些发展在从代数组合学到代数几何和理论物理的几个领域的影响目前正在深入研究中。到目前为止，纯科学和应用科学的研究人员和教育工作者很少真正认识到符号处理软件和现代计算机的处理能力相结合所提供的广阔的新视野。数学的几个领域已经真正成为实验科学。在计算机探索中，掌握了这门艺术的研究人员每天都会惊讶地看到猜测和定理文字化地从屏幕上跳出来。在计算机时代的早期，在60年代末和70年代初，参与了喷气推进实验室进行的空间探索工作，首席研究员敏锐地意识到需要开发能够将理论发现转化为实用计算方法的工具。这导致了一场疯狂的努力，试图用“算法”论证取代纯粹的“存在”证明。然而，能够连续几天手工进行大量计算而不犯一个错误的能力，是多年来只有少数人才有的一种能力。很明显，高斯和欧拉等巨人的力量在很大程度上是基于这种罕见的能力。然而，可以说，如今，任何有天赋的数学研究生只要拥有一台带有Maple或MATHEMICA的400 MHz PC，就可以轻松地在几分钟内超过高斯和欧拉数周的计算。然而，仍有大量工作要做。不断添加到这些符号操作包中的几个实用程序远远没有耗尽所有可用的可能性。令人惊讶的是，很少有研究人员意识到对称函数理论作为一种符号操纵工具的力量。后者源于这样一个事实，即非线性问题可以通过引入无限多的变量来线性化。它发展了对称函数理论的基矩阵的变化可以用来“模拟”有限装置内无穷大的存在，从而允许许多计算问题的线性化。这一点很容易看出，对对称函数理论进行研究是多么重要，它扩展和深化了该理论的计算能力。这是本项目的首要目标。

9970709The investigator and his associates study the connections between Representation Theory and the Theory of Special functions. He bridge for this connection is provided by the Frobenius map which relates group characters to symmetric functions. In the ten years since its discovery, the Macdonald symmetric function basis has progressively emerged has a central element in this connection. Efforts at proving a variety of conjectures surrounding the Macdonald basis, have led to some truly remarkable discoveries in the Theory of Symmetric functions as well as in Representation Theory. In particular this led the principal investigator and his associates to the discovery of new and efficient tools to carry out calculations (both at the theoretical as well as practical levels) within the theory of symmetric functions. The surprising development is that there is a family of plethystic operators which play a remarkable role within the theory of symmetric functions and unravel the complexity of the Macdonald basis. Investigations, made possible by these discoveries reveal that this basis encodes some truly surprising properties of the diagonal action of the symmetric group Sn on polynomials on two sets of variables x1,...,xn and y1,...,yn. The implications of these developments in several areas which range from Algebraic Combinatorics to Algebraic Geometry and Theoretical Physics are presently under intensive investigation.To this date very few researchers and educators in the pure as well as the applied sciences truly appreciate the vastness of new horizons offered by the combination of symbolic manipulation software and processing power of present day computers. Several areas of mathematics have truly become experimental sciences. In computer explorations, investigators that have mastered this art, are being daily amazed to see conjectures and theorems literately jump out of the screen. In the early days of the computer era, in the late sixties and early seventies involvement with the space exploration efforts carried out at the Jet Propulsion Laboratories, the principal investigator was made keenly aware of the needs to develop tools that enabled the translation of theoretical discoveries into practical computational methods. This led to a frantic effort towards the replacement of pure "existence" proofs by "algorithmic" arguments. Nevertheless, the ability to carry extensive computations by hand for days without committing a single mistake is a quality that has been reserved to only a handful of humans over the ages. It has become clear that the power of such giants as Gauss and Euler was substantially based on this rare ability. However, it is not irreverent to say that, nowadays, any talented mathematics graduate student in possession of a 400MHz PC with MAPLE or MATHEMATICA can easily computationally outdo weeks of both Gauss and Euler ombined in a matter of minutes. Yet a great deal remains to be done. The few utilities that are added constantly to these symbolic manipulation packages, nowhere near exhaust the possibilities that have become available. Surprisingly, very few investigators realize the power of the theory of symmetric functions as a symbolic manipulation tool. The latter stems from the fact that non linear problems may be linearized by the introduction of an infinite number of variables. It develops that the change of bases matrices of symmetric function theory may be used to "mock" the presence of infinities within a finite device, thereby permitting the linearization of many a computational problem. This given it is easily seen 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.