Representation Theoretical Methods in the Theory of Special Functions

9970709调查员和他的同事研究表征理论和特殊功能理论之间的联系。这种连接的桥梁是由Frobenius映射提供的，它将群字符与对称函数联系起来。自麦克唐纳对称函数基被发现以来的十年间，它逐渐在这方面发挥了中心作用。在证明围绕麦克唐纳基础的各种猜想的努力中，在对称函数理论和表征理论中产生了一些真正了不起的发现。特别是，这使得首席研究员和他的同事们发现了在对称函数理论中进行计算（理论和实践层面）的新的有效工具。令人惊讶的发展是，在对称函数理论中有一组多谐算子发挥了显著的作用，并揭示了麦克唐纳基的复杂性。通过这些发现进行的研究表明，这个基编码了对称群Sn对两组变量x1，…上的多项式的对角线作用的一些真正令人惊讶的性质。，xn和y1，…，yn。这些发展在从代数组合学到代数几何和理论物理等几个领域的影响目前正在深入研究中。直到今天，在纯科学和应用科学领域，很少有研究人员和教育工作者真正认识到符号操作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，精通这门艺术的研究人员每天都惊讶地看到各种猜想和定理从屏幕上跳出来。在计算机时代的早期，在60年代末和70年代初参与喷气推进实验室进行的空间探索工作时，首席研究员敏锐地意识到需要开发工具，使理论发现能够转化为实际的计算方法。这导致人们疯狂地试图用“算法”论证取代纯粹的“存在”证明。尽管如此，这种连续数天用手进行大量计算而不犯任何错误的能力，多年来一直只有少数人拥有。很明显，像高斯和欧拉这样的巨人的力量基本上是基于这种罕见的能力。然而，可以毫不夸张地说，如今，任何一个有才华的数学研究生，只要拥有一台400MHz的PC，安装MAPLE或MATHEMATICA，就可以在几分钟内轻松地计算出高斯和欧拉数周的总和。然而，还有很多工作要做。不断添加到这些符号操作包中的少数实用程序，远远没有用尽可用的可能性。令人惊讶的是，很少有研究者意识到对称函数理论作为符号操作工具的力量。后者源于这样一个事实，即非线性问题可以通过引入无限数量的变量而线性化。发展了对称函数理论中基的变换矩阵可以用来“模拟”有限装置中无穷大的存在，从而允许许多计算问题的线性化。鉴于此，很容易看出，在对称函数理论中进行研究是多么重要，它扩展和深化了理论的计算能力。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.