Representation Theoretical Methods in the Theory of Special Functions

特殊函数理论中的表示理论方法

• 批准号: 9970709
• 负责人: Adriano Garsia
• 金额: $ 9.2万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2002-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970709&HistoricalAwards=false
• 关键词: Representation; Theoretical; Methods; Theory; Special

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.

9970709研究者和他的同事们研究了表征理论和特殊函数理论之间的联系。 他的桥梁，这种联系是由弗罗贝纽斯映射，其中涉及组字符的对称功能。 自发现以来的十年中，麦克唐纳对称函数基逐渐出现，在这方面有一个中心元素。 围绕麦克唐纳基的各种证明的努力，导致了对称函数理论和表示论中一些真正了不起的发现。 特别是这导致主要研究人员和他的同事发现新的和有效的工具进行计算（无论是在理论和实践水平）内的理论对称函数。 令人惊讶的发展是，有一个家庭的plethystic运营商发挥了显着的作用，在理论的对称函数和解开的复杂性麦克唐纳基础。 这些发现使研究成为可能，揭示了这个基础编码了对称群Sn对两组变量x1，...，x2的多项式的对角作用的一些真正令人惊讶的性质。xn和y1，...，恩。 这些发展的影响，在几个领域，范围从代数组合代数几何和理论物理目前正在深入调查。到目前为止，很少有研究人员和教育工作者在纯科学以及应用科学真正欣赏广阔的新视野相结合的符号操作软件和处理能力，今天的计算机。 数学的一些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惊讶地看到图表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能够将理论发现转化为实际计算方法的工具。 这导致了一个疯狂的努力，以取代纯粹的“存在”证明的“算法”的论点。然而，能够连续几天进行大量的手工计算而不犯一个错误的能力，是一种长期以来只有少数人才拥有的品质。 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，毫不夸张地说，如今，任何一个拥有400 MHz PC和MAPLE或MATHEMATICA的天才数学研究生都可以在几分钟内轻松地在计算上超过高斯和欧拉的结合。 然而，仍有大量工作要做。 不断添加到这些符号操作包中的少数实用程序几乎没有耗尽可用的可能性。 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穷多个变量来线性化。 它的发展，对称函数理论的基矩阵的变化可以用来“模拟”有限设备内的无穷大的存在，从而允许线性化的许多计算问题。 这给它是很容易看到它是多么重要，以追求调查的理论对称函数，扩大和深化的计算能力的理论。 这是本项目的首要目标。