FRG: Collaborative Research: Atlas of Lie Groups and Representations: Unitary Representations

FRG：协作研究：李群和表示图集：酉表示

• 批准号: 0967393
• 负责人: John Stembridge
• 金额: $ 8.16万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0967393&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Atlas; Lie

This project has two primary goals. The first is to solve the problem of the unitary dual: to describe the irreducible unitary representations of real reductive Lie groups. The primary tool is an algorithm to compute the unitary dual of any given group, which we are implementing inside the "atlas" software. We plan to use this information to prove results about the unitary dual, beginning with the unitarity of Arthur's unipotent representations. The second primary goal is to make information about representation theory of real groups accessible to non-specialists, via the software, a web site, public workshops, and other means. The atlas software is freely available on the atlas web site, and will continue to be maintained there indefinitely.The idea of using symmetry to study problems in mathematics and science dates back to Fourier's work on heat nearly two hundred years ago. In the hands of Hermann Weyl, Eugene Wigner, and Andre Weil, symmetry has come to play a central role in quantum mechanics and in number theory. Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying symmetry. Representation theory studies all of the ways a given symmetry, or Lie group, can manifest itself. The problem of understanding all "unitary" representations (in which the symmetry operations preserve lengths) is one of the most important unsolved problems in the subject, and has potential applications in many areas; for example, it is an abstract version of the question, "what quantum mechanical systems can admit a certain kind of symmetry?"

该项目有两个主要目标。 首先是解决酉对偶问题：描述真实的约化李群的不可约酉表示。主要的工具是一个算法来计算任何给定的群的酉对偶，我们正在“atlas”软件中实现。我们计划使用这些信息来证明酉对偶的结果，从亚瑟的幂幺表示的酉性开始。第二个主要目标是通过软件、网站、公共研讨会和其他手段，使非专业人员能够获得有关真实的群体的表征理论的信息。Atlas软件在Atlas网站上免费提供，并将继续无限期地维护。利用对称性研究数学和科学问题的想法可以追溯到近两百年前傅立叶关于热的工作。在赫尔曼·外尔、尤金·维格纳和安德烈·魏尔的研究中，对称性在量子力学和数论中扮演了核心角色。李群，以挪威数学家Sophus Lie的名字命名，是对称性背后的数学对象。 表示论研究一个给定的对称性或李群可以表现自己的所有方式。理解所有的“酉”表示（其中对称操作保持长度不变）的问题是该学科中最重要的未解决问题之一，并且在许多领域都有潜在的应用;例如，它是问题的抽象版本，“什么量子力学系统可以承认某种对称性？”"