Collaborative Research: Atlas of Lie Groups and Representations: Computational Aspects

协作研究：李群和表示图集：计算方面

• 批准号: 1317503
• 负责人: Annegret Paul
• 金额: $ 6.59万
• 依托单位: Western Michigan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1317503&HistoricalAwards=false
• 关键词: Collaborative; Research; Atlas; Lie; Groups

The primary goal of The Atlas of Lie Groups and Representations project is to solve a fundamental problem in representation theory of Lie groups: the classification of the irreducible unitary representations of real reductive Lie groups. The contributors are taking a computational approach to the problem. They have developed the mathematical theory required to reduce the problem to a finite computation. This required rethinking representation theory from the ground up from this new point of view. The key tool is a modified notion of invariant Hermitian form, called the c-Hermitian form, which has critical uniqueness properties lacking in the usual Hermitian form. This allows the formulation of an algorithm to compute the sign of c-Hermitian and Hermitian forms, in terms of a new family of Kazhdan-Lusztig-Vogan polynomials.Symmetry plays a fundamental role in mathematics and the sciences. In the late 19th century Sophus Lie showed that the symmetry of a system can be captured in an abstract mathematical object; these are now known as Lie (pronounced Lee) groups. The ways in which a particular symmetry (i.e. Lie) group can manifest itself are known as unitary representations. The main goal of the Atlas of Lie Groups and Representations is to understand all such representations. This has applications to physics, as well as many areas of mathematics, including number theory and geometry. The approach is computational. Both the mathematical and computational challenges are great, and the project brings together mathematicians and computer scientists.

李群和表示图集项目的主要目的是解决李群表示理论中的一个基本问题：实可约李群的不可约酉表示的分类。贡献者正在采取计算方法来解决这个问题。他们发展了将问题简化为有限计算所需的数学理论。这就需要从这个新的角度重新思考表征理论。关键的工具是不变厄米特形式的修改概念，称为c-厄米特形式，它具有通常的厄米特形式所缺乏的关键唯一性性质。这允许用一族新的Kazhdan-Lusztig-Vogan多项式来表示计算c-Hermite形式和Hermite形式的符号的算法。对称性在数学和科学中起着基本的作用。在19世纪末，Sophus Lie证明了系统的对称性可以在抽象的数学对象中捕捉到；这些现在被称为Lie(发音为Lee)群。一个特定的对称(即李)群的表现方式称为么正表示。李群和表象图集的主要目标是理解所有这些表象。这适用于物理学，以及数学的许多领域，包括数论和几何。这种方法是计算性的。数学和计算的挑战都是巨大的，该项目将数学家和计算机科学家聚集在一起。