Collaborative Research: Atlas of Lie Groups and Representation Theory: Computational Aspects

李群和表示图集项目的主要目标是解决李群表示理论中的一个基本问题：真实的约化李群的不可约酉表示的分类。投稿人正在用计算方法来解决这个问题。他们已经发展了把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。关键的工具是一个修改后的概念不变的厄米特形式，称为c-厄米特形式，它具有关键的唯一性属性缺乏通常的厄米特形式。这使得一个算法的公式计算的符号的c-埃尔米特和埃尔米特形式，在一个新的家庭Kazhdan-Lusztig-Vogan多项式。对称性在数学和科学中起着重要的作用。世纪后期，索菲斯·李（Sophus Lie）证明了一个系统的对称性可以用一个抽象的数学对象来描述，这些对象现在被称为李群。一个特定的对称（即李群）可以表现自己的方式被称为酉表示。李群和表示图集的主要目标是理解所有这样的表示。这在物理学以及数学的许多领域都有应用，包括数论和几何。这种方法是计算性的。数学和计算方面的挑战都很大，该项目汇集了数学家和计算机科学家。

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.