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

李群与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：实约李群的不可约酉表示的分类。贡献者正在采用计算方法来解决这个问题。他们已经发展了将这个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。关键工具是不变厄米形式的修改概念，称为c-厄米形式，它具有通常厄米形式所缺乏的关键唯一性。这允许用新的Kazhdan-Lusztig-Vogan多项式族来计算c-厄米和厄米形式的符号的算法的公式。对称在数学和科学中扮演着重要的角色。19世纪晚期，索夫斯·李证明了系统的对称性可以用抽象的数学对象来描述；这些现在被称为李氏群。一个特定的对称群（即李群）能够表现自身的方式被称为酉表示。李群与表示图集的主要目标是理解所有这样的表示。这适用于物理，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，这个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.