Groups and Representations Conference; March 25-27, 2004; Eugene, OR

团体和代表会议；

• 批准号: 0244651
• 负责人: Alexander Kleshchev
• 金额: $ 1.28万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-01-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244651&HistoricalAwards=false
• 关键词: Groups; Representations; Conference; March; 25

Principal Investigator: Alexander KleshchevProposal Number: DMS- 0244651Institution: University of Oregon, EugeneTitle: Groups and Representations conference Abstract:The subject of this meeting will be recent developments in the structure theory of simple algebraic and finite groups, their representation theory, and the interplay between these theories. The meeting will cover three main, inter-related areas. The first is the representation theory of simple algebraic groups in defining characteristic. There is no known character formula for the irreducible modules, but some years ago, Lusztigproposed a conjectural formula, which has become the main focus of attention. While Lusztig's conjecture remains far from proved, attempts to prove it have led to spectacular progress in the area in recent years. The second area is modular representation theory in non-defining characteristic. Here one considers the finite groups of Lie type over a field of characteristic p, and studies representations over fields of characteristic different from p. One should also include in this area the representation theory of Coxeter groups (such as the symmetric groups), which is intimately related to that of groups of Lie type. Again there have been spectacular developments in recent years. Fundamental conjectures have been formulated by Alperin, Dade and Broue, and while these are again nowhere near proved, many special cases have been solved, leading to a much deeper general understanding of this field. The third area is the structure theory of simple algebraic and finite groups, particularly the subgroup structure, and its relationship with the representation theory discussed above. Powerful parallel theories for subgroups of both the finite and the algebraic simple groups have been developed, using representation theory as one of the main tools, via the actions of classical groups on their natural modules, and of exceptional groups on their adjoint modules.The symmetry of any system, physical or mathematical, abstract or concrete, is encapsulated in its symmetry group. Thus, the theory of groups finds many applications, both in mathematics and in the physical sciences. Much of group theory is concerned with the study of the actions of groups on spaces of various kinds. The study of group actions on vector spaces is known as representation theory, and that of group actions on sets as permutation group theory, and the focus for this meeting will be on these two areas and their applications. The building blocks of all finite groups are the so-called simple groups, and most of these arise in a natural way from simple algebraic groups (such as SL(n,K), the group of n x n determinant 1 matrices over an algebraically closed field K). Consequently, most attention is devoted to the representation and permutation group theory of these simple groups. These areas are alive with basic conjectures, such as those of Lusztig, Alperin, Broue and Dade. While these are all far from proved, attempts to prove them have led to spectacular progress in the subject in recent years. The meeting will focus on this progress and its applications. There is a healthy number of graduate students working in these areas, and one of the goals is to stimulate interaction between graduate students, young researchers and some of the established leaders in the field.

首席研究人员：亚历山大·克莱斯·乔彻·普罗帕森（Alexander Kleshchevpropopals）编号：DMS- 0244651INDSTITITION：俄勒冈大学Eugenetitle：团体和代表会议摘要：这次会议的主题将是简单代数和有限群体的结构理论中的最新发展，其代表理论，其代表理论以及这些理论之间的相互竞争。会议将涵盖三个主要相互关联的地区。首先是在定义特征中简单代数群体的表示理论。不可简化的模块没有已知的字符公式，但是几年前，Lusztigproppropproppropppropproppropppproped构思公式已成为关注的主要重点。尽管Lusztig的猜想尚未得到证明，但近年来该地区的尝试已取得了惊人的进步。第二个领域是非定义特征中的模块化表示理论。在这里，人们认为有限的谎言类型组在特征p的领域上，并研究了与p不同的特征领域的表示形式。一个人还应在这方面包括与谎言类型组密切相关的Coxeter群体的表示理论（例如对称组）。近年来，再次发生了壮观的发展。基本的猜想是由Alperin，Dade和Broue提出的，尽管这些猜想尚未得到证明，但已经解决了许多特殊案例，从而使人们对该领域有更深入的一般了解。第三个领域是简单代数和有限群的结构理论，尤其是亚组结构，及其与上面讨论的表示理论的关系。有限和代数简单组的亚组的强大平行理论是通过代表理论作为主要工具之一，通过经典组在其自然模块上的作用以及其伴随模块上的特殊组的作用。因此，群体理论在数学和物理科学中都发现了许多应用。小组理论的大部分内容都涉及研究群体对各种空间的行为的研究。对矢量空间的小组行动的研究称为表示理论，并且作为置换群体理论集体行动的研究将集中在这两个领域及其应用上。所有有限基团的构建块是所谓的简单组，其中大多数以自然的方式来自简单的代数组（例如SL（N，K），N x N n x N decitionant 1矩阵的组在代数封闭的字段k上）。因此，大多数关注都致力于这些简单群体的表示和置换群体理论。这些领域充满了基本猜想，例如卢斯蒂格，阿尔珀林，布鲁伊和达德的猜想。尽管这些都是远未得到证明的，但近年来，试图证明它们已取得了惊人的进步。会议将重点关注这一进度及其申请。在这些领域工作的研究生数量健康，目标之一是刺激研究生，年轻研究人员和该领域的一些已建立领导者之间的互动。