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

项目负责人：Alexander kleshchevv提案编号：DMS- 02446551机构：俄勒冈大学标题：群与表示会议摘要：本次会议的主题将是简单代数和有限群的结构理论的最新发展，它们的表示理论，以及这些理论之间的相互作用。会议将涵盖三个相互关联的主要领域。第一部分是简单代数群在定义特征方面的表示理论。不可约模没有已知的特征公式，但几年前，lusztig提出了一个推测公式，成为人们关注的焦点。尽管Lusztig的猜想还远未被证明，但近年来，证明它的尝试已经导致该领域取得了惊人的进展。第二个领域是非定义特征的模表示理论。这里考虑特征为p的域上的李型有限群，并研究特征不同于p的域上的表示。在这个领域中还应该包括与李型群密切相关的Coxeter群（如对称群）的表示理论。近年来又有了惊人的发展。Alperin、Dade和Broue已经提出了一些基本的猜想，虽然这些猜想还没有得到证明，但许多特殊情况已经得到解决，从而使人们对这个领域有了更深入的了解。第三个领域是简单代数和有限群的结构理论，特别是子群结构，以及它与上面讨论的表示理论的关系。利用表示理论作为主要工具之一，通过经典群对其自然模的作用和例外群对其伴随模的作用，发展了有限群和代数单群的子群的强大的平行理论。任何系统的对称性，无论是物理的还是数学的，抽象的还是具体的，都被封装在它的对称群中。因此，群理论在数学和物理科学中都有许多应用。群论的许多内容都是关于群在各种空间上的作用的研究。对向量空间上群行为的研究被称为表示理论，而对集合上群行为的研究被称为置换群理论，本次会议的重点将放在这两个领域及其应用上。所有有限群的组成部分都是所谓的简单群，其中大多数是由简单代数群自然产生的（例如SL(n，K)，代数闭域K上n x n个行列式矩阵的群）。因此，这些简单群的表示和置换群论成为人们关注的焦点。这些领域充满了基本的猜想，比如Lusztig、Alperin、Broue和Dade的猜想。虽然这些都远未得到证实，但近年来，证明它们的努力已导致该学科取得了惊人的进展。会议将集中讨论这一进展及其应用。有相当数量的研究生在这些领域工作，目标之一是促进研究生、年轻研究人员和该领域一些知名领导者之间的互动。

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.