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

主要研究人员：Alexander KleshchevProposal Number：DMS-0244651机构：俄勒冈大学，尤金标题：群与表示会议摘要：这次会议的主题将是简单代数和有限群的结构理论、它们的表示理论以及这些理论之间的相互作用的最新发展。会议将涉及三个相互关联的主要领域。第一种是定义特征的单代数群的表示理论。不可约模的特征标公式尚无定论，但几年前，Lusztig提出了一个猜想公式，并成为人们关注的主要焦点。虽然Lusztig的猜想仍远未得到证实，但近年来证明它的尝试已经在该领域取得了惊人的进展。第二个领域是非定义性的模块表示理论。本文考虑了特征为p的域上的Lie型有限群，研究了特征不同于p的域上的表示，其中还应包括与Lie型群密切相关的Coxeter群(如对称群)的表示理论。近年来，又一次出现了惊人的发展。Alperin、Dade和Broue提出了基本猜想，虽然这些猜想还远未得到证明，但许多特殊情况已经解决，导致对这个领域有了更深层次的普遍理解。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。通过经典群在其自然模上的作用和例外群在其伴随模上的作用，以表示论为主要工具之一，发展了关于有限单群和代数单群的子群的强大的并行理论。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，都封装在它的对称群中。因此，群论在数学和物理科学中都有很多应用。群论的很大一部分内容是研究群在各种空间中的作用。对向量空间上的群作用的研究称为表示论，对集合上的群作用的研究称为置换群理论，本次会议的重点将集中在这两个领域及其应用上。所有有限群的基础是所谓的单群，其中大多数是以自然的方式从简单代数群(如SL(n，K)，代数闭域K上的n×n行列式1矩阵组成的群)产生的。因此，人们主要关注这些单群的表示和置换群理论。这些地区充满了基本的猜想，如Lusztig、Alperin、Broue和Dade的猜想。虽然这些都还远未得到证实，但近年来，证明它们的努力已经导致了这一学科的惊人进步。会议将重点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.