Representations of Finite Groups

有限群的表示

• 批准号: 0070630
• 负责人: Robert Boltje
• 金额: $ 6.09万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070630&HistoricalAwards=false
• 关键词: Representations; Finite; Groups

Abstract for R. Boltje's NSF Proposal 00706301. The principal investigator is doing research in the field of representation theory of finite groups. More precisely, he works on questions related to the conjectures of Alperin, Dade, and Broue. These conjectures state that certain invariants of a finite group and a fixed prime number can be determined by the same invariants of subgroups which are again related to that prime. The first conjectures are concerned with invariants that are integers, whereas the third conjecture predicts equivalences of categories. The first two conjectures and consequences of the third have been verified for a convincing number of examples. It is likely that these three conjectures are consequences of a single feature which is still hidden behind the scene. Probably more important than solving these particular conjectures would be the discovery of this feature. The principal investigator works on general ideas of how to approach these conjectures, partially in collaboration with B. K"ulshammer. A graduate student is involved in this effort by writing computer programs which will either discard or give more evidence to the applicability of one of the suggested approaches.2. The principal investigator studies mysterious coincidences in the representation theory of groups which have been discovered about fifteen years ago but could not be explained so far. This research is not aimed at immediate applications outside mathematics. However, in the history of the interplay between mathematics and other sciences, in particular physics, it is a repeated pattern that theories and results which were considered as important within the edifice of mathematics became precisely what was needed in the other sciences in order to describe our real world. For example, group representations which have been studied a century ago became the right tool to describe particles in nuclear physics.

R. Boltje 的 NSF 提案 00706301 的摘要。主要研究者正在有限群表示论领域进行研究。 更准确地说，他研究与阿尔佩林、戴德和布鲁的猜想相关的问题。 这些猜想指出，有限群和固定素数的某些不变量可以由再次与该素数相关的子群的相同不变量来确定。 第一个猜想涉及整数的不变量，而第三个猜想预测类别的等价性。 前两个猜想和第三个猜想的结果已被大量令人信服的例子所验证。 这三个猜想很可能是仍然隐藏在幕后的单一特征的结果。 也许比解决这些特定猜想更重要的是这个特征的发现。 首席研究员致力于研究如何处理这些猜想的一般想法，部分与 B. K"ulshammer 合作。一名研究生通过编写计算机程序参与这项工作，该程序将丢弃或提供更多证据来证明以下之一的适用性2. 首席研究员研究了大约十五年前发现但迄今为止无法解释的群表示论中的神秘巧合，但这项研究的目的并不是在数学之外的直接应用。数学与数学之间的相互作用对于其他科学，特别是物理学来说，在数学大厦中被认为重要的理论和结果正是其他科学所需要的，以描述我们的现实世界，这是一种重复的模式。一个世纪前的研究成为描述核物理中粒子的正确工具。