Representations of Finite Groups

R. Boltje的NSF提案00706301摘要。首席研究员正在研究有限群的表示理论。更准确地说，他研究的是与Alperin、Dade和Broue的猜想有关的问题。这些猜想表明，有限群和固定素数的某些不变量可以由与该素数相关的子群的相同不变量决定。第一个猜想是关于整数不变量的，而第三个猜想预测了范畴的等价。前两个猜想和第三个猜想的结果已经通过大量令人信服的例子得到了验证。这三种猜想很可能是隐藏在场景背后的单一特征的结果。也许比解决这些特殊的猜想更重要的是发现这个特征。首席研究员与b·K·乌尔沙默（B. K . ulshammer）合作，研究如何处理这些猜想的一般思路。研究生通过编写计算机程序来参与这项工作，这些程序将抛弃或提供更多证据来证明所建议的方法之一的适用性。首席研究员研究了群体表征理论中神秘的巧合，这些巧合大约在15年前就被发现了，但迄今为止还无法解释。这项研究并不针对数学以外的直接应用。然而，在数学与其他科学，特别是物理学相互作用的历史中，有一种重复的模式，即在数学大厦中被认为是重要的理论和结果，恰好成为其他科学为了描述我们的真实世界所需要的东西。例如，一个世纪前研究的群表示成为描述核物理中粒子的正确工具。

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.