Representations of Finite Groups

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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.