Representation Theory of Finite Groups

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The principal investigator is working in the field ofrepresentation theory of finite groups. Mainly he studiesquestions related to the conjectures of Alperin, Dade, and Broue.These conjectures state that certain invariants of a finite groupand a fixed prime number can be determined by the same invariantsof subgroups which are again related to that prime. The firstconjectures 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 beenverified for a convincing number of examples. It is likely thatthese three conjectures are consequences of a single fact, object,theory, or construction which is still hidden. Probably moreimportant than solving these particular conjectures would be thediscovery of this hidden feature. The principal investigator workson general ideas of how to approach these conjectures. Graduatestudents are involved in this effort by writing computer programswhich will either discard or give more evidence to theapplicability of one of the suggested approaches.The principal investigator studies representations of groups.Groups can be viewed as mathematical abstractions of the notion ofsymmetry. They are among the most basic notions in many fields ofmathematics and are applied widely, for example in cryptology,physics, chemistry, and computer science. Representations ofgroups are manifestations of a particular type of symmetry onother mathematical objects, as for example the four rotations andfour reflections, that a square in the plane allows, form arepresentation of a group with 8 elements on a two-dimensionalvector space. More specifically, the principal investigatorstudies mysterious coincidences in the representation theory offinite groups which have been discovered about fifteen years agobut could not be explained so far. This research is not aimed atimmediate applications outside mathematics. However, in thehistory of the interplay between mathematics and other sciences,in particular with physics, it is a repeated pattern that theoriesand results which were considered as important within the edificeof mathematics became precisely what was needed in the othersciences in order to describe our real world. For example, grouprepresentations which have been studied a century ago became muchlater the right tool to describe particles in nuclear physics.