Global-Local Methods in Representation Theory (GLoRy)
Global-Local Methods in Representation Theory
Start date: Apr 1, 2012,
End date: Mar 31, 2016
Symmetry phenomena are well known to play a remarkable role in mathematics and its applications to chemistry, physics, and many other branches of natural sciences. It is, therefore, important to develop mathematical models to analyze symmetries in a unified way.Group theory provides such models in form of permutation representations, that is,groups acting on sets. After linearizing, these actions gain more structure, which can be encodedin matrices, facilitating both abstract investigations and concrete calculations.A natural question to ask then is, how far global properties of such linear representations of somecomplicated group are determined locally by better behaved subgroups.All of these are objectives of representation theory.The proposed project will contribute to this central problem of understanding the interplayof global and local representation theory. It aims, in particular, at gaining new evidence forthe long-standing conjectures of Broue and Feit, and at developing methods to subsequently attack these in greater generality. This will be achieved by a combination of very classical methods from representation theory and homological algebra, powerful up-to-date methods from computational algebra, and the promising novel concept of bisets and double Burnside rings.Since the proposed project addresses a wide range of cutting-edge problems that are of great interest to representation theorists, and moreover, have applications to related areas of mathematics as well, it carries a high potential for long-term cooperations with further research groups in Europe, and worldwide. Consequently, it comprises knowledge transfer within and into the EU, the introduction of a novel research approach, and the long-term integration of a high-potential researcher into the European science community.
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