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Representation of finite groups: conjectures, reductions, and applications

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In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

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Acknowledgements

The author gratefully acknowledges the support of the NSF (Grants DMS-0901241 and DMS-1201374).

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Pham Huu Tiep

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Correspondence to Pham Huu Tiep.

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This survey is based in part on the plenary addresses given by the author at the 2012 Spring Western Section Meeting of the American Mathematical Society (University of Hawaii at Manoa, Honolulu, HI, March 3–4, 2012) and the 8th Congress of Vietnamese Mathematicians (Nhatrang, Vietnam, Aug. 10–14, 2013), as well as the lectures given by the author at the Annual Meeting of the Deutsche Forschungsgemeinschaft (DFG) Priority Programme on Representation Theory SPP 1388, Bad Boll, Germany, March 25–28, 2013. It is a pleasure to thank the National Science Foundation, the Deutsche Forschungsgemeinschaft, and the Vietnam Institute of Advanced Study in Mathematics for partial support.

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Tiep, P.H. Representation of finite groups: conjectures, reductions, and applications. Acta Math Vietnam. 39, 87–109 (2014). https://doi.org/10.1007/s40306-013-0043-y

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