TY - JOUR

T1 - Representation of finite groups

T2 - conjectures, reductions, and applications

AU - Tiep, Pham Huu

N1 - Funding Information:
The author gratefully acknowledges the support of the NSF (Grants DMS-0901241
Funding Information:
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.
Publisher Copyright:
© 2014, Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.

PY - 2014/3/1

Y1 - 2014/3/1

N2 - 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.

AB - 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.

KW - Adequate groups

KW - Alperin weight conjecture

KW - Brauer height zero conjecture

KW - Crepant resolutions

KW - Finite groups

KW - Kollár–Larsen problem

KW - Larsen’s conjecture

KW - Low-dimensional representations

KW - Representation theory

KW - Waring problem

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U2 - 10.1007/s40306-013-0043-y

DO - 10.1007/s40306-013-0043-y

M3 - Article

AN - SCOPUS:84957575671

VL - 39

SP - 87

EP - 109

JO - Acta Mathematica Vietnamica

JF - Acta Mathematica Vietnamica

SN - 0251-4184

IS - 1

ER -