### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 87-109 |

Number of pages | 23 |

Journal | Acta Mathematica Vietnamica |

Volume | 39 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2014 |

### Fingerprint

### Keywords

- Adequate groups
- Alperin weight conjecture
- Brauer height zero conjecture
- Crepant resolutions
- Finite groups
- Kollár–Larsen problem
- Larsen’s conjecture
- Low-dimensional representations
- Representation theory
- Waring problem

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Representation of finite groups : conjectures, reductions, and applications.** / Tiep, Pham Huu.

Research output: Contribution to journal › Article

*Acta Mathematica Vietnamica*, vol. 39, no. 1, pp. 87-109. https://doi.org/10.1007/s40306-013-0043-y

}

TY - JOUR

T1 - Representation of finite groups

T2 - conjectures, reductions, and applications

AU - Tiep, Pham Huu

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

UR - http://www.scopus.com/inward/record.url?scp=84957575671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957575671&partnerID=8YFLogxK

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 -