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# Irreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjectures

- Persistent Link:
- http://hdl.handle.net/10150/293407
- Title:
- Irreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjectures
- Author:
- Issue Date:
- 2013
- Publisher:
- Rights:
- Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
- Abstract:
- In this thesis, we investigate various problems in the representation theory of finite groups of Lie type. In Chapter 2, we hope to make sense of the last statement - we will introduce some background and notation that will be useful for the remainder of the thesis. In Chapter 3, we find bounds for the largest irreducible representation degree of a finite unitary group. In Chapter 4, we describe the block distribution and Brauer characters in cross characteristic for Sp₆(2ᵃ) in terms of the irreducible ordinary characters. This will be useful in Chapter 5 and Chapter 7, which focus primarily on the group Sp₆(2ᵃ) and contain the main results of this thesis, which we now summarize. Given a subgroup H ≤ G and a representation V for G, we obtain the restriction V|H of V to H by viewing V as an FH-module. However, even if V is an irreducible representation of G, the restriction V|H may (and usually does) fail to remain irreducible as a representation of H. In Chapter 5, we classify all pairs (V, H), where H is a proper subgroup of G = Sp₆(q) or Sp₄(q) with q even, and V is an l-modular representation of G for l ≠ 2 which is absolutely irreducible as a representation of H. This problem is motivated by the Aschbacher-Scott program on classifying maximal subgroups of finite classical groups. The local-global philosophy plays an important role in many areas of mathematics. In the representation theory of finite groups, the so-called "local-global" conjectures would relate the representation theory of G to that of certain proper subgroups, such as the normalizer of a Sylow subgroup. One might hope that these conjectures could be proven by showing that they are true for all simple groups. Though this turns out not quite to be the case, some of these conjectures have been reduced to showing that a finite set of stronger conditions hold for all finite simple groups. In Chapter 7, we show that Sp₆(q) and Sp₄(q), q even, are "good" for these reductions.
- Type:
- text; Electronic Dissertation
- Keywords:
- Degree Name:
- Ph.D.
- Degree Level:
- doctoral
- Degree Program:
- Degree Grantor:
- University of Arizona
- Advisor:

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.language.iso | en | en_US |

dc.title | Irreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjectures | en_US |

dc.creator | Schaeffer Fry, Amanda | en_US |

dc.contributor.author | Schaeffer Fry, Amanda | en_US |

dc.date.issued | 2013 | - |

dc.publisher | The University of Arizona. | en_US |

dc.rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. | en_US |

dc.description.abstract | In this thesis, we investigate various problems in the representation theory of finite groups of Lie type. In Chapter 2, we hope to make sense of the last statement - we will introduce some background and notation that will be useful for the remainder of the thesis. In Chapter 3, we find bounds for the largest irreducible representation degree of a finite unitary group. In Chapter 4, we describe the block distribution and Brauer characters in cross characteristic for Sp₆(2ᵃ) in terms of the irreducible ordinary characters. This will be useful in Chapter 5 and Chapter 7, which focus primarily on the group Sp₆(2ᵃ) and contain the main results of this thesis, which we now summarize. Given a subgroup H ≤ G and a representation V for G, we obtain the restriction V|H of V to H by viewing V as an FH-module. However, even if V is an irreducible representation of G, the restriction V|H may (and usually does) fail to remain irreducible as a representation of H. In Chapter 5, we classify all pairs (V, H), where H is a proper subgroup of G = Sp₆(q) or Sp₄(q) with q even, and V is an l-modular representation of G for l ≠ 2 which is absolutely irreducible as a representation of H. This problem is motivated by the Aschbacher-Scott program on classifying maximal subgroups of finite classical groups. The local-global philosophy plays an important role in many areas of mathematics. In the representation theory of finite groups, the so-called "local-global" conjectures would relate the representation theory of G to that of certain proper subgroups, such as the normalizer of a Sylow subgroup. One might hope that these conjectures could be proven by showing that they are true for all simple groups. Though this turns out not quite to be the case, some of these conjectures have been reduced to showing that a finite set of stronger conditions hold for all finite simple groups. In Chapter 7, we show that Sp₆(q) and Sp₄(q), q even, are "good" for these reductions. | en_US |

dc.type | text | en_US |

dc.type | Electronic Dissertation | en_US |

dc.subject | irreducible restriction | en_US |

dc.subject | local-global | en_US |

dc.subject | maximal subgroup | en_US |

dc.subject | representation theory | en_US |

dc.subject | Mathematics | en_US |

dc.subject | finite classical group | en_US |

thesis.degree.name | Ph.D. | en_US |

thesis.degree.level | doctoral | en_US |

thesis.degree.discipline | Graduate College | en_US |

thesis.degree.discipline | Mathematics | en_US |

thesis.degree.grantor | University of Arizona | en_US |

dc.contributor.advisor | Tiep, Pham H. | en_US |

dc.contributor.committeemember | Tiep, Pham H. | en_US |

dc.contributor.committeemember | Lux, Klaus | en_US |

dc.contributor.committeemember | Savitt, David | en_US |

dc.contributor.committeemember | Thakur, Dinesh | en_US |

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