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:
Schaeffer Fry, Amanda
Issue Date:
2013
Publisher:
The University of Arizona.
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:
irreducible restriction; local-global; maximal subgroup; representation theory; Mathematics; finite classical group
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Tiep, Pham H.

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleIrreducible Representations of Finite Groups of Lie Type: On the Irreducible Restriction Problem and Some Local-Global Conjecturesen_US
dc.creatorSchaeffer Fry, Amandaen_US
dc.contributor.authorSchaeffer Fry, Amandaen_US
dc.date.issued2013-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.abstractIn 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.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectirreducible restrictionen_US
dc.subjectlocal-globalen_US
dc.subjectmaximal subgroupen_US
dc.subjectrepresentation theoryen_US
dc.subjectMathematicsen_US
dc.subjectfinite classical groupen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorTiep, Pham H.en_US
dc.contributor.committeememberTiep, Pham H.en_US
dc.contributor.committeememberLux, Klausen_US
dc.contributor.committeememberSavitt, Daviden_US
dc.contributor.committeememberThakur, Dineshen_US
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