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In Search of Bounds on the Dimension of Ext Between Irreducible Modules for Finite Groups of Lie Type

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In Search of Bounds on the Dimension of Ext Between Irreducible Modules for Finite Groups of Lie Type In Search of Bounds on the Dimension of Ext between Irreducible Modules for Finite Groups of Lie Type Veronica Shalotenko Sterling, Virginia B.S., Cooper Union for the Advancement of Science and Art, 2012 A Dissertation presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics University of Virginia May, 2018 Contents 0 Preface 5 I Background 9 1 The Ext Functor and Group Cohomology 9 1.1 The Ext Functor . .9 1.2 Ext and Extensions . 10 1.3 Some Properties of the Ext Functor . 11 1.3.1 The Long Exact Sequence in Ext . 11 1.3.2 The Eckmann-Shapiro Lemma . 11 1.3.3 Ext1 between Irreducible Modules . 13 1.4 Group Cohomology . 15 1.4.1 Group Cohomology via the Standard Resolution . 15 1.4.2 Group Cohomology via Cochains . 16 1.4.3 Group Cohomology in Low Degree . 17 1.4.4 The inflation homomorphism on group cohomology. 17 1.5 The Restriction and Corestriction Homomorphisms on Group Cohomology . 18 1.5.1 The Restriction Homomorphism . 18 1.5.2 The Corestriction Homomorphism . 18 1.6 Composition of Restriction and Corestriction . 19 2 Groups with a BN-pair 21 2.1 The Axioms of a Group with a BN-pair . 21 2.2 Parabolic Subgroups . 22 2.3 Finite Groups with a Split BN-Pair . 23 3 Finite groups of Lie type 24 3.1 Steinberg endomorphisms . 24 3.2 The action of F on the character group of T .................. 26 3.3 The classification of the finite groups of Lie type . 28 3.4 The Root System and Root Subgroups of the Finite Group of Lie Type GF . 30 3.5 Construction of a BN-pair for the Finite Group of Lie Type GF ....... 31 3.6 The Levi Decomposition of a Parabolic Subgroup of a Finite Group of Lie Type G with a Split BN-Pair . 32 3.7 An Order Formula for a Finite Group of Lie Type Arising from a Simple Linear Algebraic Group . 33 4 Harish-Chandra Series 34 4.1 Harish-Chandra Induction and Restriction . 34 4.2 Some Properties of Harish-Chandra Induction and Restriction . 35 4.3 Cuspidal Modules and Harish-Chandra Series. 35 4.4 The Principal Series Representations . 38 1 5 The Steinberg Module 39 5.1 Steinberg's Original Results on the \Steinberg Module" . 39 5.1.1 A Basis for the Steinberg Module . 39 5.1.2 An Irreducibility Criterion for the Steinberg Module . 40 5.2 A New Proof of the Irreducibility Criterion for the Steinberg Module . 43 5.2.1 Introduction . 43 5.2.2 The Proof of the Irreducibility Criterion . 43 6 Hecke Algebras 46 6.1 The Hecke algebra associated to a Coxeter system . 46 6.2 Parabolic Subalgebras of He ........................... 46 6.3 Specializing the Generic Hecke Algebra . 47 6.4 The \Double Coset" Hecke Algebra . 48 6.4.1 Definition of a Hecke Algebra given in [12] . 48 6.4.2 The case of a group with a BN-pair . 49 6.5 Hecke Algebra Associated with a Harish Chandra Series . 53 7 Highest Weight Categories and Quasi-hereditary Algebras 57 7.1 Preliminaries . 57 7.2 Homological Properties of the Standard and Costandard Objects . 58 7.3 The Ringel Dual . 59 7.4 Global Dimension of a Quasi-hereditary Algebra . 59 7.5 Heredity Ideals . 61 8 The q-Schur algebra 63 9 The Indexing of the Irreducible kGLn(q)-modules 65 9.1 The First Indexing of the Irreducible kGLn(q)-Modules (CPS) . 65 9.1.1 A Key Theorem . 65 9.1.2 A decomposition of OGLn(q) into sums of blocks . 66 9.1.3 An OG-module which satisfies the hypotheses of the key theorem . 66 9.1.4 A Morita Equivalence for GLn(q)..................... 68 9.1.5 An Indexing of the Irreducible kGLn(q)-modules . 69 9.2 The Second Indexing of the Irreducible kGLn(q)-Modules . 70 9.3 The Third Indexing of the Irreducible kGLn(q)-Modules . 71 9.4 Indexing of the Irreducible kGLn(q)-Modules belonging to the Unipotent Prin- cipal Series . 72 II Generalizing the Results of Guralnick and Tiep to Find Bounds on the Dimension of Ext1 between Irreducible Modules 73 10 A Summary of the Results of Guralnick and Tiep [35] 73 10.1 Introduction . 73 10.2 Irreducible Modules V with V B 6=0....................... 75 10.3 Irreducible Modules V with V B =0....................... 76 2 11 A Bound on the Dimension of Ext1 between Irreducibles in the Unipotent Principal Series 80 11.1 Introduction . 80 11.2 A Preliminary Result . 80 11.3 Case I: r - jBj ................................... 81 11.4 Case II: r j jBj .................................. 82 12 A New Proof of Guralnick and Tiep's Bound on the Dimension of H1(G; V ) in the case that V B = 0 85 12.1 Introduction . 85 1 12.2 A first result on the dimension of H (G; V ) when V2 = Irrk(GjB). 85 12.3 Groups of Rank 1 . 86 12.4 Groups of Higher Rank . 88 13 A Bound on the Dimension of Ext1 Between Principal Series Irreducible Modules in Cross Characteristic 91 13.1 Introduction . 91 1 13.2 A Bound on the Dimension of ExtkG(Y; V ) when Y and V belong to distinct Principal Series . 92 1 13.3 A Bound on the Dimension of ExtkG(Y; V ) when Y and V belong to the same Principal Series . 94 13.3.1 Case I: r - jBj ............................... 95 13.3.2 Case II: r j jBj .............................. 95 14 A Bound on the Dimension of Ext1 between a Unipotent Principal Series Representation and an Irreducible Outside the Unipotent Principal Series 98 14.1 Introduction . 98 14.2 Some Preliminaries from [29, Section 4]: The Steinberg Module and Harish- Chandra Series . 98 14.2.1 Introduction . 98 14.2.2 The BN-Pair of G ............................ 99 14.2.3 An r-modular system. 99 14.2.4 An OG-lattice which yields the Steinberg module . 100 14.2.5 The Gelfand-Graev module . 100 14.2.6 Harish-Chandra series arising from the regular character σ ...... 101 1 14.3 A Bound on dim ExtkG(Y; V ).......................... 103 III Explicit Computations of Bounds on the Dimension of Ext1 Between Irreducible Modules for GLn(q) in Cross Characteristic106 15 Introduction 106 16 The Harish-Chandra Vertex of the Module D2(1; λ) 107 17 Some Examples of Bounds on the Dimension of Ext1 109 3 IV Generalizing the Results of Cline, Parshall, and Scott [9] to Calculate Ext Groups between Irreducible kGLn(q)-Modules 112 18 Self-Extensions of Irreducible Modules for GLn(q) in Certain Cross Char- acteristics 112 18.1 Introduction . 112 1 18.2 ExtkG(D(1; λ);D(1; λ)) in the case that λ ` n is l-restricted. 113 19 Calculations of Higher Ext Groups Between Certain Modules for GLn(q) in Cross Characteristic 116 19.1 Introduction . 116 19.2 A Generalization of [9, Theorem 12.4] . 116 V Conclusion 124 VI Appendices 126 20 Cohomology of Cyclic r-Groups 126 21 The Steinberg Module in Defining Characteristic 128 22 The Character of the Steinberg Module in Characteristic 0 130 22.1 Introduction . 130 22.2 A generalized character of G ........................... 130 22.3 The character of StC ............................... 132 4 0 Preface In the second half of the 20th century, one of the biggest open problems in mathematics was the classification of the finite simple groups. Research which would eventually serve as the basis for this classification emerged in the 1950s, but there was no organized effort to classify the finite simple groups until the 1970s. A major breakthrough toward the classification came in 1963 with the Feit-Thompson Theorem, which showed that every finite group of odd order is solvable. In 1972, Daniel Gorenstein gave a series of lectures at the University of Chicago in which he outlined a program for the classification of the finite simple groups [27]. The mathematical community responded to Gorenstein's plan with great enthusiasm. In the years after Gorenstein presented his program, over 100 mathematicians across the world worked to implement his ideas. In 1981, Gorenstein announced that the proof of the classification theorem was complete. However, upon review, several gaps were discovered in the proof, and work on the classification theorem continued. The problems with the proof of the classification theorem were highly non-trivial; it took Aschbacher and Smith seven years (and two books) to resolve the last gap [27]. Ashbacher and Smith finished their work in 2004, and the consensus now is that the proof of the classification theorem is complete. (Goren- stein died in 1992, twelve years before the proof of the classification theorem was completed.) The current proof of the classification theorem is immense - it consists of over 10,000 pages across about 500 journal articles [27]. Due to the scope of this work, few mathe- maticians understand the full proof of the classification theorem. In the 1980s, Gorenstein initiated an effort to address this issue and began working together with Lyons and Solomon to revise and shorten the proof of the classification theorem. After Gorenstein's death, Lyons and Solomon continued working on this \revisionism" project. The American Mathematical Society is publishing their work in a multi-volume series [32]; the first six volumes of this series were published between the years of 1994 and 2005 (the first three are introductory), and more are expected in the future [27].
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