Representations of elementary abelian p-groups and vector bundles:
Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unp...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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Cambridge
Cambridge University Press
2017
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| Schriftenreihe: | Cambridge tracts in mathematics
208 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 UBR01 Volltext Inhaltsverzeichnis |
| Zusammenfassung: | Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory |
| Beschreibung: | 1 Online-Ressource (xvii, 328 Seiten) |
| ISBN: | 9781316795699 |
| DOI: | 10.1017/9781316795699 |
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| 520 | |a Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory | ||
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Datensatz im Suchindex
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| adam_text | Titel: Representations of elementary Abelian p-groups and vector bundles
Autor: Benson, David J
Jahr: 2017
Representations of Elementary Abelian p-Groups and Vector Bundles DAVID J^BENSON University of Aberdeen Cambridge UNIVERSITY PRESS
Contents Preface page xi Introduction xiii 1 Modular Representations and Elementary Abelian Groups 1 1.1 Introduction 1 1.2 Representation Type 1 1.3 Shifted Subgroups 3 1.4 The Language of n -Points 5 1.5 The Stable Module Category 6 1.6 The Derived Category 8 1.7 Singularity Categories 10 1.8 Cohomology of Elementary Abelian p-Groups 12 1.9 Chouinard’s Theorem, Dade’s Lemma and Rank Varieties 15 1.10 Carlson’s Modules, and a Matrix Version 18 1.11 Diagrams for Modules 20 1.12 Tensor Products 24 1.13 Duality 26 1.14 Symmetric and Exterior Powers 28 1.15 Schur Functions 29 1.16 Schur Functors 31 1.17 Radical Layers of kE 33 1.18 Twisted Versions of k E 35 2 Cyclic Groups of Order p 38 2.1 Introduction 38 2.2 Modules for Z/ p 38 2.3 Tensor Products 39 2.4 Gaussian Polynomials 41 2.5 Generalised Gaussian Polynomials and a Hook Formula 42 v
45 47 50 52 54 54 55 57 59 60 62 63 64 65 69 72 72 75 76 78 83 86 89 89 94 98 101 103 106 110 112 114 116 118 119 120 123 127 129 129 131 Contents 2.6 À-Rings and Representations of SL( 2, C) 2.7 The Representation Ring of Z/p 2.8 Symmetric and Exterior Powers of Jordan Blocks 2.9 Schur Functors for SL( 2, C) and Z/p Background from Algebraic Geometry 3.1 Affine Space and Affine Varieties 3.2 Generic Points and Closed Points 3.3 Projective Space and Projective Varieties 3.4 Tangent Spaces 3.5 Presheaves and Sheaves 3.6 Stalks and Sheafification 3.7 The Language of Schemes 3.8 Sheaves of Modules 3.9 Coherent Sheaves on Projective Varieties 3.10 Cohomology of Sheaves Jordan Type 4.1 Nilvarieties 4.2 Matrices and Tangent Spaces 4.3 A Theorem of Gerstenhaber 4.4 Dominance Order and Nilpotent Jordan Types 4.5 Generic and Maximal Jordan Type 4.6 Tensor Products Modules of Constant Jordan Type 5.1 Introduction and Definitions 5.2 Homogeneous Modules 5.3 An Exact Category 5.4 Endotrivial Modules 5.5 Wild Representation Type 5.6 The Constant Image Property 5.7 The Generic Kernel 5.8 The Subquotient Rad _l .fi(jW)/Rad 2 .fi(A/) 5.9 The Constant Kernel Property 5.10 The Generic Image 5.11 W-Modules 5.12 Constant Jordan type with One Non-Projective Block 5.13 Rickard’s Conjecture 5.14 Consequences and Variations 5.15 Further Conjectures Vector Bundles on Projective Space 6.1 Definitions and First Properties 6.2 Tests for Vector Bundles
Contents vii 6.3 Vector Bundles on Projective Space 134 6.4 The Tangent Bundle and the Euler Sequence 136 6.5 Homogeneity and Uniformity 136 6.6 Monads and Subquotients 138 6.7 The Null Correlation Bundle 139 6.8 The Examples of Tango 140 6.9 Cohomology of Projective Space 141 6.10 Differential Forms and Bott’s Theorem 142 6.11 Simplicity 143 6.12 Hilbert’s Syzygy Theorem 146 Chern Classes 149 7.1 Chern Classes of Graded Modules 149 7.2 Chern Classes of Coherent Sheaves on P -1 151 7.3 Some Computations 154 7.4 Restriction of Vector Bundles 155 7.5 Chern Numbers of Twists and Duals 157 7.6 Chern Roots 159 7.7 Power Sums 160 7.8 The Hirzebruch-Riemann-Roch Theorem 166 7.9 Chern Numbers and the Frobenius Map 169 Modules of Constant Jordan Type and Vector Bundles 172 8.1 Introduction 172 8.2 The Operator 6 173 8.3 The Action of 0 on Fibres 174 8.4 The Functors 3~, and j 176 8.5 Twists and Syzygies 181 8.6 Chern Numbers of 3j(M) 183 8.7 The Construction: p = 2 185 8.8 The Construction: p Odd 187 8.9 Proof of the Realisation Theorem 189 8.10 Functoriality 192 8.11 Tensor Products 192 8.12 Negative Tate Cohomology 194 8.13 The BGG Correspondence 195 Examples 199 9.1 Modules for (Z/2) 2 199 9.2 Modules for (Z/p) 2 201 9.3 Larger Rank 203 9.4 Nilvarieties 204 9.5 The Tangent and Cotangent Bundles 205 9.6 The Null Correlation Bundle, p = 2 207
Contents viii 9.7 The Null Correlation Bundle, p Odd 208 9.8 Instanton Bundles 210 9.9 Schwarzenberger’s Bundles 212 9.10 The Examples of Tango 214 9.11 The Horrocks-Mumford Bundle 215 9.12 Automorphisms of the Horrocks-Mumford Bundle 218 9.13 Realising the Horrocks-Mumford Bundle 223 9.14 The Horroeks Parent Bundle and the Tango Bundle 226 10 Restrictions Coming from Chern Numbers 232 10.1 Matrices of Constant Rank 232 10.2 Congruences on Chern Numbers 235 10.3 Restrictions on Stable Jordan Type, p Odd 237 10.4 Eliminating More Stable Jordan Types 238 10.5 Restrictions on Jordan Type for p = 2 240 10.6 Applying Hirzebruch-Riemann-Roch for p = 2: The Case in = 0 244 10.7 Bypassing Hirzebruch-Riemann-Roch 246 10.8 Applying and Bypassing Hirzebruch-Riemann-Roch for p = 2: The Case I in r — 3 246 10.9 Nilvarieties of Constant Jordan Type [p] n for p 3 249 10.10 Nilvarieties with a Single Jordan Block 251 10.11 Babylonian Towers 253 11 Orlov’s Correspondence 255 11.1 Introduction 255 11.2 Maximal Cohen-Macaulay Modules 257 11.3 The Orlov Correspondence 259 11.4 The Functors 261 11.5 An Example 263 11.6 The Bidirectional Koszul Complex 264 11.7 A Bimodule Resolution 267 11.8 The Adjunction 269 11.9 The Equivalence 269 11.10 The Trivial Module 270 11.11 Computer Algebra 272 11.12 Cohomology 272 11.13 Twisted Versions nfkE 274 12 Phenomenology of Modules over Elementary Abelian p-Groups 276 12.1 Introduction 276 12.2 Module Constructions 278 12.3 Odd Primes Are More Difficult 280 12.4 Relative Cohomology 281
Contents IX 12.5 Small Modules for Quadrics, p — 2 283 12.6 Small Modules for Quadrics, p Odd 285 12.7 Trying to Understand the Specht Module S (3,) 286 12.8 Modules with Small Loewy Length 287 12.9 Small Modules for (L/p) 1 296 12.10 The Bound is Close to Sharp 299 Appendix A Modules for Z/ p 301 Appendix B Problems 308 References 312 Index 324
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| spelling | Benson, D. J. (David J.) 1955- Verfasser aut Representations of elementary abelian p-groups and vector bundles David J. Benson, University of Aberdeen Cambridge Cambridge University Press 2017 1 Online-Ressource (xvii, 328 Seiten) txt rdacontent c rdamedia cr rdacarrier Cambridge tracts in mathematics 208 Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory Abelian p-groups Abelian groups Vector bundles Erscheint auch als Druck-Ausgabe 978-1-107-17417-7 https://doi.org/10.1017/9781316795699 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029449724&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Benson, D. J. (David J.) 1955- Representations of elementary abelian p-groups and vector bundles Abelian p-groups Abelian groups Vector bundles |
| title | Representations of elementary abelian p-groups and vector bundles |
| title_auth | Representations of elementary abelian p-groups and vector bundles |
| title_exact_search | Representations of elementary abelian p-groups and vector bundles |
| title_full | Representations of elementary abelian p-groups and vector bundles David J. Benson, University of Aberdeen |
| title_fullStr | Representations of elementary abelian p-groups and vector bundles David J. Benson, University of Aberdeen |
| title_full_unstemmed | Representations of elementary abelian p-groups and vector bundles David J. Benson, University of Aberdeen |
| title_short | Representations of elementary abelian p-groups and vector bundles |
| title_sort | representations of elementary abelian p groups and vector bundles |
| topic | Abelian p-groups Abelian groups Vector bundles |
| topic_facet | Abelian p-groups Abelian groups Vector bundles |
| url | https://doi.org/10.1017/9781316795699 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029449724&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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