Representations of Algebras : 17th Workshop and International Conference on Representation of Algebras, August 10-19, 2016, Syracuse University, Syracuse, New York

This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 10-19, 2016, at Syracuse University, Syracuse, NY. Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras. Read more... Abstract: This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 10-19, 2016, at Syracuse University, Syracuse, NY. Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative resolutions, twisted commutative algebras, and upper cluster algebras Cover 1 Title page 4 Contents 6 Preface 8 Commutative algebraic groups up to isogeny. II 10 1. Introduction 10 2. A construction of hereditary categories 12 2.1. Two preliminary results 12 2.2. Torsion pairs 13 2.3. The category of extensions 16 2.4. Universal extensions 18 2.5. Relation to module categories 21 3. Applications to commutative algebraic groups 24 3.1. Some isogeny categories 24 3.2. More isogeny categories 30 3.3. Functors of points 36 3.4. Finiteness conditions for Hom and Ext groups 38 3.5. Finiteness representation type: an example 40 References 43 Noncommutative resolutions of discriminants 46 1. Introduction 46 2. Reflection groups 47 3. (Noncommutative) resolutions of singularities 50 4. The classical McKay correspondence 52 5. NCRs of discriminants 56 6. Further questions 58 7. Acknowledgements 58 References 58 Polyhedral models for tensor product multiplicities 62 Introduction 62 1. Graded Upper Cluster Algebras 64 2. Auslander-Reiten theory of Presentations 65 3. Cluster Character from Quiver with Potential 70 4. iARt QPs 71 5. Remarks on Non-simply Laced Cases 74 Acknowledgment 76 References 76 Special multiserial algebras, Brauer configuration algebras and more: A survey 78 1. Introduction 78 2. Multiserial and special multiserial algebras 79 3. Algebras defined by cycles 80 4. Brauer configurations and Brauer configuration algebras 80 5. Connection results 81 6. Examples 82 7. Almost gentle algebras 83 8. Representations of multiserial algebras 84 9. Radical cubed zero 84 References 85 Nakayama-type phenomena in higher Auslander–Reiten theory 88 1. Introduction 88 2. Preliminaries 89 3. Higher Nakayama algebras 93 4. Obstructions to an alternative definition of higher Nakayama algebras 97 5. Cluster categories of type A_{n} and A_{∞} 100 References 105 K-polynomials of type A quiver orbit closures and lacing diagrams 108 1. Background and context 108 2. Lacing diagrams 110 3. K-polynomials of quiver orbit closures 114 4. The component formula 117 5. Open problems 119 References 120 Krull–Gabriel dimension and the Ziegler spectrum 124 1. Purity in categories of modules 125 2. The Krull–Gabriel dimension of R 130 3. Examples 133 References 137 On the K-theory of weighted projective curves 140 Introduction 140 1. Coherent sheaves on a smooth projective curve 141 1.1. The Euler form 142 1.2. Shift action associated to a point 145 1.3. The divisor sequence 146 2. Coherent sheaves on a weighted projective curve 148 2.1. The category of p-cycles 148 2.2. The reduced (or numerical) Grothendieck group 150 2.3. Attaching tubes 151 2.4. Orbifold Euler characteristic and weighted Riemann-Roch 152 2.5. Impact of the Euler characteristic 153 2.6. Shift action, weighted divisor group and weighted Picard group 155 2.7. The localization sequence 158 Appendix A. Multiplicative structure 159 Acknowledgements 161 References 161 Finite-dimensional algebras arising as blocks of finite group algebras 164 Introduction 164 1. Properties of blocks of finite group algebras 165 2. Basic algebras of dimension at most 12 of blocks 166 3. Defect groups of block algebras 172 4. Some finiteness conjectures 174 5. Hochschild cohomology background 177 6. HH0(B) 179 7. HH1(B) 180 8. Integrable derivations 183 9. Invariance properties of integrable derivations 185 10. Separably equivalent algebras 187 11. Finitistic and dominant dimensions 189 12. Fusion and algebra structure of blocks 191 References 194 Kronecker modules generated by modules of length 2 198 1. Introduction 198 2. Preliminaries 200 3. The bristled modules for n=1 and n=2. 202 4. Proof of the main theorem. 203 5. Further considerations. 216 Appendix A. The bristle variety β(M) of a Kronecker module M. 219 Appendix B. Bristled modules for arbitrary artin algebras. 220 Appendix C. A non-finiteness assertion for tame algebras. 222 References 223 Noetherian properties in representation theory 224 1. Introduction 224 2. Two examples 225 3. Twisted commutative algebras 226 4. Noetherian properties 229 5. What next? 231 References 232 Thick tensor ideals of right bounded derived categories of commutative rings 234 Introduction 234 1. Tensor triangulated categories and Balmer spectra 235 2. Compactly and cocompactly generated thick tensor ideals of \dm(R) 241 3. The Balmer spectrum of \dm(R) and classifications of thick tensor ideals 246 4. The case of discrete valuation rings and applications 256 Acknowledgments 259 References 259 Stability, shards, and preprojective algebras 260 1. Semistable subcategories 260 2. Preprojective algebras 262 3. Weyl groups 263 4. Shards 264 5. Join-irreducibles of W and bricks of Π 267 6. Technical Lemmas 268 7. Proof of Main Theorem 269 8. Connection to other work 270 Acknowledgements 270 References 270 Computations and applications of some homological constants for polynomial representations of GL_{n} 272 1. Introduction 272 2. Applications of the constants i(F,r) and p(F,r) 275 3. Computations of i(F,r) and p(F,r) 282 References 289 Back Cover 293 Cover......Page 1 Title page......Page 4 Contents......Page 6 Preface......Page 8 1. Introduction......Page 10 2.1. Two preliminary results......Page 12 2.2. Torsion pairs......Page 13 2.3. The category of extensions......Page 16 2.4. Universal extensions......Page 18 2.5. Relation to module categories......Page 21 3.1. Some isogeny categories......Page 24 3.2. More isogeny categories......Page 30 3.3. Functors of points......Page 36 3.4. Finiteness conditions for Hom and Ext groups......Page 38 3.5. Finiteness representation type: an example......Page 40 References......Page 43 1. Introduction......Page 46 2. Reflection groups......Page 47 3. (Noncommutative) resolutions of singularities......Page 50 4. The classical McKay correspondence......Page 52 5. NCRs of discriminants......Page 56 References......Page 58 Introduction......Page 62 1. Graded Upper Cluster Algebras......Page 64 2. Auslander-Reiten theory of Presentations......Page 65 3. Cluster Character from Quiver with Potential......Page 70 4. iARt QPs......Page 71 5. Remarks on Non-simply Laced Cases......Page 74 References......Page 76 1. Introduction......Page 78 2. Multiserial and special multiserial algebras......Page 79 4. Brauer configurations and Brauer configuration algebras......Page 80 5. Connection results......Page 81 6. Examples......Page 82 7. Almost gentle algebras......Page 83 9. Radical cubed zero......Page 84 References......Page 85 1. Introduction......Page 88 2. Preliminaries......Page 89 3. Higher Nakayama algebras......Page 93 4. Obstructions to an alternative definition of higher Nakayama algebras......Page 97 5. Cluster categories of type _{} and _{∞}......Page 100 References......Page 105 1. Background and context......Page 108 2. Lacing diagrams......Page 110 3. K-polynomials of quiver orbit closures......Page 114 4. The component formula......Page 117 5. Open problems......Page 119 References......Page 120 Krull–Gabriel dimension and the Ziegler spectrum......Page 124 1. Purity in categories of modules......Page 125 2. The Krull–Gabriel dimension of R......Page 130 3. Examples......Page 133 References......Page 137 Introduction......Page 140 1. Coherent sheaves on a smooth projective curve......Page 141 1.1. The Euler form......Page 142 1.2. Shift action associated to a point......Page 145 1.3. The divisor sequence......Page 146 2.1. The category of -cycles......Page 148 2.2. The reduced (or numerical) Grothendieck group......Page 150 2.3. Attaching tubes......Page 151 2.4. Orbifold Euler characteristic and weighted Riemann-Roch......Page 152 2.5. Impact of the Euler characteristic......Page 153 2.6. Shift action, weighted divisor group and weighted Picard group......Page 155 2.7. The localization sequence......Page 158 Appendix A. Multiplicative structure......Page 159 References......Page 161 Introduction......Page 164 1. Properties of blocks of finite group algebras......Page 165 2. Basic algebras of dimension at most 12 of blocks......Page 166 3. Defect groups of block algebras......Page 172 4. Some finiteness conjectures......Page 174 5. Hochschild cohomology background......Page 177 6. 0()......Page 179 7. 1()......Page 180 8. Integrable derivations......Page 183 9. Invariance properties of integrable derivations......Page 185 10. Separably equivalent algebras......Page 187 11. Finitistic and dominant dimensions......Page 189 12. Fusion and algebra structure of blocks......Page 191 References......Page 194 1. Introduction......Page 198 2. Preliminaries......Page 200 3. The bristled modules for =1 and =2.......Page 202 4. Proof of the main theorem.......Page 203 5. Further considerations.......Page 216 Appendix A. The bristle variety () of a Kronecker module .......Page 219 Appendix B. Bristled modules for arbitrary artin algebras.......Page 220 Appendix C. A non-finiteness assertion for tame algebras.......Page 222 References......Page 223 1. Introduction......Page 224 2. Two examples......Page 225 3. Twisted commutative algebras......Page 226 4. Noetherian properties......Page 229 5. What next?......Page 231 References......Page 232 Introduction......Page 234 1. Tensor triangulated categories and Balmer spectra......Page 235 2. Compactly and cocompactly generated thick tensor ideals of \dm()......Page 241 3. The Balmer spectrum of \dm() and classifications of thick tensor ideals......Page 246 4. The case of discrete valuation rings and applications......Page 256 References......Page 259 1. Semistable subcategories......Page 260 2. Preprojective algebras......Page 262 3. Weyl groups......Page 263 4. Shards......Page 264 5. Join-irreducibles of and bricks of Π......Page 267 6. Technical Lemmas......Page 268 7. Proof of Main Theorem......Page 269 References......Page 270 1. Introduction......Page 272 2. Applications of the constants (,) and (,)......Page 275 3. Computations of (,) and (,)......Page 282 References......Page 289 Back Cover......Page 293 "This volume contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held from August 1-19, 2016, at Syracuse University, Syracuse, New York. Included are three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories. Other articles represent contributions to areas in and related to representation theory, such as noncommutative algebras, and upper cluster algebras."--Page 4 of cover Contains the proceedings of the 17th Workshop and International Conference on Representations of Algebras (ICRA 2016), held in August 2016, at Syracuse University. This volume includes three survey articles based on short courses in the areas of commutative algebraic groups, modular group representation theory, and thick tensor ideals of bounded derived categories.