Graded blocks of group algebras
In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group...
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| Format: | Thesis |
| Language: | English |
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2010
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| _version_ | 1797105342854725632 |
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| author | Bogdanic, D |
| author2 | Erdmann, K |
| author_facet | Erdmann, K Bogdanic, D |
| author_sort | Bogdanic, D |
| collection | OXFORD |
| description | In this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group. |
| first_indexed | 2024-03-07T06:46:09Z |
| format | Thesis |
| id | oxford-uuid:faeaaeab-1fe6-46a9-8cbb-f3f633131a73 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:46:09Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:faeaaeab-1fe6-46a9-8cbb-f3f633131a732022-03-27T13:09:55ZGraded blocks of group algebrasThesishttp://purl.org/coar/resource_type/c_db06uuid:faeaaeab-1fe6-46a9-8cbb-f3f633131a73MathematicsHomological algebraRepresentation theory of finite groups and associative algebrasGroup theory and generalizations (mathematics)EnglishOxford University Research Archive - Valet2010Bogdanic, DErdmann, KRouquier, RIn this thesis we study gradings on blocks of group algebras. The motivation to study gradings on blocks of group algebras and their transfer via derived and stable equivalences originates from some of the most important open conjectures in representation theory, such as Broue’s abelian defect group conjecture. This conjecture predicts the existence of derived equivalences between categories of modules. Some attempts to prove Broue’s conjecture by lifting stable equivalences to derived equivalences highlight the importance of understanding the connection between transferring gradings via stable equivalences and transferring gradings via derived equivalences. The main idea that we use is the following. We start with an algebra which can be easily graded, and transfer this grading via derived or stable equivalence to another algebra which is not easily graded. We investigate the properties of the resulting grading. In the first chapter we list the background results that will be used in this thesis. In the second chapter we study gradings on Brauer tree algebras, a class of algebras that contains blocks of group algebras with cyclic defect groups. We show that there is a unique grading up to graded Morita equivalence and rescaling on an arbitrary basic Brauer tree algebra. The third chapter is devoted to the study of gradings on tame blocks of group algebras. We study extensively the class of blocks with dihedral defect groups. We investigate the existence, positivity and tightness of gradings, and we classify all gradings on these blocks up to graded Morita equivalence. The last chapter deals with the problem of transferring gradings via stable equivalences between blocks of group algebras. We demonstrate on three examples how such a transfer via stable equivalences is achieved between Brauer correspondents, where the group in question is a TI group. |
| spellingShingle | Mathematics Homological algebra Representation theory of finite groups and associative algebras Group theory and generalizations (mathematics) Bogdanic, D Graded blocks of group algebras |
| title | Graded blocks of group algebras |
| title_full | Graded blocks of group algebras |
| title_fullStr | Graded blocks of group algebras |
| title_full_unstemmed | Graded blocks of group algebras |
| title_short | Graded blocks of group algebras |
| title_sort | graded blocks of group algebras |
| topic | Mathematics Homological algebra Representation theory of finite groups and associative algebras Group theory and generalizations (mathematics) |
| work_keys_str_mv | AT bogdanicd gradedblocksofgroupalgebras |