{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:52285"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:52285","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Group rings over the p-adic integers","abstract":"The thesis \"Group Rings over the p-Adic Integers\" is concerned with the representation theory of finite groups over the ring of p-adic integers (or, usually, an algebraic extension thereof). It deals with blocks of special linear groups of degree two, blocks of dihedral defect and blocks of symmetric groups. In each case the aim is to describe basic orders of such blocks, that is, orders of minimal rank in the Morita equivalence class of the order in question. This thesis combines arithmetic methods from the theory of orders in semi simple algebras with methods from modular representation theory and representation theory of algebras (in particular derived equivalences). The thesis is structured as follows: The first chapter deals with the necessary representation theoretic and order theoretic prerequisites. In the second chapter, a new method is introduced which allows a reduction of the problem of lifting an algebra over a field of characteristic p to an order over an appropriate p-adic ring to the analogous problem for an algebra which is derived equivalent to the original one. In the third chapter the aforementioned method is applied to blocks of dihedral defect. There is a classification of such blocks over an algebraically closed field of characteristic two. Some of the algebras occurring in this classification can be lifted to an essentially unique order by elementary means. By making use of derived equivalences between different algebras in the classification, a description for all blocks of dihedral defect with more than one simple module is obtained. This solves an open problem concerning certain unknown scalars occurring in the classification of blocks of dihedral defect over an algebraically closed field of characteristic two. The third chapter then applies the method developed in the second chapter to blocks of special linear groups of degree two in defining characteristic. It is shown that such blocks have an essentially unique lift to an order over an appropriate p-adic ring, provided the field of coefficients for the group ring is sufficiently large. The crucial fact that is exploited here is that the blocks under consideration are derived equivalent to blocks of group rings of upper triangular unipotent matrices. As a corollary, an open conjecture of Nebe concerning the basic orders of blocks of special linear groups of degree two is proved. The last chapter of the thesis deals with blocks of symmetric groups. It contains a modified version of Scopes' reduction and a description of basic orders of defect two blocks of symmetric groups.","abstract_html":"The thesis "Group Rings over the p-Adic Integers" is concerned with the representation theory of finite groups over the ring of p-adic integers (or, usually, an algebraic extension thereof). It deals with blocks of special linear groups of degree two, blocks of dihedral defect and blocks of symmetric groups. In each case the aim is to describe basic orders of such blocks, that is, orders of minimal rank in the Morita equivalence class of the order in question. This thesis combines arithmetic methods from the theory of orders in semi simple algebras with methods from modular representation theory and representation theory of algebras (in particular derived equivalences). The thesis is structured as follows: The first chapter deals with the necessary representation theoretic and order theoretic prerequisites. In the second chapter, a new method is introduced which allows a reduction of the problem of lifting an algebra over a field of characteristic p to an order over an appropriate p-adic ring to the analogous problem for an algebra which is derived equivalent to the original one. In the third chapter the aforementioned method is applied to blocks of dihedral defect. There is a classification of such blocks over an algebraically closed field of characteristic two. Some of the algebras occurring in this classification can be lifted to an essentially unique order by elementary means. By making use of derived equivalences between different algebras in the classification, a description for all blocks of dihedral defect with more than one simple module is obtained. This solves an open problem concerning certain unknown scalars occurring in the classification of blocks of dihedral defect over an algebraically closed field of characteristic two. The third chapter then applies the method developed in the second chapter to blocks of special linear groups of degree two in defining characteristic. It is shown that such blocks have an essentially unique lift to an order over an appropriate p-adic ring, provided the field of coefficients for the group ring is sufficiently large. The crucial fact that is exploited here is that the blocks under consideration are derived equivalent to blocks of group rings of upper triangular unipotent matrices. As a corollary, an open conjecture of Nebe concerning the basic orders of blocks of special linear groups of degree two is proved. The last chapter of the thesis deals with blocks of symmetric groups. It contains a modified version of Scopes' reduction and a description of basic orders of defect two blocks of symmetric groups.","abstract_has_math":false,"creators":["Eisele, Florian"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Nebe, Gabriele"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-30T19:40:50Z","subjects":["info:eu-repo/classification/ddc/510","Ganzzahlige Darstellungstheorie","Darstellungstheorie","Mathematik","integral representation theory","representation theory"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114521%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114521%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114521%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/52285","outbound_label":"Repository record","outbound_source":"dc:identifier"},"source_record":{"url":"https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A52285","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nebe, Gabriele"]},{"key":"dc:creator","label":"Author","values":["Eisele, Florian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-40425"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Ganzzahlige Darstellungstheorie","Darstellungstheorie","Mathematik","integral representation theory","representation theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/52285","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114521%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The thesis \"Group Rings over the p-Adic Integers\" is concerned with the representation theory of finite groups over the ring of p-adic integers (or, usually, an algebraic extension thereof). It deals with blocks of special linear groups of degree two, blocks of dihedral defect and blocks of symmetric groups. In each case the aim is to describe basic orders of such blocks, that is, orders of minimal rank in the Morita equivalence class of the order in question. This thesis combines arithmetic methods from the theory of orders in semi simple algebras with methods from modular representation theory and representation theory of algebras (in particular derived equivalences). The thesis is structured as follows: The first chapter deals with the necessary representation theoretic and order theoretic prerequisites. In the second chapter, a new method is introduced which allows a reduction of the problem of lifting an algebra over a field of characteristic p to an order over an appropriate p-adic ring to the analogous problem for an algebra which is derived equivalent to the original one. In the third chapter the aforementioned method is applied to blocks of dihedral defect. There is a classification of such blocks over an algebraically closed field of characteristic two. Some of the algebras occurring in this classification can be lifted to an essentially unique order by elementary means. By making use of derived equivalences between different algebras in the classification, a description for all blocks of dihedral defect with more than one simple module is obtained. This solves an open problem concerning certain unknown scalars occurring in the classification of blocks of dihedral defect over an algebraically closed field of characteristic two. The third chapter then applies the method developed in the second chapter to blocks of special linear groups of degree two in defining characteristic. It is shown that such blocks have an essentially unique lift to an order over an appropriate p-adic ring, provided the field of coefficients for the group ring is sufficiently large. The crucial fact that is exploited here is that the blocks under consideration are derived equivalent to blocks of group rings of upper triangular unipotent matrices. As a corollary, an open conjecture of Nebe concerning the basic orders of blocks of special linear groups of degree two is proved. The last chapter of the thesis deals with blocks of symmetric groups. It contains a modified version of Scopes' reduction and a description of basic orders of defect two blocks of symmetric groups."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 129 S. (2012). = Aachen, Techn. Hochsch., Diss., 2012"]},{"key":"dc:title","label":"Title","values":["Group rings over the p-adic integers"]}]}],"canonical_facts":{"dc:contributor":["Nebe, Gabriele"],"dc:coverage":["DE"],"dc:creator":["Eisele, Florian"],"dc:date":["2012"],"dc:description":["The thesis \"Group Rings over the p-Adic Integers\" is concerned with the representation theory of finite groups over the ring of p-adic integers (or, usually, an algebraic extension thereof). It deals with blocks of special linear groups of degree two, blocks of dihedral defect and blocks of symmetric groups. In each case the aim is to describe basic orders of such blocks, that is, orders of minimal rank in the Morita equivalence class of the order in question. This thesis combines arithmetic methods from the theory of orders in semi simple algebras with methods from modular representation theory and representation theory of algebras (in particular derived equivalences). The thesis is structured as follows: The first chapter deals with the necessary representation theoretic and order theoretic prerequisites. In the second chapter, a new method is introduced which allows a reduction of the problem of lifting an algebra over a field of characteristic p to an order over an appropriate p-adic ring to the analogous problem for an algebra which is derived equivalent to the original one. In the third chapter the aforementioned method is applied to blocks of dihedral defect. There is a classification of such blocks over an algebraically closed field of characteristic two. Some of the algebras occurring in this classification can be lifted to an essentially unique order by elementary means. By making use of derived equivalences between different algebras in the classification, a description for all blocks of dihedral defect with more than one simple module is obtained. This solves an open problem concerning certain unknown scalars occurring in the classification of blocks of dihedral defect over an algebraically closed field of characteristic two. The third chapter then applies the method developed in the second chapter to blocks of special linear groups of degree two in defining characteristic. It is shown that such blocks have an essentially unique lift to an order over an appropriate p-adic ring, provided the field of coefficients for the group ring is sufficiently large. The crucial fact that is exploited here is that the blocks under consideration are derived equivalent to blocks of group rings of upper triangular unipotent matrices. As a corollary, an open conjecture of Nebe concerning the basic orders of blocks of special linear groups of degree two is proved. The last chapter of the thesis deals with blocks of symmetric groups. It contains a modified version of Scopes' reduction and a description of basic orders of defect two blocks of symmetric groups."],"dc:identifier":["https://publications.rwth-aachen.de/record/52285","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114521%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-40425"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 129 S. (2012). = Aachen, Techn. Hochsch., Diss., 2012"],"dc:subject":["info:eu-repo/classification/ddc/510","Ganzzahlige Darstellungstheorie","Darstellungstheorie","Mathematik","integral representation theory","representation theory"],"dc:title":["Group rings over the p-adic integers"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:40:50Z"}