{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:50390"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:50390","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Hecke-Algebren zu unimodularen und unitären Matrixgruppen über Dedekind-Ringen","abstract":"In this thesis the structure of the Hecke algebras related to unimodular matrix groups over Dedekind domains and to unitary matrix groups over the ring of integers of an imaginary quadratic number field is being analysed. The first main result of the analysis of the unimodular groups over a Dedekind domain o is a measure formula which in the case of (2 x 2) matrices allows to calculate the number of right cosets of type GL_2(o) A_i contained in a double coset of type GL_2(o) A GL_2(o) by means of the determinantal divisors of the matrix A. As an application of this measure formula to the product in the Hecke algebra one obtains a reduction formula. The second central result is a tensor product decomposition. Denote by H^* the Hecke algebra on the set of all matrices from o^{n x n} with determinant in Z {0} with respect to the subgroup GL_n(o) and by H^*_p for every prime p in N the Hecke algebra on the set of all matrices from o^{n x n} with a determinant of type p^k or -p^k for some k in N_0, likewise with respect to the subgroup GL_n(o). Then H^* is isomorphic to the tensor product of the H^*_p for the primes p in N. Furthermore, it will be proven that the H^*_p are always finitely generated algebras. For the case n = 2 the explicit construction of some systems of generators and the analysis of minimality show that in those cases in which p o has the prime ideal decomposition p' or (p')^2 over o there even exist algebraically independent systems of generators -- more precisely, that H^*_p is isomorphic to C[X_1, X_2] in those cases --, and that in those cases of prime ideal composition p o = p' q' there always exists a set of relators R(p) such that H^*_p is isomorphic to C[X_1, X_2, X_3, X_4] / R(p), which in the case that o is the ring of integers of a quadratic number field yields a representation of H^* as tensor product where the isomorphism type of H^*_p for the inert, ramified and split primes p can always be given explicitly. The analysis of the unitary matrix groups yields distinguished representatives of right cosets (those whose C-block is the zero matrix) and the existence of simultanous sets of representatives of right and left cosets. The main result is a weak version of a tensor product decomposition. As a negative result it is proven that in general a tensor product decomposition similar to the one for the case of a unimodular group does not exist here.","abstract_html":"In this thesis the structure of the Hecke algebras related to unimodular matrix groups over Dedekind domains and to unitary matrix groups over the ring of integers of an imaginary quadratic number field is being analysed. The first main result of the analysis of the unimodular groups over a Dedekind domain o is a measure formula which in the case of (2 x 2) matrices allows to calculate the number of right cosets of type GL_2(o) A_i contained in a double coset of type GL_2(o) A GL_2(o) by means of the determinantal divisors of the matrix A. As an application of this measure formula to the product in the Hecke algebra one obtains a reduction formula. The second central result is a tensor product decomposition. Denote by H^* the Hecke algebra on the set of all matrices from o^{n x n} with determinant in Z {0} with respect to the subgroup GL_n(o) and by H^*_p for every prime p in N the Hecke algebra on the set of all matrices from o^{n x n} with a determinant of type p^k or -p^k for some k in N_0, likewise with respect to the subgroup GL_n(o). Then H^* is isomorphic to the tensor product of the H^*_p for the primes p in N. Furthermore, it will be proven that the H^*_p are always finitely generated algebras. For the case n = 2 the explicit construction of some systems of generators and the analysis of minimality show that in those cases in which p o has the prime ideal decomposition p' or (p')^2 over o there even exist algebraically independent systems of generators -- more precisely, that H^*_p is isomorphic to C[X_1, X_2] in those cases --, and that in those cases of prime ideal composition p o = p' q' there always exists a set of relators R(p) such that H^*_p is isomorphic to C[X_1, X_2, X_3, X_4] / R(p), which in the case that o is the ring of integers of a quadratic number field yields a representation of H^* as tensor product where the isomorphism type of H^*_p for the inert, ramified and split primes p can always be given explicitly. The analysis of the unitary matrix groups yields distinguished representatives of right cosets (those whose C-block is the zero matrix) and the existence of simultanous sets of representatives of right and left cosets. The main result is a weak version of a tensor product decomposition. As a negative result it is proven that in general a tensor product decomposition similar to the one for the case of a unimodular group does not exist here.","abstract_has_math":false,"creators":["Ensenbach, Marc"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Krieg, Aloys"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-30T19:40:25Z","subjects":["info:eu-repo/classification/ddc/510","Hecke-Algebra","Matrizengruppe","Mathematik","Hecke algebra","matrix group"],"languages":["ger"],"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-112937%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112937%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112937%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/50390","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%3A50390","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Krieg, Aloys"]},{"key":"dc:creator","label":"Author","values":["Ensenbach, Marc"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2008"]},{"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-25809"]},{"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","Hecke-Algebra","Matrizengruppe","Mathematik","Hecke algebra","matrix group"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["ger"]},{"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/50390","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112937%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis the structure of the Hecke algebras related to unimodular matrix groups over Dedekind domains and to unitary matrix groups over the ring of integers of an imaginary quadratic number field is being analysed. The first main result of the analysis of the unimodular groups over a Dedekind domain o is a measure formula which in the case of (2 x 2) matrices allows to calculate the number of right cosets of type GL_2(o) A_i contained in a double coset of type GL_2(o) A GL_2(o) by means of the determinantal divisors of the matrix A. As an application of this measure formula to the product in the Hecke algebra one obtains a reduction formula. The second central result is a tensor product decomposition. Denote by H^* the Hecke algebra on the set of all matrices from o^{n x n} with determinant in Z {0} with respect to the subgroup GL_n(o) and by H^*_p for every prime p in N the Hecke algebra on the set of all matrices from o^{n x n} with a determinant of type p^k or -p^k for some k in N_0, likewise with respect to the subgroup GL_n(o). Then H^* is isomorphic to the tensor product of the H^*_p for the primes p in N. Furthermore, it will be proven that the H^*_p are always finitely generated algebras. For the case n = 2 the explicit construction of some systems of generators and the analysis of minimality show that in those cases in which p o has the prime ideal decomposition p' or (p')^2 over o there even exist algebraically independent systems of generators -- more precisely, that H^*_p is isomorphic to C[X_1, X_2] in those cases --, and that in those cases of prime ideal composition p o = p' q' there always exists a set of relators R(p) such that H^*_p is isomorphic to C[X_1, X_2, X_3, X_4] / R(p), which in the case that o is the ring of integers of a quadratic number field yields a representation of H^* as tensor product where the isomorphism type of H^*_p for the inert, ramified and split primes p can always be given explicitly. The analysis of the unitary matrix groups yields distinguished representatives of right cosets (those whose C-block is the zero matrix) and the existence of simultanous sets of representatives of right and left cosets. The main result is a weak version of a tensor product decomposition. As a negative result it is proven that in general a tensor product decomposition similar to the one for the case of a unimodular group does not exist here."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 179 S. (2008). = Aachen, Techn. Hochsch., Diss., 2008"]},{"key":"dc:title","label":"Title","values":["Hecke-Algebren zu unimodularen und unitären Matrixgruppen über Dedekind-Ringen"]}]}],"canonical_facts":{"dc:contributor":["Krieg, Aloys"],"dc:coverage":["DE"],"dc:creator":["Ensenbach, Marc"],"dc:date":["2008"],"dc:description":["In this thesis the structure of the Hecke algebras related to unimodular matrix groups over Dedekind domains and to unitary matrix groups over the ring of integers of an imaginary quadratic number field is being analysed. The first main result of the analysis of the unimodular groups over a Dedekind domain o is a measure formula which in the case of (2 x 2) matrices allows to calculate the number of right cosets of type GL_2(o) A_i contained in a double coset of type GL_2(o) A GL_2(o) by means of the determinantal divisors of the matrix A. As an application of this measure formula to the product in the Hecke algebra one obtains a reduction formula. The second central result is a tensor product decomposition. Denote by H^* the Hecke algebra on the set of all matrices from o^{n x n} with determinant in Z {0} with respect to the subgroup GL_n(o) and by H^*_p for every prime p in N the Hecke algebra on the set of all matrices from o^{n x n} with a determinant of type p^k or -p^k for some k in N_0, likewise with respect to the subgroup GL_n(o). Then H^* is isomorphic to the tensor product of the H^*_p for the primes p in N. Furthermore, it will be proven that the H^*_p are always finitely generated algebras. For the case n = 2 the explicit construction of some systems of generators and the analysis of minimality show that in those cases in which p o has the prime ideal decomposition p' or (p')^2 over o there even exist algebraically independent systems of generators -- more precisely, that H^*_p is isomorphic to C[X_1, X_2] in those cases --, and that in those cases of prime ideal composition p o = p' q' there always exists a set of relators R(p) such that H^*_p is isomorphic to C[X_1, X_2, X_3, X_4] / R(p), which in the case that o is the ring of integers of a quadratic number field yields a representation of H^* as tensor product where the isomorphism type of H^*_p for the inert, ramified and split primes p can always be given explicitly. The analysis of the unitary matrix groups yields distinguished representatives of right cosets (those whose C-block is the zero matrix) and the existence of simultanous sets of representatives of right and left cosets. The main result is a weak version of a tensor product decomposition. As a negative result it is proven that in general a tensor product decomposition similar to the one for the case of a unimodular group does not exist here."],"dc:identifier":["https://publications.rwth-aachen.de/record/50390","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-112937%22"],"dc:language":["ger"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-25809"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 179 S. (2008). = Aachen, Techn. Hochsch., Diss., 2008"],"dc:subject":["info:eu-repo/classification/ddc/510","Hecke-Algebra","Matrizengruppe","Mathematik","Hecke algebra","matrix group"],"dc:title":["Hecke-Algebren zu unimodularen und unitären Matrixgruppen über Dedekind-Ringen"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:40:25Z"}