{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:118"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:118","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups","abstract":"Let \\(G\\) be a simple linear algebraic group over the algebraically closed field \\(k\\). Assume \\(p\\) = char \\(k\\) > 0 is good for \\(G\\) and that \\(G\\) is defined and split over the prime field \\(\\char{bbold10}{0x46}_p\\). For a power \\(q\\) of \\(p\\), we write \\(G(q)\\) for the Chevalley group consisting of the \\(\\char{bbold10}{0x46}_q\\)-rational points of \\(G\\). Let \\(F : G \\rightarrow G\\) be the standard Frobenius morphism such that \\(G^F\\)= \\(G(q)\\). Let \\(B\\) be an \\(F\\)-stable Borel subgroup of \\(G\\); write \\(U\\) for the unipotent radical of \\(B\\) and \\(\\char{eufm10}{0x75}\\) for its Lie algebra. We note that \\(U\\) and \\(\\char{eufm10}{0x75}\\) are \\(F\\)-stable and that \\(U(q)\\) is a Sylow \\(p\\)-subgroup of \\(G(q)\\). We study the adjoint orbits of \\(U\\) and show that the conjugacy classes of \\(U(q)\\) are in correspondence with the \\(F\\)-stable adjoint orbits of \\(U\\). This allows us to deduce results about the conjugacy classes of \\(U(q)\\). We are also interested in the adjoint orbits of \\(B\\) in \\(\\char{eufm10}{0x75}\\) and the \\(B(q)\\)-conjugacy classes in \\(U(q)\\). In particular, we consider the question of when \\(B\\) acts on a \\(B\\)-submodule of \\(\\char{eufm10}{0x75}\\) with a Zariski dense orbit. For our study of the adjoint orbits of \\(U\\) we require the existence of \\(B\\)-equivariant isomorphisms of varieties \\(U/M \\rightarrow\\) \\(\\char{eufm10}{0x75}\\)/\\(\\char{eufm10}{0x6d}\\), where \\(M\\) is a unipotent normal subgroup of \\(B\\) and \\(\\char{eufm10}{0x6d}\\) = Lie\\(M\\). We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all \\(M\\).","abstract_html":"Let \\(G\\) be a simple linear algebraic group over the algebraically closed field \\(k\\). Assume \\(p\\) = char \\(k\\) &gt; 0 is good for \\(G\\) and that \\(G\\) is defined and split over the prime field <span class=\"etd-inline-math\">\\char{bbold10}{0x46}<sub>p</sub></span>. For a power \\(q\\) of \\(p\\), we write \\(G(q)\\) for the Chevalley group consisting of the <span class=\"etd-inline-math\">\\char{bbold10}{0x46}<sub>q</sub></span>-rational points of \\(G\\). Let \\(F : G \\rightarrow G\\) be the standard Frobenius morphism such that <span class=\"etd-inline-math\">G<sup>F</sup></span>= \\(G(q)\\). Let \\(B\\) be an \\(F\\)-stable Borel subgroup of \\(G\\); write \\(U\\) for the unipotent radical of \\(B\\) and \\(\\char{eufm10}{0x75}\\) for its Lie algebra. We note that \\(U\\) and \\(\\char{eufm10}{0x75}\\) are \\(F\\)-stable and that \\(U(q)\\) is a Sylow \\(p\\)-subgroup of \\(G(q)\\). We study the adjoint orbits of \\(U\\) and show that the conjugacy classes of \\(U(q)\\) are in correspondence with the \\(F\\)-stable adjoint orbits of \\(U\\). This allows us to deduce results about the conjugacy classes of \\(U(q)\\). We are also interested in the adjoint orbits of \\(B\\) in \\(\\char{eufm10}{0x75}\\) and the \\(B(q)\\)-conjugacy classes in \\(U(q)\\). In particular, we consider the question of when \\(B\\) acts on a \\(B\\)-submodule of \\(\\char{eufm10}{0x75}\\) with a Zariski dense orbit. For our study of the adjoint orbits of \\(U\\) we require the existence of \\(B\\)-equivariant isomorphisms of varieties \\(U/M \\rightarrow\\) \\(\\char{eufm10}{0x75}\\)/\\(\\char{eufm10}{0x6d}\\), where \\(M\\) is a unipotent normal subgroup of \\(B\\) and \\(\\char{eufm10}{0x6d}\\) = Lie\\(M\\). We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all \\(M\\).","abstract_has_math":true,"creators":["Goodwin, Simon Mark"],"institution":"University of Birmingham","degree_name":"d_ph","degree_level":"d_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2005,"date_issued":"2005-07","date_published":"2005-07","updated_at":"2026-07-24T01:10:46Z","subjects":["QA Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["epsrc"]},{"key":"dc:creator","label":"Author","values":["Goodwin, Simon Mark"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2005-07-11"]},{"key":"dc:date.issued","label":"Date","values":["2005-07"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["School of Mathematics & Statistics","Mathematics and Statistics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Birmingham"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["http://etheses.bham.ac.uk//id/eprint/118/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["d_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["d_ph"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://etheses.bham.ac.uk//id/eprint/118/1/Goodwin05PhD.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let \\(G\\) be a simple linear algebraic group over the algebraically closed field \\(k\\). Assume \\(p\\) = char \\(k\\) > 0 is good for \\(G\\) and that \\(G\\) is defined and split over the prime field \\(\\char{bbold10}{0x46}_p\\). For a power \\(q\\) of \\(p\\), we write \\(G(q)\\) for the Chevalley group consisting of the \\(\\char{bbold10}{0x46}_q\\)-rational points of \\(G\\). Let \\(F : G \\rightarrow G\\) be the standard Frobenius morphism such that \\(G^F\\)= \\(G(q)\\). Let \\(B\\) be an \\(F\\)-stable Borel subgroup of \\(G\\); write \\(U\\) for the unipotent radical of \\(B\\) and \\(\\char{eufm10}{0x75}\\) for its Lie algebra. We note that \\(U\\) and \\(\\char{eufm10}{0x75}\\) are \\(F\\)-stable and that \\(U(q)\\) is a Sylow \\(p\\)-subgroup of \\(G(q)\\). We study the adjoint orbits of \\(U\\) and show that the conjugacy classes of \\(U(q)\\) are in correspondence with the \\(F\\)-stable adjoint orbits of \\(U\\). This allows us to deduce results about the conjugacy classes of \\(U(q)\\). We are also interested in the adjoint orbits of \\(B\\) in \\(\\char{eufm10}{0x75}\\) and the \\(B(q)\\)-conjugacy classes in \\(U(q)\\). In particular, we consider the question of when \\(B\\) acts on a \\(B\\)-submodule of \\(\\char{eufm10}{0x75}\\) with a Zariski dense orbit. For our study of the adjoint orbits of \\(U\\) we require the existence of \\(B\\)-equivariant isomorphisms of varieties \\(U/M \\rightarrow\\) \\(\\char{eufm10}{0x75}\\)/\\(\\char{eufm10}{0x6d}\\), where \\(M\\) is a unipotent normal subgroup of \\(B\\) and \\(\\char{eufm10}{0x6d}\\) = Lie\\(M\\). We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all \\(M\\)."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups"]}]}],"canonical_facts":{"dc:contributor.sponsor":["epsrc"],"dc:creator":["Goodwin, Simon Mark"],"dc:date":["2005-07-11"],"dc:date.issued":["2005-07"],"dc:description.abstract":["Let \\(G\\) be a simple linear algebraic group over the algebraically closed field \\(k\\). Assume \\(p\\) = char \\(k\\) > 0 is good for \\(G\\) and that \\(G\\) is defined and split over the prime field \\(\\char{bbold10}{0x46}_p\\). For a power \\(q\\) of \\(p\\), we write \\(G(q)\\) for the Chevalley group consisting of the \\(\\char{bbold10}{0x46}_q\\)-rational points of \\(G\\). Let \\(F : G \\rightarrow G\\) be the standard Frobenius morphism such that \\(G^F\\)= \\(G(q)\\). Let \\(B\\) be an \\(F\\)-stable Borel subgroup of \\(G\\); write \\(U\\) for the unipotent radical of \\(B\\) and \\(\\char{eufm10}{0x75}\\) for its Lie algebra. We note that \\(U\\) and \\(\\char{eufm10}{0x75}\\) are \\(F\\)-stable and that \\(U(q)\\) is a Sylow \\(p\\)-subgroup of \\(G(q)\\). We study the adjoint orbits of \\(U\\) and show that the conjugacy classes of \\(U(q)\\) are in correspondence with the \\(F\\)-stable adjoint orbits of \\(U\\). This allows us to deduce results about the conjugacy classes of \\(U(q)\\). We are also interested in the adjoint orbits of \\(B\\) in \\(\\char{eufm10}{0x75}\\) and the \\(B(q)\\)-conjugacy classes in \\(U(q)\\). In particular, we consider the question of when \\(B\\) acts on a \\(B\\)-submodule of \\(\\char{eufm10}{0x75}\\) with a Zariski dense orbit. For our study of the adjoint orbits of \\(U\\) we require the existence of \\(B\\)-equivariant isomorphisms of varieties \\(U/M \\rightarrow\\) \\(\\char{eufm10}{0x75}\\)/\\(\\char{eufm10}{0x6d}\\), where \\(M\\) is a unipotent normal subgroup of \\(B\\) and \\(\\char{eufm10}{0x6d}\\) = Lie\\(M\\). We define relative Springer isomorphisms which are certain maps of the above form and prove that they exist for all \\(M\\)."],"dc:format":["application/pdf"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/118/1/Goodwin05PhD.pdf"],"dc:publisher.department":["School of Mathematics & Statistics","Mathematics and Statistics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/118/"],"dc:subject":["QA Mathematics"],"dc:title":["Relative Springer isomorphisms and the conjugacy classes in Sylow p-subgroups of Chevalley groups"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["d_ph"],"dc:type.qualificationname":["d_ph"]},"updated_at":"2026-07-24T01:10:46Z"}