{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22442"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22442","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic","abstract":"Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\\Phi$, and a set of simple roots $\\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties.","abstract_html":"Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\\Phi$, and a set of simple roots $\\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties.","abstract_has_math":true,"creators":["Wenzel, Christian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Haboush, William J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:39:57Z","date_published":"2011-05-07T13:39:57Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Wenzel, Christian"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114457","(UMI)AAI9114457"],"render_values":[{"text":"AAI9114457","href":null,"code":true},{"text":"(UMI)AAI9114457","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22442","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Haboush, William J."]},{"key":"dc:creator","label":"Author","values":["Wenzel, Christian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:39:57Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Wenzel, Christian"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9114457","(UMI)AAI9114457","http://hdl.handle.net/2142/22442"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\\Phi$, and a set of simple roots $\\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties.","In my thesis I generalize to subgroup-schemes containing B. They are group-schemes, but not necessarily varieties; their algebras of functions might have nilpotent elements, i.e. they might not be reduced. In my thesis I show that in characteristic p $>$ 0, there are infinitely many whenever G $\\not=$ 1, I exhibit their structure, and I classify them.","I show that in characteristic p $>$ 3, the subgroup-schemes containing B correspond to $\\tilde\\Delta$, the set of all maps from $\\Delta$ to $\\rm I\\!N \\cup \\{\\infty\\},$ in such a way that it extends the classical classification of parabolic subgroups in terms of subsets of $\\Delta$. To each $\\varphi$ there is a parabolic P$\\sb\\varphi$ with $\\rm P\\sb\\varphi = U\\sb\\varphi\\cdot P\\sb{I(\\varphi)},$ I$(\\varphi) = \\{\\alpha\\in\\Delta\\mid\\varphi(\\alpha)=\\infty\\}$, $\\rm P\\sb{I(\\varphi)} = (P\\sb\\varphi)\\sb{red}$ = Spec(K (P$\\sb\\varphi$) /nilrad), U$\\sb\\varphi$ being a certain local unipotent subgroup-scheme. In characteristic 2,3 the situation is more complicated.","Furthermore I give a construction of G/P also for non-reduced P. I show that G/P is a rational projective variety, whenever char (K) $>$ 3.","Made available in DSpace on 2011-05-07T13:39:57Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114457.pdf: 2003410 bytes, checksum: 2508c8e7f65c047fb5ab06fb44c8fb0f (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:57:39Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic"]}]}],"canonical_facts":{"dc:contributor":["Haboush, William J."],"dc:creator":["Wenzel, Christian"],"dc:date":["2011-05-07T13:39:57Z","10000-01-01","1990"],"dc:description":["Given a semisimple linear algebraic group G over an algebraically closed field K, we fix a Borel subgroup B and a maximal torus T. This determines a root system $\\Phi$, and a set of simple roots $\\Delta$. The subgroups containing B are called parabolic subgroups. They correspond to subsets of $\\Delta$. Thus there are finitely many. In this classical context, parabolic subgrous are understood to be varieties.","In my thesis I generalize to subgroup-schemes containing B. They are group-schemes, but not necessarily varieties; their algebras of functions might have nilpotent elements, i.e. they might not be reduced. In my thesis I show that in characteristic p $>$ 0, there are infinitely many whenever G $\\not=$ 1, I exhibit their structure, and I classify them.","I show that in characteristic p $>$ 3, the subgroup-schemes containing B correspond to $\\tilde\\Delta$, the set of all maps from $\\Delta$ to $\\rm I\\!N \\cup \\{\\infty\\},$ in such a way that it extends the classical classification of parabolic subgroups in terms of subsets of $\\Delta$. To each $\\varphi$ there is a parabolic P$\\sb\\varphi$ with $\\rm P\\sb\\varphi = U\\sb\\varphi\\cdot P\\sb{I(\\varphi)},$ I$(\\varphi) = \\{\\alpha\\in\\Delta\\mid\\varphi(\\alpha)=\\infty\\}$, $\\rm P\\sb{I(\\varphi)} = (P\\sb\\varphi)\\sb{red}$ = Spec(K (P$\\sb\\varphi$) /nilrad), U$\\sb\\varphi$ being a certain local unipotent subgroup-scheme. In characteristic 2,3 the situation is more complicated.","Furthermore I give a construction of G/P also for non-reduced P. I show that G/P is a rational projective variety, whenever char (K) $>$ 3.","Made available in DSpace on 2011-05-07T13:39:57Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9114457.pdf: 2003410 bytes, checksum: 2508c8e7f65c047fb5ab06fb44c8fb0f (MD5) Previous issue date: 1990","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:57:39Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:27:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9114457","(UMI)AAI9114457","http://hdl.handle.net/2142/22442"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Wenzel, Christian"],"dc:subject":["Mathematics"],"dc:title":["Classification of all parabolic subgroup schemes of a semisimple linear algebraic group over an algebraically closed field of positive characteristic"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}