{"id":{"repo_id":"windsor","oai_identifier":"oai:uwindsor.scholaris.ca:20.500.14776/8462"},"canonical_url":"https://search.dev.ndltd.org/etd/windsor/oai:uwindsor.scholaris.ca:20.500.14776/8462","repository":{"repo_id":"windsor","name":"University of Windsor","base_url":"https://uwindsor.scholaris.ca/server/oai/request"},"display":{"title":"Unitary Irreducible Highest Weight Modules of su(2,1) of Highest Weight λ - ρ Where λ is Integral Regular","abstract":"In this thesis, the following problem is tackled: for the real form su(2, 1) of sl3(C), compute chsL(x(−λ)) for λ ∈ Λ+ regular (i.e., λ integral, dominant, regular weights) and x ∈ W, and determine when the representations are unitary (i.e. when chsL(x(−λ)) = chL(x(−λ))). Here, L(x(−λ)) is the unique irreducible quotient of the Verma module M(x(−λ)) of highest weight x(−λ) − ρ. One of the most important open problems in Mathematics is called The Unitary Dual Problem: classify the irreducible unitary representations of a group. This was introduced in the 1930s by Israel Gelfand. In the realm of unitary representations, the initial commencement often involves portraying the Hermitian representations, those that harbor an invariant Hermitian form. Following this classification, one embarks on calculating the signatures of these invariant forms within the Hermitian tapestry. The final act in this symphony of classification involves discerning which of these forms embrace positivity or negativity in their definiteness. For the problem at hand, we heavily rely on the machineries developed particularly in [Yee19]. For the enthusiast, it is recommended to explore the arcane landscapes the author had ventured into. For our purposes, we explicitly deal with computations. Much of the chapters included are self-contained. Since the key players in this Mathematical fabula scaenica are “weight spaces”, “Weyl groups” and “affine Weyl groups”, their masquerading nature had been unveiled through elucidation. Whenever new notations had risen, the meaning associated with it had been expounded on. For now, we let the sea advance in silence.","abstract_html":"In this thesis, the following problem is tackled: for the real form su(2, 1) of sl3(C), compute chsL(x(−λ)) for λ ∈ Λ+ regular (i.e., λ integral, dominant, regular weights) and x ∈ W, and determine when the representations are unitary (i.e. when chsL(x(−λ)) = chL(x(−λ))). Here, L(x(−λ)) is the unique irreducible quotient of the Verma module M(x(−λ)) of highest weight x(−λ) − ρ. One of the most important open problems in Mathematics is called The Unitary Dual Problem: classify the irreducible unitary representations of a group. This was introduced in the 1930s by Israel Gelfand. In the realm of unitary representations, the initial commencement often involves portraying the Hermitian representations, those that harbor an invariant Hermitian form. Following this classification, one embarks on calculating the signatures of these invariant forms within the Hermitian tapestry. The final act in this symphony of classification involves discerning which of these forms embrace positivity or negativity in their definiteness. For the problem at hand, we heavily rely on the machineries developed particularly in [Yee19]. For the enthusiast, it is recommended to explore the arcane landscapes the author had ventured into. For our purposes, we explicitly deal with computations. Much of the chapters included are self-contained. Since the key players in this Mathematical fabula scaenica are “weight spaces”, “Weyl groups” and “affine Weyl groups”, their masquerading nature had been unveiled through elucidation. Whenever new notations had risen, the meaning associated with it had been expounded on. For now, we let the sea advance in silence.","abstract_has_math":false,"creators":["Mobassir, Yasin"],"institution":"University of Windsor","degree_name":"M.Sc.","degree_level":null,"degree_discipline":"Mathematics and Statistics","degree_department":null,"school":null,"contributors":["scholarship@uwindsor.ca"],"advisors":["Wai Ling Yee"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-06-20","date_published":"2024-06-20","updated_at":"2026-07-27T22:04:47Z","subjects":[],"languages":["en_CA"],"rights":[],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/20.500.14776/8462","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["scholarship@uwindsor.ca"]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Wai Ling Yee"]},{"key":"dc:creator","label":"Author","values":["Mobassir, Yasin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-07-03 14:19"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-11-14 13:03","2025-07-03T18:19:50Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-06-20"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/masterThesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics and Statistics","Mathematics, Physical Sciences and Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.Sc."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Windsor"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_CA"]},{"key":"dc:rights","label":"Dc Rights","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/20.500.14776/8462"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, the following problem is tackled: for the real form su(2, 1) of sl3(C), compute chsL(x(−λ)) for λ ∈ Λ+ regular (i.e., λ integral, dominant, regular weights) and x ∈ W, and determine when the representations are unitary (i.e. when chsL(x(−λ)) = chL(x(−λ))). Here, L(x(−λ)) is the unique irreducible quotient of the Verma module M(x(−λ)) of highest weight x(−λ) − ρ. One of the most important open problems in Mathematics is called The Unitary Dual Problem: classify the irreducible unitary representations of a group. This was introduced in the 1930s by Israel Gelfand. In the realm of unitary representations, the initial commencement often involves portraying the Hermitian representations, those that harbor an invariant Hermitian form. Following this classification, one embarks on calculating the signatures of these invariant forms within the Hermitian tapestry. The final act in this symphony of classification involves discerning which of these forms embrace positivity or negativity in their definiteness. For the problem at hand, we heavily rely on the machineries developed particularly in [Yee19]. For the enthusiast, it is recommended to explore the arcane landscapes the author had ventured into. For our purposes, we explicitly deal with computations. Much of the chapters included are self-contained. Since the key players in this Mathematical fabula scaenica are “weight spaces”, “Weyl groups” and “affine Weyl groups”, their masquerading nature had been unveiled through elucidation. Whenever new notations had risen, the meaning associated with it had been expounded on. For now, we let the sea advance in silence."]},{"key":"dc:title","label":"Title","values":["Unitary Irreducible Highest Weight Modules of su(2,1) of Highest Weight λ - ρ Where λ is Integral Regular"]}]}],"canonical_facts":{"dc:contributor":["scholarship@uwindsor.ca"],"dc:contributor.advisor":["Wai Ling Yee"],"dc:creator":["Mobassir, Yasin"],"dc:date.accessioned":["2025-07-03 14:19"],"dc:date.available":["2024-11-14 13:03","2025-07-03T18:19:50Z"],"dc:date.issued":["2024-06-20"],"dc:description.abstract":["In this thesis, the following problem is tackled: for the real form su(2, 1) of sl3(C), compute chsL(x(−λ)) for λ ∈ Λ+ regular (i.e., λ integral, dominant, regular weights) and x ∈ W, and determine when the representations are unitary (i.e. when chsL(x(−λ)) = chL(x(−λ))). Here, L(x(−λ)) is the unique irreducible quotient of the Verma module M(x(−λ)) of highest weight x(−λ) − ρ. One of the most important open problems in Mathematics is called The Unitary Dual Problem: classify the irreducible unitary representations of a group. This was introduced in the 1930s by Israel Gelfand. In the realm of unitary representations, the initial commencement often involves portraying the Hermitian representations, those that harbor an invariant Hermitian form. Following this classification, one embarks on calculating the signatures of these invariant forms within the Hermitian tapestry. The final act in this symphony of classification involves discerning which of these forms embrace positivity or negativity in their definiteness. For the problem at hand, we heavily rely on the machineries developed particularly in [Yee19]. For the enthusiast, it is recommended to explore the arcane landscapes the author had ventured into. For our purposes, we explicitly deal with computations. Much of the chapters included are self-contained. Since the key players in this Mathematical fabula scaenica are “weight spaces”, “Weyl groups” and “affine Weyl groups”, their masquerading nature had been unveiled through elucidation. Whenever new notations had risen, the meaning associated with it had been expounded on. For now, we let the sea advance in silence."],"dc:identifier.uri":["https://hdl.handle.net/20.500.14776/8462"],"dc:language.iso":["en_CA"],"dc:rights":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:title":["Unitary Irreducible Highest Weight Modules of su(2,1) of Highest Weight λ - ρ Where λ is Integral Regular"],"dc:type":["info:eu-repo/semantics/masterThesis"],"thesis:degree_discipline":["Mathematics and Statistics","Mathematics, Physical Sciences and Mathematics"],"thesis:degree_name":["M.Sc."],"thesis:institution_name":["University of Windsor"]},"updated_at":"2026-07-27T22:04:47Z"}