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postgraduate thesis: On a Deodhar-type decomposition and a Poisson structure on double Bott-Samelson varieties

TitleOn a Deodhar-type decomposition and a Poisson structure on double Bott-Samelson varieties
Authors
Advisors
Advisor(s):Lu, J
Issue Date2013
PublisherThe University of Hong Kong (Pokfulam, Hong Kong)
Citation
Mouquin, V. F.. (2013). On a Deodhar-type decomposition and a Poisson structure on double Bott-Samelson varieties. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5153701
AbstractFlag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role. The Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure.
DegreeDoctor of Philosophy
SubjectSchubert varieties
Poisson manifolds
Decomposition (Mathematics)
Dept/ProgramMathematics
Persistent Identifierhttp://hdl.handle.net/10722/195988

 

DC FieldValueLanguage
dc.contributor.advisorLu, J-
dc.contributor.authorMouquin, Victor Fabien-
dc.date.accessioned2014-03-21T03:50:03Z-
dc.date.available2014-03-21T03:50:03Z-
dc.date.issued2013-
dc.identifier.citationMouquin, V. F.. (2013). On a Deodhar-type decomposition and a Poisson structure on double Bott-Samelson varieties. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5153701-
dc.identifier.urihttp://hdl.handle.net/10722/195988-
dc.description.abstractFlag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role. The Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure.-
dc.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.subject.lcshSchubert varieties-
dc.subject.lcshPoisson manifolds-
dc.subject.lcshDecomposition (Mathematics)-
dc.titleOn a Deodhar-type decomposition and a Poisson structure on double Bott-Samelson varieties-
dc.typePG_Thesis-
dc.identifier.hkulb5153701-
dc.description.thesisnameDoctor of Philosophy-
dc.description.thesislevelDoctoral-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b5153701-

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