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dc.contributor.authorBasu, Saugata
dc.contributor.authorRiener, Cordian
dc.date.accessioned2019-03-19T14:09:10Z
dc.date.available2019-03-19T14:09:10Z
dc.date.issued2018-04-30
dc.description.abstractWe consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant <i>d</i>. We prove that if a Specht module, S<sup>λ</sup>⁠, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition λ is bounded by O(<i>d</i>). This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the abovementioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov, and Zell.en_US
dc.description.sponsorshipNational Science Foundationen_US
dc.descriptionThis is a pre-copyedited, author-produced version of an article accepted for publication in <i>International Mathematics Research Notices</i> following peer review. The version of record Basu, S. & Riener, C. (2018). On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities. <i>International mathematics research notices</i>, rny062, is available online at: <a href=https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny062/4989853#116372541>https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny062/4989853#116372541</a>.en_US
dc.identifier.citationBasu, S. & Riener, C. (2018). On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities. <i>International mathematics research notices</i>, rny062. https://doi.org/10.1093/imrn/rny062en_US
dc.identifier.cristinIDFRIDAID 1582634
dc.identifier.doi10.1093/imrn/rny062
dc.identifier.issn1073-7928
dc.identifier.issn1687-0247
dc.identifier.urihttps://hdl.handle.net/10037/15026
dc.language.isoengen_US
dc.publisherOxford University Pressen_US
dc.relation.journalInternational mathematics research notices
dc.rights.accessRightsopenAccessen_US
dc.subjectVDP::Mathematics and natural science: 400::Mathematics: 410::Algebra/algebraic analysis: 414en_US
dc.subjectVDP::Matematikk og Naturvitenskap: 400::Matematikk: 410::Algebra/algebraisk analyse: 414en_US
dc.subjectSymmetric groupen_US
dc.subjectIsotypic decompositionen_US
dc.subjectSemi-algebraic setsen_US
dc.subjectSpecht modulesen_US
dc.titleOn the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicitiesen_US
dc.typeJournal articleen_US
dc.typeTidsskriftartikkelen_US
dc.typePeer revieweden_US


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