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On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities

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https://hdl.handle.net/10037/15026
DOI
https://doi.org/10.1093/imrn/rny062
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Date
2018-04-30
Type
Journal article
Tidsskriftartikkel
Peer reviewed

Author
Basu, Saugata; Riener, Cordian
Abstract
We 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 d. We prove that if a Specht module, Sλ⁠, 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(d). 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.
Description
This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices 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. International mathematics research notices, rny062, is available online at: https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rny062/4989853#116372541.
Publisher
Oxford University Press
Citation
Basu, S. & Riener, C. (2018). On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities. International mathematics research notices, rny062. https://doi.org/10.1093/imrn/rny062
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