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Equivariant Hodge-Deligne polynomials of symmetric products of algebraic groups

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Abstract

Let X be a complex quasi-projective algebraic variety. In this paper we study the mixed Hodge structures of the symmetric products \(\mathrm {Sym}^{n}X\) when the cohomology of X is given by exterior products of cohomology classes with odd degree. We obtain an expression for the equivariant mixed Hodge polynomials \(\mu _{X^{n}}^{S_{n}}\left( t,u,v\right) \), codifying the permutation action of \(S_{n}\) as well as its subgroups. This allows us to deduce formulas for the mixed Hodge polynomials of its symmetric products \(\mu _{\mathrm {Sym}^{n}X}\left( t,u,v\right) \). These formulas are then applied to the case of linear algebraic groups.

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Notes

  1. I would like to thank the referee for pointing out this generalization.

  2. Actually the cohomology - or homology, since Hopf algebras are self-dual - of Lie groups were the inspiration behind the definition of Hopf algebra.

References

  1. Cartier, P.: A primer of Hopf algebras, Frontiers in number theory, physics, and geometry, pp. 537–615. Springer, Berlin, II (2007)

  2. Cheah, J.: On the cohomology of Hilbert schemes of points. J. Algebraic Geom. 5, 479–511 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Danilov, V.I., Khovanski, A.G.: Newton polyhedra and an algorithm for computing Hodge- Deligne numbers. Math. USSR Izvestiya 29, 279–298 (1987)

    Article  Google Scholar 

  4. Deligne, P.: Théorie de Hodge II. Publ. Math. I.H.E.S. 40, 5–55 (1971)

    Article  Google Scholar 

  5. Deligne, P.: Théorie de Hodge III. Publ. Math. I.H.E.S. 44, 5–77 (1974)

    Article  Google Scholar 

  6. Dimca, A., Lehrer, G.I: Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements. In: Bjorner, A., Cohen, F., De Concini, C., Procesi, C., Salvetti, M. (eds.) Configuration Spaces. CRM Series. pp. 231–253, Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-431-1_10

  7. Florentino, C., Lawton, S.: Topology of character varieties of Abelian groups. Topol. Appl. 173, 32–58 (2014)

    Article  MathSciNet  Google Scholar 

  8. Florentino, C., Silva, J.: Hodge-Deligne polynomials of character varieties of free abelian groups. Open Math. (2021). https://doi.org/10.1515/math-2021-0038

    Article  MathSciNet  MATH  Google Scholar 

  9. Grothendieck, A.: Sur quelques points dalgébre homologique I. Tohoku Math. J. 9, 119–221 (1957)

    MathSciNet  MATH  Google Scholar 

  10. Hausel, T., Rodriguez-Villegas, F.: Mixed Hodge polynomials of character varieties. Invent. Math. 174, 555–624 (2008)

    Article  MathSciNet  Google Scholar 

  11. Hopf, H.: Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen. Ann. Math. 42, 22–52 (1941)

    Article  MathSciNet  Google Scholar 

  12. Logares, M., Muñoz, V., Newstead, P.E.: Hodge polynomials of SL(2, C)-character varieties for curves of small genus. Rev. Mat. Complut. 26, 635–703 (2013)

    Article  MathSciNet  Google Scholar 

  13. Maxim, L., Schürmann, J.: Twisted genera of symmetric products. Selecta Math. New Ser. 18, 283–317 (2012)

    Article  MathSciNet  Google Scholar 

  14. Maxim, L., Schürmann, J.: Plethysm and cohomology representations of external and symmetric products. Adv. Math. 375, 107373 (2020)

    Article  MathSciNet  Google Scholar 

  15. Macdonald, I.G.: The Poincaré Polynomial of a Symmetric Product. Math. Proc. Cambridge Philos. Soc. 58, 163–177 (1962)

    Article  Google Scholar 

  16. Macdonald, I.G.: Symmetric products of an algebraic curve. Topology 1, 319–343 (1962)

    Article  MathSciNet  Google Scholar 

  17. Mumford, D.B., Fogarty, J.: Geometric Invariant Theory. Springer, Berlin (1982)

    Book  Google Scholar 

  18. Sikora, A.S.: Character varieties of abelian groups. Math. Z. 277, 241–256 (2014)

    Article  MathSciNet  Google Scholar 

  19. Peters, C., Steenbrink, J.: Mixed Hodge Structures. Springer, Berlin (2008)

    MATH  Google Scholar 

  20. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Can. J. Math. 6, 274–304 (1954)

    Article  MathSciNet  Google Scholar 

  21. Stafa, M.: Poincaré series of character varieties for Nilpotent Groups. J. Group Theory 22, 419–440 (2019)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

I would like to thank P. Boavida and E. Costa for many useful conversations around the topic of mixed Hodge structures. I would also like to give a special thanks to C. Florentino, for the same reason but also for his many suggestions to the article. Finally, I would like to thank the anonymous referee, who did a very complete report leading to several improvements throughout the article.

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  1. Área Departamental de Matemática, ISEL, Rua Conselheiro Emídio Navarro, 1, 1959-007, Lisbon, Portugal

    Jaime D. Silva

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Correspondence to Jaime D. Silva.

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This work was supported by the project PTDC/MAT-GEO/2823/2014.

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Silva, J.D. Equivariant Hodge-Deligne polynomials of symmetric products of algebraic groups. manuscripta math. 169, 33–50 (2022). https://doi.org/10.1007/s00229-021-01314-6

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