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Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 11–22 | Cite as

Symmetric quotient stacks and Heisenberg actions

  • Andreas Krug
Article
  • 82 Downloads

Abstract

For every smooth projective variety X, we construct an action of the Heisenberg algebra on the direct sum of the Grothendieck groups of all the symmetric quotient stacks \([X^n/{\mathfrak {S}}_n]\) which contains the Fock space as a subrepresentation. The action is induced by functors on the level of the derived categories which form a weak categorification of the action.

Notes

Acknowledgements

The author was financially supported by the research Grant KR 4541/1-1 of the DFG. He thanks Sabin Cautis, Daniel Huybrechts, Ciaran Meachan, David Ploog, Miles Reid, Pawel Sosna, and the referee for helpful comments.

References

  1. 1.
    Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001). (Electronic) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, X.: A note on separable functors and monads. arXiv:1403.1332 (2014)
  3. 3.
    Cautis, S., Licata, A.: Heisenberg categorification and Hilbert schemes. Duke Math. J. 161(13), 2469–2547 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Elagin, A.D.: On equivariant triangulated categories. arXiv:1403.7027 (2014)
  5. 5.
    Feigin, B.L., Tsymbaliuk, A.I.: Equivariant \(K\)-theory of Hilbert schemes via shuffle algebra. Kyoto J. Math. 51(4), 831–854 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ganter, N., Kapranov, M.: Symmetric and exterior powers of categories. Transform. Groups 19(1), 57–103 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grojnowski, I.: Instantons and affine algebras. I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3(2), 275–291 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001). (electronic) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khovanov, M.: Heisenberg algebra and a graphical calculus. Fund. Math. 225, 169–210 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Krug, A.: \({\mathbb{P}}\)-functor versions of the Nakajima operators. arXiv:1405.1006 (2014)
  11. 11.
    Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. Math. (2) 145(2), 379–388 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University Lecture Series, vol. 18. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  13. 13.
    Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the \(K\)-theory of the Hilbert scheme of \({\mathbb{A}}^2\). Duke Math. J. 162(2), 279–366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vistoli, A.: Higher equivariant \(K\)-theory for finite group actions. Duke Math. J. 63(2), 399–419 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of MarburgMarburgGermany

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