Mathematische Zeitschrift

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

Symmetric quotient stacks and Heisenberg actions

Article

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) MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ganter, N., Kapranov, M.: Symmetric and exterior powers of categories. Transform. Groups 19(1), 57–103 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Grojnowski, I.: Instantons and affine algebras. I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3(2), 275–291 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001). (electronic) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Khovanov, M.: Heisenberg algebra and a graphical calculus. Fund. Math. 225, 169–210 (2014)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University Lecture Series, vol. 18. American Mathematical Society, Providence (1999)MATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Vistoli, A.: Higher equivariant \(K\)-theory for finite group actions. Duke Math. J. 63(2), 399–419 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of MarburgMarburgGermany

Personalised recommendations