Skip to main content
Log in

SYMMETRIC AND EXTERIOR POWERS OF CATEGORIES

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We define symmetric and exterior powers of categories, fitting into categorified Koszul complexes. We discuss examples and calculate the effect of these power operations on the categorical characters of matrix 2-representations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Barratt, S. Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972), 1-14.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149-168.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Bartlett, The geometry of unitary 2-representations of finite groups and their 2-characters, Appl. Categ. Structures 19 (2011), no. 1, 175-232.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Bakalov, A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series, Vol. 21, American Mathematical Society, Providence, RI, 2001.

    Google Scholar 

  5. А. Бондал, М. Капранов, Оснащенные триангулированные категории, Матем. сб. 181 (1990), вып. 5, 669-683. Engl. transl.: A. Bondal, M. Kapranov, Framed triangulated categories, Math. USSR-Sb. 70 (1991), no. 1, 93-107.

  6. A. Bondal, M. Larsen, V. A. Lunts, Grothendieck ring of pretriangulated categories, Int. Math. Res. Not. 29 (2004), 1461-1495.

    Article  MathSciNet  Google Scholar 

  7. T. Bridgeland, Flops and derived categories, Invent. Math. 147 (2002), 613-632.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Brundan, A. Kleshchev, Projective representations of symmetric groups via Sergeev duality, Math. Zeitschrift 239 (2002), 27-68.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Deligne, Le determinant de la cohomologie, in: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985), Contemp. Math., Vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93-177.

  10. P. Deligne, Catégories tannakiennes, in: The Grothendieck Festschrift, Vol. II, Progress in Mathematics, Vol. 87, Birkhäuser Boston, Boston, MA, 1990, p. 111-195.

  11. P. Deligne, Action du groupe des tresses sur une catégorie, Invent. Math. 128 (1997), 159-175.

    Article  MATH  MathSciNet  Google Scholar 

  12. J. A. Devoto, Equivariant elliptic homology and finite groups, Michigan Math. J. 43 (1996), 3-32.

    Article  MATH  MathSciNet  Google Scholar 

  13. R. Dijkgraaf, Discrete torsion and symmetric products, preprint hep-th/9912101.

  14. R. Dijkgraaf, G. Moore, E. Verlinde, H. Verlinde, Elliptic genera of symmetric products and second quantized strings, Comm. Math. Phys. 185 (1997), no. 1, 197-209.

    Article  MATH  MathSciNet  Google Scholar 

  15. V. Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, in: The Unity of Mathematics, Birkhauser, Boston, 2006.

    Google Scholar 

  16. I. B. Frenkel, N. Jing, W. Wang, Vertex representations via finite groups and the McKay correspondence, Intern. Math. Research Notices, 2000, no. 4, 195-222.

  17. I. B. Frenkel, N. Jing, W. Wang, Twisted vertex representations via spin groups and the McKay correspondence, Duke Math. J. 111 (2002), 51-96.

    Article  MATH  MathSciNet  Google Scholar 

  18. N. Ganter, Inner products of 2-representations, arXiv:1110.1711.

  19. N. Ganter, M. Kapranov, Representation and character theory in 2-categories, Adv. in Math. 217 (2008), 2268-2300.

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Ganter, Global Mackey functors and n-special λ-rings, arXiv:1301.4616.

  21. J. Greenough, Monoidal 2-structure of bimodule categories, J. Algebra 324 (2010), no. 8, 1818-1859.

    Article  MATH  MathSciNet  Google Scholar 

  22. I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275-291.

    Article  MATH  MathSciNet  Google Scholar 

  23. F. Hirzebruch, T. Berger, R. Jung, Manifolds and Modular Forms, with appendices by Nils-Peter Skoruppa and by Paul Baum, Aspects of Mathematics, E20, Friedr. Viehweg & Sohn, Braunschweig, 1992.

    Google Scholar 

  24. M. J. Hopkins, I. M. Singer, Quadratic functions in geometry, topology, and M-theory, J. Differential Geom. 70 (2005) 329-452.

    MATH  MathSciNet  Google Scholar 

  25. N. Jing, Vertex operators, symmetric functions and the spin group Γ n , J. Algebra 138 (1991), 340-398.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Johnson, A. M. Osorno, Modeling stable one-types, Theory and Appl. of Categ. 20 (2012), 520-537.

    MathSciNet  Google Scholar 

  27. T. Jozefiak, Semisimple super-algebras, in: Algebra — Some Current Trends (Varna, 1986), Lecture Notes in Math., Vol. 1352, Springer, Berlin, 1988, pp. 96-113.

  28. T. Jozefiak, Characters of projective representations of symmetric groups, Exposition. Math. 7 (1989), 193-247.

    MATH  MathSciNet  Google Scholar 

  29. V. G. Kac, D. H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 3308-3312.

    Article  MATH  MathSciNet  Google Scholar 

  30. V. G. Kac, A. K. Raina, Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific, Teaneck, NJ, 1987.

    MATH  Google Scholar 

  31. M. M. Kapranov, V. A. Voevodsky, Braided monoidal 2-categories and Zamolodchikov tetrahedra equations, in: Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Part 2, Amer. Math. Soc., Providence, RI, 1994, pp. 177-259.

  32. A. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge Tracts in Mathematics, Vol. 163, Cambridge University Press, Cambridge, 2005.

    Book  Google Scholar 

  33. A. Lascoux, B. Leclerc, J.-Y. Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205-263.

    Article  MATH  MathSciNet  Google Scholar 

  34. V. Lyubashenko, External tensor product of categories of perverse sheaves, Ukrainian Math. J. 53 (2001), 354-367.

    Article  MathSciNet  Google Scholar 

  35. Y. I. Manin, Gauge Fields and Complex Geometry, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  36. J. Milnor, Introduction to Algebraic K-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Russian transl.: Дж. Милнор, Введение в алгебраическую К-теорию, Мир, M., 1974.

  37. H. B. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), 379-388.

    Google Scholar 

  38. L. Previdi, Sato Grassmannians for generalized Tate spaces, preprint arXiv:1002.4863.

  39. S. Priddy, On Ω S and the infinite symmetric group, in: Algebraic Topology (Univ. Wisconsin, Madison, Wis., 1970), Proc. Sympos. Pure Math., Vol. XXII, Amer. Math. Soc., Providence, R.I., 1971, pp. 217-220.

  40. C. Rezk, Quasi elliptic λ-rings, in preparation.

  41. I. Schur, Über die Darstellung der symmetrischen and der alternierenden Gruppen durch gebrochene lineare Substitutionen, J. reine und angew. Math. 139 (1911), 155-250.

    MATH  Google Scholar 

  42. S.G.A. 4, Théorie des topos et cohomologie étale des schémas, t. III, Lecture Notes in Mathematics, Vol. 305, Springer Verlag, Berlin, 1973.

  43. B. Totaro, Chern numbers for singular varieties and elliptic homology, Ann. of Math. 151 (2000), 757-791.

    Article  MATH  MathSciNet  Google Scholar 

  44. W. Wang, Equivariant K-theory, generalized symmetric products and twisted Heisenberg algebra, Comm. Math. Phys. 234 (2003), 101-127.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to MIKHAIL KAPRANOV.

Additional information

Much of this paper was written while Ganter held a position at Colby College, Maine, and was supported by the NSF grant DMS-0504539.

Kapranov was supported by NSF Grant DMS-0801198.

Rights and permissions

Reprints and permissions

About this article

Cite this article

GANTER, N., KAPRANOV, M. SYMMETRIC AND EXTERIOR POWERS OF CATEGORIES. Transformation Groups 19, 57–103 (2014). https://doi.org/10.1007/s00031-014-9255-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-014-9255-z

Keywords

Navigation