Communications in Mathematical Physics

, Volume 340, Issue 2, pp 833–849 | Cite as

A Higher Frobenius–Schur Indicator Formula for Group-Theoretical Fusion Categories

  • Peter SchauenburgEmail author


Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: a finite group G endowed with a three-cocycle ω, and a subgroup \({H\subset G}\) endowed with a two-cochain whose coboundary is the restriction of ω. The objects of the category are G-graded vector spaces with suitably twisted \({H}\)-actions; the associativity of tensor products is controlled by ω. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in H of right H-cosets in G, with respect to two-cocycles defined by the initial data. We derive and study general formulas that express the higher Frobenius–Schur indicators of simple objects in a group-theoretical fusion category in terms of the group-theoretical and cohomological data defining the category and describing its simples.


Hopf Algebra Module Category Simple Object Monoidal Category Tensor Category 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bantay P.: The Frobenius–Schur indicator in conformal field theory. Phys. Lett. B 394(1–2), 87–88 (1997) ISSN: 0370-2693ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dijkgraaf, R., Pasquier, V., Roche, P.: Quasi Hopf algebras, group cohomology and orbifold models. In: Binétruy, P., Girardi, G., Sorba, P. (eds.) Nuclear Phys. B Proc. Suppl. 18B (1990). Recent Advances in Field Theory (Annecy-le-Vieux, 1990), pp. 60–72 (1991). ISSN: 0920-5632Google Scholar
  3. 3.
    Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann.Math.(2) 162(2), 581–642 (2005) ISSN:0003-486XMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fuchs, J., et al.: S 4 symmetry of 6j symbols and Frobenius–Schur indicators in rigid monoidal C * categories. J. Math. Phys. 40(1), 408–426 (1999). ISSN: 0022-2488Google Scholar
  5. 5.
    Goff C., Mason G., Ng S.-H.: On the gauge equivalence of twisted quantum doubles of elementary abelian and extra-special 2-groups. J. Algebra 312(2), 849–875 (2007) ISSN: 0021-8693MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Karpilovsky, G.: Group representations, vol. 2. North-Holland Mathematics Studies, vol. 177, pp. xvi+902. North-Holland Publishing Co., Amsterdam (1993). ISBN: 0-444-88726-1Google Scholar
  7. 7.
    Karpilovsky, G.: Group representations, vol. 3. North-Holland Mathematics Studies, vol. 180, pp. xvi+907. North-Holland Publishing Co., Amsterdam (1994). ISBN: 0-444-87433-XGoogle Scholar
  8. 8.
    Kashina, Y., Sommerhäuser, Y., Zhu, Y.: On higher Frobenius–Schur indicators. Mem. Am. Math. Soc. 181(855), viii+65 (2006). ISSN: 0065-9266Google Scholar
  9. 9.
    Linchenko, V., Montgomery, S.: A Frobenius–Schur theorem for Hopf algebras. Algebras Represent. Theory 3(4), 347–355 (2000). ISSN: 1386-923X. (Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday)Google Scholar
  10. 10.
    Majid S.: Quantum double for quasi-Hopf algebras. Lett. Math. Phys. 45(1), 1–9 (1998). ISSN: 0377-9017MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mason, G., Ng, S.-H.: Central invariants and Frobenius–Schur indicators for semisimple quasi-Hopf algebras. Adv. Math. 190(1), 161–195 (2005). ISSN: 0001-8708Google Scholar
  12. 12.
    Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989). ISSN: 0010-3616Google Scholar
  13. 13.
    Natale, S.: Frobenius–Schur indicators for a class of fusion categories. Pac. J. Math. 221(2), 353–377 (2005). ISSN: 0030-8730Google Scholar
  14. 14.
    Ng S.-H., Schauenburg P.: Central invariants and higher indicators for semisimple quasi-Hopf algebras. Trans. Am. Math. Soc. 360(4), 1839–1860 (2008) ISSN: 0002-9947MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ng, S.-H., Schauenburg, P.: Frobenius–Schur indicators and exponents of spherical categories. Adv. Math. 211(1), 34–71 (2007). ISSN: 0001-8708Google Scholar
  16. 16.
    Ng, S.-H., Schauenburg, P.: Higher Frobenius–Schur indicators for pivotal categories. In: Kauffman, L.H., Radford, D.E., Souza, F.J.O. (eds.) Hopf Algebras and Generalizations. Contemp. Math., vol. 441, pp. 63–90. AMS, Providence (2007)Google Scholar
  17. 17.
    Nikshych, D.: Morita equivalence methods in classification of fusion categories. In: Andruskiewitsch, N., Cuadra, J., Torrecillas, B. (eds.) Hopf Algebras and Tensor Categories. Contemp. Math., vol. 585, pp. 289–325. American Mathematical Society, Providence (2013)Google Scholar
  18. 18.
    Nikshych D.: Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories. Sel. Math. (N.S.) 14(1), 145–161 (2008) ISSN: 1022-1824MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ostrik, V.: Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. 27, 1507–1520 (2003). ISSN: 1073-7928Google Scholar
  20. 20.
    Schauenburg, P.: Computing higher Frobenius–Schur indicators in fusion categories constructed from inclusions of finite groups. Pac. J. Math. arXiv:1502.02314 [math.QA]
  21. 21.
    Schauenburg P.: Hopf bimodules, coquasibialgebras, and an exact sequence of Kac. Adv. Math. 165(2), 194–263 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schauenburg, P.: Hopf modules and the double of a quasi-Hopf algebra. Trans. Am. Math. Soc. 354(8), 3349–3378 (2002)Google Scholar
  23. 23.
    Schauenburg, P.: The monoidal center construction and bimodules. J. Pure Appl. Algebra 158(2–3), 325-346 (2001). ISSN: 0022-4049Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne, UMR 5584 CNRSUniversité Bourgogne Franche-ComtéDijonFrance

Personalised recommendations