Higher Frobenius-Schur Indicators in Group-Theoretical Fusion Categories

Speaker:  Peter Schauenburg - Institut de Mathématiques de Bourgogne
  Wednesday, March 18, 2015 at 3:30 PM 15:15 rinfresco; 15:30 inizio seminario

 Fusion categories are semisimple abelian categories with an abstract tensor product structure. Such categories arise in connection with various topics in mathematics and mathematical physics such as conformal field theory, subfactor theory and low-dimensional topology. Considered in their own right, they are intricate finite-dimensional (in some sense) algebraic structures meeting combinatorial and arithmetical constraints. The first examples that "everybody" knows are the categories of representations of finite groups. In the same way examples arise from representations of quantum groups or Hopf algebras. Group-theoretical fusion categories form a class of fusion categories that can be constructed from certain rather classical data involving a finite group, a subgroup, and some complications given in terms of group cohomology. They are considered as a well-understood class (in the classification program for fusion categories outright "trivial"), while their structure in detail is quite complicated; in fact group-theoretical categories includethe representation categories of many interesting semisimple Hopf algebras. Frobenius-Schur indicators were first considered (in degree 2) by their namesakes for irreducible complex characters of finite groups not long after the latter were first invented. They indicate whether a complex representation can be realized by matrices over the real numbers. Generalized beyond the representation category of a finite group, and to higher degrees, Frobenius-Schur indicators give a system of numerical invariants for fusion categories, more precisely for each of their (simple) objects. These invariants are a useful theoretical tool in the structure and classification theory of fusion categories, but they are not always straightforward to calculate in concrete examples. In the case of group-theoretical categories, however, the higher Frobenius-Schur indicators can be computed by very concrete formulas in terms of the rather classical data defining those categories and describing their simples, to wit, finite groups, certain (stabilizer) subgroups, and their irreducible (projective)


Contact person
Lidia Angeleri

Publication date
March 5, 2015