618 research outputs found

    Some quasitensor autoequivalences of Drinfeld doubles of finite groups

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    We report on two classes of autoequivalences of the category of Yetter-Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the rr-th power operation, with rr relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter-Drinfeld modules. Both autoequivalences preserve higher Frobenius-Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.Comment: 18 page

    Computing Higher Frobenius-Schur Indicators in Fusion Categories Constructed from Inclusions of Finite Groups

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    We consider a subclass of the class of group-theoretical fusion categories: To every finite group GG and subgroup HH one can associate the category of GG-graded vector spaces with a two-sided HH-action compatible with the grading. We derive a formula that computes higher Frobenius-Schur indicators for the objects in such a category using the combinatorics and representation theory of the groups involved in their construction. We calculate some explicit examples for inclusions of symmetric groups.Comment: 29 page

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

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    Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group GG endowed with a three-cocycle ω\omega, and a subgroup H⊂GH\subset G endowed with a two-cochain whose coboundary is the restriction of ω\omega. The objects of the category are GG-graded vector spaces with suitably twisted HH-actions; the associativity of tensor products is controlled by ω\omega. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in HH of right HH-cosets in GG, 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.Comment: 21 page

    On the Frobenius-Schur indicators for quasi-Hopf algebras

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    Mason and Ng have given a generalization to semisimple quasi-Hopf algebras of Linchenko and Montgomery's generalization to semisimple Hopf algebras of the classical Frobenius-Schur theorem for group representations. We give a simplified proof, in particular a somewhat conceptual derivation of the appropriate form of the Frobenius-Schur indicator that indicates if and in which of two possible fashions a given simple module is self-dual.Comment: 7 page

    Frobenius-Schur indicators for some fusion categories associated to symmetric and alternating groups

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    We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups H⊂GH\subset G, where more concretely GG is symmetric or alternating, and HH is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusion categories not associated to Hopf algebras.Comment: 15 page
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