66 research outputs found
Some quasitensor autoequivalences of Drinfeld doubles of finite groups
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 -th power operation, with 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
We consider a subclass of the class of group-theoretical fusion categories:
To every finite group and subgroup one can associate the category of
-graded vector spaces with a two-sided -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
Group-theoretical fusion categories are defined by data concerning finite
groups and their cohomology: A finite group endowed with a three-cocycle
, and a subgroup endowed with a two-cochain whose
coboundary is the restriction of .
The objects of the category are -graded vector spaces with suitably
twisted -actions; the associativity of tensor products is controlled by
. Simple objects are parametrized in terms of projective
representations of finite groups, namely of the stabilizers in of right
-cosets in , 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
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
We calculate Frobenius-Schur indicator values for some fusion categories
obtained from inclusions of finite groups , where more concretely
is symmetric or alternating, and 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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