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 G is symmetric or alternating, and H 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.