Fiber Functors of Equivariantizations of Finite Tensor Categories
Let $G$ be a finite group acting on a finite tensor category $\mathcal{C}$. We classify fiber functors on the equivariantization $\mathcal{C}^G$ in terms of equivariant exact module categories over $\mathcal{C}$, indexed by subgroups of $G$. The data are a subgroup $H\subseteq G$ and an $H$-equivariant $\mathcal{C}$-module category $\mathcal{M}$ whose underlying $\mathcal{C}$-module category is indecomposable, exact, and semisimple; they give a fiber functor precisely when $H$ acts transitively on the simple objects of $\mathcal{M}$ and the stabilizer cocycle of one, hence every, simple object is non-degenerate. Through Tannaka-Krein reconstruction this describes realizations of $\mathcal{C}^G$ as the representation category of a finite-dimensional Hopf algebra, with no semisimplicity hypothesis on $\mathcal{C}$. As applications, for odd primes $p$ we determine the fiber functors on $\mathrm{Rep}(H_p)$, where $H_p$ denotes Nikshych's semisimple Hopf algebra of dimension $4p^2$: there is one equivalence class if $p\equiv 3\pmod 4$ and two if $p\equiv 1\pmod 4$. We also use the classification for gaugings to determine which non-pointed entries in the small-dimensional list of Green and Nikshych are representation categories of semisimple factorizable Hopf algebras.