Finite Symmetric Integral Tensor Categories with the Chevalley Property with an Appendix by Kevin Coulembier and Pavel Etingof

We prove that every finite symmetric integral tensor category C with the Chevalley property over an algebraically closed field k of characteristic p > 2 admits a symmetric fiber functor to the category of supervector spaces. This proves Ostrik's conjecture [25, Conjecture 1.3] in this case. Equivalently, we prove that there exists a unique finite supergroup scheme G over k and a grouplike element epsilon is an element of kG of order <= 2, whose action by conjugation on G coincides with the parity automorphism of G, such that C is symmetric tensor equivalent to Rep(G, epsilon). In particular, when C is unipotent, the functor lands in Vec, so C is symmetric tensor equivalent to Rep(U) for a unique finite unipotent group scheme U over k. We apply our result and the results of [17] to classify certain finite dimensional triangular Hopf algebras with the Chevalley property over k (e.g., local), in group scheme-theoretical terms. Finally, we compute the Sweedler cohomology of restricted enveloping algebras over an algebraically closed field k of characteristic p > 0, classify associators for their duals, and study finite dimensional (not necessarily triangular) local quasi-Hopf algebras and finite (not necessarily symmetric) unipotent tensor categories over an algebraically closed field k of characteristic p > 0. The appendix by K. Coulembier and P. Etingof gives another proof of the above classification results using the recent paper [4], and more generally, shows that the maximal Tannakian and super-Tannakian subcategory of a symmetric tensor category over a field of characteristic not equal 2 is always a Serre subcategory.