Abstract. We study properties of symmetric fusion categories in characteristic p. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object X of such a category, and derive an explicit formula for the Verlinde fiber functor F (X) of X (defined by the second author) in terms of the usual and super Frobenius-Perron dimensions of X. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic. IntroductionLet k be an algebraically closed field of characteristic p > 0. Let C be a symmetric fusion category over k. Then, according to the main result of [O], C admits a symmetric tensor functor F : C → Ver p into the Verlinde category Ver p (the quotient of Rep k (Z/pZ) by negligible morphisms); we prove that this functor is unique. We derive an explicit formula for the decomposition of F (X) into simple objects for each X ∈ C; it turns out that this decomposition is completely determined by just two parameters -the Frobenius-Perron dimension FPdim(X) and the super Frobenius-Perron dimension SFPdim(X) which we introduce in this paper. We use this formula to find the decomposition of symmetric powers of objects in Ver p , and in particular find the invariants in them -the "fusion" analog of the classical formula for polynomial invariants of binary forms. We also relate the super Frobenius-Perron dimension to the second Adams operation and to the p-adic dimension introduced in [EHO], and classify symmetric categorifications of fusion rings of rank 2. Finally, we classify triangular semisimple Hopf algebras in an arbitrary characteristic, generalizing the result of [EG] for characteristic zero, and classify braided fusion categories of rank two.The paper is organized as follows. Section 2 contains preliminaries. In Section 3 we prove the uniqueness of the Verlinde fiber functor. In Section 4, we define the super Frobenius-Perron dimension of an object 1 of a symmetric fusion category, and give a formula for this dimension in terms of the second Adams operation; here we also classify braided fusion categories of rank two. In Section 5, we prove a decomposition formula for the Verlinde fiber functor of an object X of a symmetric fusion category in terms of the ordinary and super Frobenius-Peron dimensions of X, and use this formula to compute the transcendence degrees of the symmetric and exterior algebra of X. In Section 6, we use the formula of Section 5 to find the decomposition of symmetric powers of objects in the Verlinde category, and in particular give a formula for their invariants. In Section 7, we compute the p-adic dimensions of an object in a symmetric fusion category. Finally, in Section 8, we classify semisimple triangular Hopf algebras in any characteristic.Acknowledgements. The work of P.E...

Abstract. In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in the case of semisimple categories and more generally in the case of categories with fiber functors to the characteristic p > 0 Verlinde category of SL 2 . We also construct a symmetric finite tensor category sVec 2 over fields of characteristic 2 and show that it is a candidate for the category of supervector spaces in this characteristic. We further show that sVec 2 does not fiber over the characteristic 2 Verlinde category of SL 2 and then prove the conjecture for any category fibered over sVec 2 .

In this article, we prove that the category of affine group schemes of finite type in the Verlinde category is equivalent to the category of Harish-Chandra pairs in the Verlinde category. Subsequently, we extend this equivalence to an equivalence between corresponding representation categories and then study some consequences of this equivalence to the representation theory of GL(L), with L a simple object in the Verlinde category. 6.4. Proof of equivalence between the categories of cocommutative Hopf algebras in Ver ind p and dual Harish-Chandra pairs in Ver p 40 7. Inverse Functor for Harish-Chandra pairs: construction via duality 40 8. Representations of affine group schemes of finite type in Ver p 47 8.1. Affine group schemes in Ver p with trivial underlying ordinary group 49 9. Representation theory of GL(X) 50 9.1. Representations of GL(L i ) for simple objects L i 54 References 56

We study properties of symmetric fusion categories in characteristic p. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object X of such a category, and derive an explicit formula for the Verlinde fiber functor F (X) of X (defined by the second author) in terms of the usual and super Frobenius-Perron dimensions of X. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique, and classify braided fusion categories of rank two and triangular semisimple Hopf algebras in any characteristic.

In this article, we construct affine group schemes GL(X) where X is any object in the Verlinde category in characteristic p and classify their irreducible representations. We begin by showing that for a simple object X of categorical dimension i, this representation category is semisimple and is equivalent to the connected component of the Verlinde category for SL i . Subsequently, we use this along with a Verma module construction to classify irreducible representations of GL(nL) for any simple object L and any natural number n. Finally, parabolic induction allows us to classify irreducible representations of GL(X) where X is any object in Verp.