Algebraic geometry and representation theory in the Verlinde category

Abstract

This thesis studies algebraic geometry and the representation theory of group schemes in the setting of symmetric tensor categories over algebraically closed fields of positive characteristic. A specific focus is paid to the Verlinde category, a symmetric fusion category in characteristic p that serves as a universal base for all such categories. Symmetric tensor categories provide a natural setting in which it makes sense to discuss the notion of a commutative, associative unital algebra. In the first third of the thesis, we prove some fundamental facts about these algebras, showing that, in the Verlinde category and any category built out of it, finitely generated algebras are Noetherian, have finitely generated invariants and are finite as a module over their invariants. Subsequently, we use this result to extend some fundamental properties of commutative algebras from the original setting of vector spaces to the more general setting of symmetric tensor categories.

The middle portion of the thesis focuses on some other applications of commutative algebra in the Verlinde category to elaborate on some work of Ostrik and to obtain important combinatorial decomposition formulas. This is a presentation of joint work by the author with Etingof and Ostrik. Symmetric tensor categories also provide a natural setting in which to discuss affine group schemes and their representations, namely the commutative Hopf algebras and their comodules. The last part of this thesis focuses on the structure and representation theory of affine group schemes of finite type in the Verlinde category. Using some of the basic commutative algebra properties proved in the thesis, we extend some results of Masuoka from the setting of super vector spaces in positive characteristic to that of the Verlinde category.

In particular, we define the notion of a Harish-Chandra pair and show that the category of affine group schemes and their representations in the Verlinde category is equivalent to the category of Harish-Chandra pairs and their representations in the Verlinde category.