Finite symmetric tensor categories and the Verlinde tower

Every finite symmetric tensor category in characteristic p > 0 admits a fibre functor to a higher Verlinde category, the finite case of the Benson-Etingof-Ostrik conjecture, after OpenAI's 2026 manuscript. Tilting modules for SL_2; an equivariant vector bundle on the plane and the restricted tilting theorem; higher Frobenius functors; norm maps and symmetrizers; rank varieties; descent through Frobenius-Perron dimensions; incompressible categories and reconstruction.

Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Finite symmetric tensor categories and fibre functors
2. Rational representations of SL₂ and tilting modules
3. Lifting representations of elementary abelian groups to the plane
4. The restricted tilting theorem
5. Higher Frobenius functors
6. Norm maps and a first nonprojective tensor power
7. Rank varieties and an exact higher Frobenius functor
8. Descent to a Verlinde category
9. Incompressible categories, uniqueness and reconstruction

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Tannakian categories · The affine Grassmannian and geometric Satake
• Induced representations and Frobenius reciprocity · Open Mathematics Courses (Representations of finite groups)
• Representations and complete reducibility · Open Mathematics Courses (Representations of finite groups)
• Representations of sl(2) · Open Mathematics Courses (Semisimple Lie algebras)
• Tensor categories: rigid symmetric tensor categories and fibre functors (course The affine Grassmannian and geometric Satake, lesson Tannakian categories); representations of finite groups and induced representations (course Representations of finite groups); representations of sl(2) in characteristic zero (course Semisimple Lie algebras); finite-dimensional algebras; coherent sheaves and vector bundles.