On geometrically reductive tensor categories

TL;DR

This paper proves that higher Verlinde categories are geometrically reductive, enabling the application of algebraic geometry results to these tensor categories.

Contribution

It confirms a key conjecture about higher Verlinde categories and reduces related conjectures to existing literature, advancing the understanding of tensor categories.

Findings

Proved higher Verlinde categories are geometrically reductive.

Reduced two conjectures on geometric reductivity to known conjectures.

Enabled algebraic geometry techniques to be applied to these tensor categories.

Abstract

We prove the conjecture that higher Verlinde categories are geometrically reductive. This is one of the two properties required in order for recent results on algebraic geometry in tensor categories to apply to these categories. We also reduce two further conjectures concerning geometric reductivity for tensor categories to other conjectures appearing in the literature.