Reductions of tensor categories modulo primes

TL;DR

This paper investigates conditions under which semisimple rigid tensor categories can be reduced modulo primes to remain semisimple, identifying good and bad primes and exploring their properties in various categories.

Contribution

It establishes criteria for good primes in group-theoretical fusion categories and discusses the complexity of these conditions in general fusion categories.

Findings

Good primes are relatively prime to the squared norm of simple objects.

For group-theoretical fusion categories, the converse condition holds.

Results are provided for Verlinde categories and other known fusion categories.

Abstract

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the M\"uger squared norm |V|^2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.