A brief history of the positivity conjecture in tensor category theory

TL;DR

This paper constructs a finite group with specific irreducible characters to disprove the positivity conjecture in rational conformal field theory and related fusion category conjectures.

Contribution

It provides explicit counterexamples to longstanding conjectures in tensor category theory and rational CFT.

Findings

Counterexamples to the positivity conjecture in rational CFT

Disproof of Zhenghan Wang's conjecture on pivotal fusion categories

Identification of finite groups with characters exhibiting specific Frobenius-Schur indicators

Abstract

We show the existence of a finite group $G$ having an irreducible character $\chi$ with Frobenius-Schur indicator $\nu_2(\chi){=}{+}1$ such that $\chi^2$ has an irreducible constituent $\varphi$ with $\nu_2(\varphi){=}{-}1$. This provides counterexamples to the positivity conjecture in rational CFT and a conjecture of Zhenghan Wang about pivotal fusion categories.