A Brief History of the Positivity Conjecture in Tensor Category

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.