An infinite family of counterexamples to a conjecture on positivity (Q2039379)

The paper provides a family of finite groups \(G\) and for each such \(G\) its complex irreducible characters \(\chi\) such that \(\chi^2\) contains a summand whose Frobenius-Schur indicator is \(-1\). The examples address a positivity question in tensor categories and is motivated by a counterexample of Geoffrey Mason to the following conjecture of Zhenghan Wang: for an object \(Y\) in a pivotal fusion category, each simple summand in \(Y\otimes Y^\vee\) has the Frobenius-Schur indicator \(1\). Like the examples produced in the current paper, the older counterexample was in the classical setting of finite group representations.