Character values at involutions

Abstract:

If ${\chi _1},{\chi _2},{\chi _3}$ are irreducible characters of a finite group $G$ satisfying $\int _G {{\chi _1}{\chi _2}{\chi _3} \ne 0}$ and $\sigma$ is an involution in $G$, then the proportions of $- 1$’s among the eigenvalues of the corresponding representations at $\sigma$ are the sides of a triangle on a sphere of circumference $2$.