Almost irreducible tensor squares

Let G be a covering group of a finite almost simple group We determine those faithful irreducible complex characters chi of G for which chi x chi* - 1 is again irreducible. This gives a classification of the quasi-simple absolutely irreducible subgroups of GL(n)(q) of order prime to q which act irreducibly on the Lie algebra of type. A(n-1) via the adjoint representation. The proof uses Lusztig's description of the degrees of irreducible characters of reductive groups and the determination of Brauer trees by Fong and Srinivasan to handle the case of groups of Lie type. It turns out that the only infinite series of examples are characters of Weyl representations for SUn(F-2) and Sp(2n)(F-3).