Almost irreducible tensor squares

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