IRREDUCIBILITY OF ALTERNATING AND SYMMETRICAL SQUARES

We investigate the question when the alternating or symmetric square o f an absolutely irreducible projective representation of a non-abelian simple group G is again irreducible. The knowledge of such representa tions is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We obtain complete results for G an alternating group and for G a projective special linear group when the given representation is in non-defining characteristic. For the pr oof we exhibit a linear composition factor in the socle of the restric tion to a large subgroup of the alternating or symmetric square of a g iven projective representation V, Assuming irreducibility this shows t hat the dimension of V has to be very small. A good knowledge of proje ctive representations of small dimension allows to rule out these case s as well.