Irreducibility of alternating and¶symmetric squares

Abstract:

We investigate the question when the alternating or symmetric square of an absolutely irreducible projective representation of a non-abelian simple group G is again irreducible. The knowledge of such representations 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 proof we exhibit a linear composition factor in the socle of the restriction to a large subgroup of the alternating or symmetric square of a given projective representation V. Assuming irreducibility this shows that the dimension of V has to be very small. A good knowledge of projective representations of small dimension allows to rule out these cases as well.