Irreducibility of alternating and symmetric squares
- 67 Downloads
We investigate the question when the alternating or symmetric square of an absolutely irreducible projective representation of a non-abelian simple groupG 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 forG an alternating group and forG 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 representationV. Assuming irreducibility this shows that the dimension ofV has to be very small. A good knowledge of projective representations of small dimension allows to rule out these cases as well.
Mathematics Subject Classification (1991)20E28 20G40 20C20
Unable to display preview. Download preview PDF.
- 1.Aschbacher, M.: Small degree representations of groups of Lie type. Preprint Caltech (1986)Google Scholar
- 2.Guralnick, R., Tiep, P.H.: Low-dimensional representations of special linear groups in cross characteristic. Preprint (1997)Google Scholar
- 9.Malle, G.: Almost irreducible tensor squares. To appear. Comm. Algebra (1997)Google Scholar