Cross Characteristic Projective Representations for Some Classical Groups

Let G(pn) be a finite simple group of Lie type and let V be a projective irreducible representation of G(pn) over a field of characteristic different from p. Lower bounds for the dimension of V have been computed in [V. Landázuri and G. M. Seitz, J. Algebra32 (1974), 418–433] and [G. M. Seitz and A. E. Zalesskii, J. Algebra158 (1993), 233–243] and have recently been improved in the case of linear groups by R. Guralnick and P. H. Tiep [Proc. London Math. Soc. (3)78 (1999), 116–138] and in the case of characteristic zero representations in [P. H. Tiep and A. Zalesskii, Comm. Algebra24 (1996), 2093–2167]. The purpose of this paper is to improve the bounds in [G. M. Seitz and A. E. Zalesskii, J. Algebra158 (1993), 233–243] for the case of orthogonal groups and to provide estimates for the gap between the smallest representations for orthogonal and unitary groups.