Cross characteristic projective representations for some classical groups
Cross characteristic projective representations of the finite simple classical groups are studied. In particular bounds are obtained for the smallest representations and some gap results are obtained as well -- i.e. results of the form -- if the dimension of an irreducible representation is not less than \(d_1\), then it must have dimension at least \(d_2\). The first major paper on the subject was by \textit{V. Landazuri} and \textit{G. M. Seitz} [J. Algebra 32, 418-443 (1974; Zbl 0325.20008)]. They obtained lower bounds for the smallest nontrivial irreducible representations of the finite Chevalley groups in all characteristics but the natural characteristic. The bounds were quite good, but not best possible. These results were improved by \textit{G. M. Seitz} and \textit{A. E. Zalesskii} [J. Algebra 158, No. 1, 233-243 (1993; Zbl 0789.20014)]. Subsequently, there has been a major effort to find the best possible bounds and classify the modules of dimension close to the bound and then prove a gap result. In characteristic zero, one can use the Deligne-Luzstig theory and this was done for the classical groups by \textit{P. H. Tiep} and \textit{A. E. Zalesskii} [Commun. Algebra 24, No. 6, 2093-2167 (1996; Zbl 0901.20031)]. This has been done for linear groups, symplectic groups in odd characteristic and unitary groups in a series of articles by Guralnick-Tiep, Guralnick-Magaard-Saxl-Tiep and Hiss-Malle. In this article, the author improves the bounds of Seitz-Zalesskii for the orthogonal groups -- obtaining essentially the best possible bounds. The key idea is to restrict to various parabolic subgroups. These types of results have had many applications, not the least of which is in the problem of classifying maximal subgroups of the classical groups.
scientific article; zbMATH DE number 1502170