Representations of the General Linear Groups Which Are Irreducible over Subgroups

1.1. Aschbacher-Scott Program. Primitive permutation groups have been studied since Galois and Jordan and have applications in many different areas. If Γ is a transitive permutation group with a point stabilizer M then Γ is primitive if and only if M < Γ is a maximal subgroup. So studying primitive permutation groups is equivalent to studying maximal subgroups. In most problems involving a finite primitive group Γ, the AschbacherO’Nan-Scott theorem [2] allows one to concentrate on the case where Γ is almost quasi-simple, i.e. L ⊳ Γ/Z(Γ) ≤ Aut(L) for a non-abelian simple group L. The results of Liebeck-Praeger-Saxl [33] and Liebeck-Seitz [34] then allow one to assume furthermore that Γ is a finite classical group. At this stage, Aschbacher’s theorem [1] severely restricts possibilities for the maximal subgroup H. Namely, if H < Γ is maximal then H ∈ S ∪⋃8i=1 Ci, (1.1)