Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules

Abstract

We call an element of a finite general linear group GL(d, q) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than d/2. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than d/2. We show that for groups G with SL(d, q) ≤ G ≤ GL(d, q) most pairs of fat elements from G generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set of all pairs in G × G, is less than q ^−d+1. We also prove that the conditional probability to obtain a pair (g ₁, g ₂) in G × G which generates a reducible subgroup, given that g ₁, g ₂ are fat elements, is less than 2q ^−d+1. Further, we show that any reducible subgroup generated by a pair of fat elements acts irreducibly on a subspace of dimension greater than d/2, and in the induced action the generating pair corresponds to a pair of fat elements.