Rank 3 permutation modules of the finite classical groups

Abstract

Journal of Algebra 291 (2005) 551-606. doi:10.1016/j.jalgebra.2005.02.005Received by publisher: 2004-09-14Harvest Date: 2016-01-04 12:19:59DOI: 10.1016/j.jalgebra.2005.02.005Page Range: 551-60

The Ore conjecture

“…The degrees of the irreducible constituents of the latter are well known (see e.g. [42]). It follows in particular that, when n = 2m + 1, some…”

Primes dividing the degrees of the real characters

“…We will use rank 3 permutation actions, cf. [27] for instance, to produce χ for several classical groups.…”

On p-Brauer characters of p′-degree and self-normalizing Sylow p-subgroups

“…Then gð1Þ 1 À2 ðmod pÞ. Moreover, g is an irreducible constituent ofâ a by [23,Corollary 6.5], of degree larger than að1Þ=p, and so we are done again.…”

“…Again we consider the rank 3 permutation action of G on the singular 1-spaces of the natural module F 2n q , and its character r. Then r ¼ 1 G þ a þ b, where a and b are unipotent, AutðGÞ-invariant, irreducible characters, of degree ðq n À 1Þðq nÀ1 þ qÞ=ðq 2 À 1Þ and ðq 2n À q 2 Þ=ðq 2 À 1Þ, respectively. It was shown in [23] thatr r contains exactly two non-trivial irreducible constituents, g of degree ðq n À 1Þðq nÀ1 þ qÞ=ðq 2 À 1Þ À k where k ¼ 2 if 2jn and 1 otherwise, and d of degree ðq n À 1Þðq nÀ1 À 1Þ=ðq þ 1Þ.…”