Character tables of the maximal parabolic subgroups of the Ree groups 2F4(q2)

Abstract

We compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of the simple Ree groups 2 F 4(q 2 ). For one of the maximal parabolic subgroups, we find an irreducible character of the unipotent radical that does not extend to its inertia subgroup.

The simple Ree groups F42(q2) are determined by the set of their character degrees

“…It follows that r a | q 8 as r a is even. As in the previous case, we have that f divides r a − 1 and since r a q 8 , we deduce that f q 8 − 1. However as q 8 …”

“…By Lemma 2.2(a), we have |G : I|μ i (1) = |G : U ||U : I|μ i (1) divides some character degree of G. Thus the index |G : U | = |G /M : U /M| of 2 F 4 (q 2 ) must divide some character degree of 2 F 4 (q 2 ) and hence by Lemma 3.2, U /M ∼ = P a or U /M ∼ = P b , where P a and P b are maximal parabolic subgroups of 2 F 4 (q 2 ). Note that both unipotent radicals [q 22 ] and [q 20 ] of the parabolic subgroups P a and P b , respectively, are nonabelian by using the commutator relations (see [8]). Let t = |U : I| and recall that if N P G and λ ∈ Irr(N), then Irr(G|λ) denotes the set of all irreducible constituents of λ G .…”

On the restriction of cross characteristic representations of 2 F 4(q) to proper subgroups

“…According to [6], the faithful character χ must be one of Pa χ 36 , Pa χ 37 or Pa χ j (k) with j ∈ {34, 35, 38, 39, 40}. The values of these characters on u and v are computed in [6] and displayed in the following Table: u Now one sees that (1) is violated.…”

“…We will consider the two long-root elements u := α 12 (1) and v := α 8 (1) of P a , in the notation of [5], [6]. Certainly, these two long-root elements are conjugate in G, so ϕ(u) = ϕ(v).…”

“…The values of these characters on u and v are computed in [6] and displayed in the following Table: u Now one sees that (1) is violated.…”