Character tables of parabolic subgroups of Steinberg's triality groups

Abstract

We compute the conjugacy classes of elements and the character tables of the parabolic subgroups of Steinberg's triality groups 3 D 4 (q), where q is a power of an odd prime. Our main tools are Clifford theory applied to the Levi decomposition of the parabolic subgroups and the decomposition of restrictions of the unipotent characters of 3 D 4 (q). For many of the calculations we use the CHEVIE package.

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

“…It is shown in [7] and [8] that for every prime power q and every parabolic subgroup P of Steinberg's simple triality group 3 D 4 (q), maximal extensibility holds with respect to U P ¢ P , where U P is the unipotent radical of P . The analogous statement for the Ree groups 2 F 4 (q 2 ) with q 2 = 2 2n+1 is not true, as shown by the following remark.…”

“…Next, we construct the irreducible characters of P b covering λ 3 . We use the notation from [7,Lemma 5.4]. By Proposition 4.1,Ī 3 is the semidirect product of K := n b and …”

“…By Proposition 4.1,Ī 4 is the semidirect product of the group K := n 4 and the normal subgroup ik ∈ C, where we use the notation from [10, Table 5]. So, again by [7,Lemma 5.4], P b has exactly (q 2 − √ 2q)/4 + 4 irreducible characters covering λ 4 : the irreducible characters (εĪ 4 …”

“…In this paper we calculate the character tables of the maximal parabolic subgroups of 2 F 4 (q 2 ). Our methods are similar to those in [7]. Up to conjugacy, 2 F 4 (q 2 ) has two maximal parabolic subgroups, which we call P a and P b .…”

On the 2-decomposition numbers of Steinberg's triality groups D43(q), q odd

“…To calculate these characters we use generic character tables of parabolic subgroups and Maple programs written by the author for inducing class functions. These character tables and programs are based on the Maple [2] part of the CHEVIE [8] package (see [11]). The use of CHEVIE also allows us to compute easily scalar products of the induced characters with the complex irreducible characters of 3 D 4 (q).…”