Some Semiclassical Parabolic Systems of Rank 4

✍ Scribed by Corinna Wiedorn

• Book ID: 102571567
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 398 KB
• Volume: 211
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7622

✦ Synopsis

The aim of this paper is to give a classification of a certain class of semiclassical parabolic systems of rank 4.

Definition 1. Let G be any group. A semiclassical parabolic system for G is a set P i i ∈ I , I = 1 n , of subgroups of G with the following properties:

(i) G = P 1 P n and G = G i = P j j ∈ I \ i for all i ∈ I.

(ii) There exists a finite subgroup S ≤ B = n i=1 P i such that S ∈ Syl 2 P i ∩ Syl 2 P ij for all i j ∈ I, where P ij = P i P j .

(iii) For all i ∈ I, P i /B P i is a rank-1-Lie group defined over a field of char 2 with Borel subgroup B/B P i .

(iv) For all i j ∈ I, i = j, either P ij = P i P j or P ij /B P ij is a rank-2-Lie group defined over a field of char 2 or P ij /B P ij ∼ = 3A 6 or 3 6 and the last case occurs at least once (otherwise the system is called classical).

(v) B G = g∈G B g = 1.

We call the subgroups P i the minimal parabolics, the G i the maximal parabolics, and n the rank of the parabolic system. * This work is part of the Ph.D. thesis of the author. 472