The semiclassical parabolic systems
Parabolic systems are used in the study of simple groups of Lie type and some sporadic simple groups. In order to include more sporadic groups semiclassical parabolic systems were introduced by \textit{M.~A. Ronan} and \textit{G. Stroth} [Eur. J. Comb. 5, 59-91 (1984; Zbl 0574.20012)]. A semiclassical parabolic system of rank \(n\) is a set of subgroups \(P_1,\ldots,P_n\) of a group \(G\) such that \(G\) is generated by these subgroups but not by fewer of them; such that \(P_i/B_{P_i}\) is a rank 1 group of Lie type in characteristic 2 with Borel subgroup \(B/B_{P_i}\), where \(B=\bigcap_{i=1}^nP_i\); such that \(B=P_i\cap B_j\) for all \(i\neq j\) and \(B_G=1\); such that for all \(i\neq j\), either \(\langle P_i,P_j\rangle=P_iP_j\) or \(\langle P_i,P_j\rangle/B_{\langle P_i,P_j\rangle}\) is a rank 2 group of Lie type in characteristic two or \(3A_6\), \(3\Sigma_6\) and the latter occurs for at least one choice of \(i\) and \(j\). Semiclassical parabolic systems of rank 3 were classified by \textit{F.~G. Timmesfeld} [J. Algebra 124, No. 1, 9-59 (1989; Zbl 0742.20039)] and those of rank 4 by \textit{A. Fukshansky} and \textit{G. Stroth} [Geom. Dedicata 70, No. 3, 305-329 (1998; Zbl 0907.20027)], and \textit{C. Wiedorn} [J. Algebra 211, No. 2, 472-513 (1999; Zbl 0923.20022)]. In the paper under review the authors complete the (local) classification of all semiclassical parabolic systems by determining those systems of rank at least 5. This is achieved by induction on the rank \(n\) and by examining the diagrams associated with semiclassical parabolic systems. The vertices of these associated diagrams are the \(P_i\) and the edges are formed as in the Dynkin diagrams of the corresponding Lie groups or the tilde geometry corresponding to \(3A_6\), \(3\Sigma_6\). The authors exclude certain subdiagrams to obtain a list of possible diagrams. For example, the results of Timmesfeld [loc.cit.] imply that no triangles occur, and, among the many steps, it is shown that no subdiagram of type \(F_4\) can occur unless \(n=4\). In most cases the groups cannot be explicitly determined, only the amalgam and it is not clear if such groups exist and if they do exist whether or not they are finite. However, the proof contains many structural information and results on representations about the groups involved.
scientific article; zbMATH DE number 1505832