Some semiclassical parabolic systems of rank \(4\)
A semiclassical parabolic system is a set of subgroups \(\{P_i,\;\in I\}\) of a group \(G\) such that (i) \(G=\langle P_i\mid i\in I\rangle\) and \(G\neq G_i=\langle P_j\mid j\in I\setminus\{i\}\rangle\) for all \(i\). (ii) There is \(S\leq B=\bigcap_{i\in I}P_i\), \(S\in\text{Syl}_2\langle P_i,P_j\rangle\cap\text{Syl}_2(P_i)\) for all \(i,j\in I\). (iii) \(P_i/B_{P_i}\) is a rank 1 Lie group in characteristic 2 with Borel subgroup \(B/B_{P_i}\), (iv) \(B_G=1\), (v) for all \(i,j\in I\), \(i\neq j\), 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 Lie group in characteristic two or \(3A_6\), \(3\Sigma_6\) and the latter occurs for at least one choice of \(i\) and \(j\). Now the paper under review treats the case \(| I|=4\). As usual, one can describe a parabolic system by a digram whose nodes are the \(P_i\) and edges are the Dynkin diagrams of the corresponding Lie groups or \(\circ\tildediagrBar\circ\) in case of \(\langle P_i,P_j\rangle/B_{\langle P_i,P_j\rangle}\cong 3A_6\) or \(3\Sigma_6\). The author classifies all such systems under the additional assumption that if \(\langle P_i,P_j,P_k\rangle\) belongs to \(\circ\diagrbar\circ\tildediagrBar\circ\) then \(\langle P_i,P_j,P_k\rangle/B_{\langle P_i,P_j,P_k\rangle}\cong 3^7\text{Sp}_6(2)\). The case that there is a diagram of this type which belongs to \(M_{24}\) or \(He\) was treated by \textit{A. Fukshansky} and the reviewer [Geom. Dedicata 70, No. 3, 305-329 (1998; Zbl 0907.20027)]. So altogether we have a complete classification in the rank four case. The result is too long to be stated here. But there is one nice example with diagram \(\circ\diagrbar\circ\tildediagrBar\circ\diagrbar\circ\) which belongs to \(3^{833}F_4(2)\). More information about this example can be found in two papers of the author [J. Algebra 195, No. 2, 337-356 (1997; Zbl 0918.51016); Beitr. Algebra Geom. 38, No. 2, 337-342 (1997; Zbl 0890.51010)].
scientific article; zbMATH DE number 1271334
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