More on Geometries of the Fischer Group Fi22

• Authors: A. A. Ivanov;C. Wiedorn
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2002

We give a new, purely combinatorial characterization of geometries \cal E with diagramc.F_4(1):\begin{picture}(45,5)\put(2,1){\makebox(0,0){$\circ$}}\put(12,1){\makebox(0,0){$\circ$}}\put(22,1){\makebox(0,0){$\circ$}}\put(32,1){\makebox(0,0){$\circ$}}\put(42,1){\makebox(0,0){$\circ$}}\put(32.5,1){\line(1,0){8.7}}\put(2.5,1){\line(1,0){8.7}}\put(12.5,1){\line(1,0){8.7}}\put(22.5,1.2){\line(1,0){8.9}}\put(22.5,0.8){\line(1,0){8.9}}\put(32.5,1){\line(1,0){8.7}}\put(7,2.5){c}\put(1,-2.5){\scriptsize {1}}\put(11,-2.5){\scriptsize{2}}\put(21,-2.5){\scriptsize{2}}\put(31,-2.5){\scriptsize{1}}\put(41,-2.5){\scriptsize{1}}\end{picture}identifying each under some “natural” conditions—but not assuming any group action a priori—with one of the two geometries \cal E(Fi22) and \cal E(3 · Fi22) related to the Fischer 3-transposition group Fi22 and its non-split central extension 3 · Fi22, respectively. As a by-product we improve the known characterization of the c-extended dual polar spaces for Fi22 and 3 · Fi22 and of the truncation of the c-extended 6-dimensional unitary polar space.