Matrix representation of 2F4(2)' in unitary U(27) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T16:46:28Z http://mathoverflow.net/feeds/question/99432 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99432/matrix-representation-of-2f42-in-unitary-u27 Matrix representation of 2F4(2)' in unitary U(27) Marek Mitros 2012-06-13T11:46:33Z 2012-09-14T08:45:32Z <p>Hello,</p> <p>I am looking for matrix representation of Tits group \$^2F_4(2)'\$ of size 17 971 200. Atlas of finite groups offer several matrix representations but they are not embedded in U(27). They are in GL(27, C). </p> <p>I am looking really for embedding of \$^2F_4(2)'\$ in \$E_6\$ compact Lie group. I tried to guess the embedding but no luck so far. I have come to idea that if I have this group generators in U(27) then I will find them in \$E_6\$ really.</p> <p>Atsuyama has defined embedding of EIII symmetric space into E6 Lie group by mapping point in EIII to "reflection". The formula for such reflection can be found in Atsuyma paper. I am hoping to find 1755 points in EIII in which reflections would be 2A conjugacy class in \$^2F_4(2)'\$.</p> <p><strong>Motivation</strong> The motivation for my research is following. Daniel Allcock has written to me that "There seems to be a friendship between \$^3D_4(2)\$ and \$Co_0\$ even though neither contains the other". That friendship can be expressed as mapping 819 reflections from 2A conjugacy class in \$^3D_4(2)\$ embedded in F4 into elements of 2A conjugacy class of \$Co_0\$.</p> <p>One could think that there might be such friendship between \$^2F_4(2)'\$ embedded in E6 Lie group and some sporadic group X.</p> <p>Regards, Marek</p> http://mathoverflow.net/questions/99432/matrix-representation-of-2f42-in-unitary-u27/107156#107156 Answer by Marek Mitros for Matrix representation of 2F4(2)' in unitary U(27) Marek Mitros 2012-09-14T08:45:32Z 2012-09-14T08:45:32Z <p>I am happy to inform that Robert Wilson have sent me the complex matrices generating the 2F4(2)' group inside the E6 compact Lie group. See the paper <a href="http://arxiv.org/pdf/1208.4221.pdf" rel="nofollow">http://arxiv.org/pdf/1208.4221.pdf</a> for details.</p> <p>Best regards, Marek</p>