Matrix representation of 2F4(2)' in unitary U(27)

Hello,

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).

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.

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 $^2F_4(2)'$.

Motivation 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 $Co_0$.

One could think that there might be such friendship between $^2F_4(2)'$ embedded in E6 Lie group and some sporadic group X.

Regards, Marek

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Marek, could you define EIII? Just to be sure I understand what you're asking: Are you looking for irreducible embeddings? If so the ATLAS says there are only two into GL(27,C) and if these don't work (as you say), then you have no chance. There are also two into GL(26,C); so the question is whether the image of these reps in GL(26,c) can in turn be embedded into U(27). (I'm not sure if this means that they must be embedded into U(26) but I would guess so...) – Nick Gill Jun 13 '12 at 12:40
I presume you are familiar with this paper: ams.org/mathscinet-getitem?mr=1653177 Griess, Robert L., Jr.; Ryba, A. J. E. Finite simple groups which projectively embed in an exceptional Lie group are classified! Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 1, 75–93. This paper gives a full list of finite simple groups lying inside exceptional Lie groups. For the ${^2F_4(2)'}$ embedding in $E_6$ they refer to this paper: Arjeh Cohen and David Wales, On finite subgroups of F4 (C) and E6 (C), Proc. London Math. Soc., (3) 74 (1997), 105-150. – Nick Gill Jun 13 '12 at 12:55
EIII is defined in Atsuyama "Projective spaces...II", 1997 as elements of complexified Jordan algebra h3O satisfying x delta x = 0 where points x and ax are identified for non zero complex number a. Paper of Cohen, Wales proves that embedding of Ree 2F4(2)' in E6 exists but does not give explicit embedding, which I am looking for. So if such embedding exists, then having E6 embedded in U(27) we must obtain 2F4(2)' in U(27). So theory say that embedding exists but in practice we don't know how to obtain matrices in U(27) generating this group. Regards, Marek – Marek Mitros Jun 13 '12 at 15:45
No more answers ? Can somebody explain why representation theory does not give embedding of finite group into O(n) orthogonal Lie group ? BTW. I discovered on June 20th that $^2F_4(2)′$ is subgroup of Fischer $Fi_{22}$ ! Any comments on this fact ? – Marek Mitros Jun 29 '12 at 20:43
A finite group representation is always unitary, and you can construct the corresponding invariant form; this amounts to some linear algebra. Whether it is orthogonal, is completely determined by its character. See e.g. Serre's little book on representation theory. – Dima Pasechnik Jun 30 '12 at 0:17