Timeline for Matrix representation of 2F4(2)' in unitary U(27)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 14, 2012 at 8:46 | vote | accept | CommunityBot | ||
| Sep 14, 2012 at 8:45 | answer | added | user21230 | timeline score: 1 | |
| Jul 6, 2012 at 6:55 | comment | added | user21230 | OK. But having representation of finite group g in GL(27,C) - can we obtain representation in U(27) ? My question was how to obtain representation of finite group in compact Lie group - O(n), U(n) or Sp(n). Atlas gives representations not compact. | |
| Jun 30, 2012 at 0:17 | comment | added | Dima Pasechnik | 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. | |
| Jun 29, 2012 at 20:43 | comment | added | user21230 | 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 ? | |
| Jun 29, 2012 at 20:38 | history | edited | user21230 | CC BY-SA 3.0 |
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| Jun 13, 2012 at 15:45 | comment | added | user21230 | 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 | |
| Jun 13, 2012 at 12:55 | comment | added | Nick Gill | 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. | |
| Jun 13, 2012 at 12:40 | comment | added | Nick Gill | 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...) | |
| Jun 13, 2012 at 11:46 | history | asked | user21230 | CC BY-SA 3.0 |