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Let G be a compact Lie group with a maximal torus T, then what does the complex representation of G $\alpha:G\rightarrow U(n)$ mean? Does it mean that regarding $\alpha(g)$ as an isometry on $C^n$ for $g\in G$?

Also, it will be good someone let me know the representations $\alpha:G\rightarrow U(n)$ for $G=G_2,F_4,E_6,E_7,E_8$? I don't have J.F.Adams's book "Lectures on exceptional lie groups"at hand.

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First question: it is a smooth homomorphism. Second question: no, the condition you wrote only specifies a set-theoretic homomorphism; you need an additional differentiability condition that relates the isometries corresponding to nearby elements. –  S. Carnahan Jun 6 '10 at 17:16
    
Closed. Please check the FAQ for more details. As usual if the answer is already on wikipedia, the question is inappropriate here. –  Scott Morrison Jun 6 '10 at 23:46
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closed as off topic by Scott Morrison Jun 6 '10 at 23:45

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2 Answers

See http://en.wikipedia.org/wiki/Unitary_representation for the representation theory. You can follow the links from http://en.wikipedia.org/wiki/Simple_Lie_group#Exceptional_cases to find external links to the dimension sequences for the irreducible representations.

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You can use Atlas of Lie groups software to compute representations.

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