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Jan 25, 2013 at 8:54 comment added Uwe Franz Thank you very much, Allen and Robert, your explanations are very helpful for me!
Jan 24, 2013 at 6:45 vote accept Uwe Franz
Jan 23, 2013 at 19:57 comment added Robert Bryant Just a comment on references: É. Cartan wrote down explicit descriptions of these quotient simplices for all of the compact simple, simply-connected Lie groups (including the exceptional ones) some time in the 1930s. These are exactly the kind of descriptions that the OP wants. I'm traveling right now and don't have a copy of his collected works handy, or else I'd supply it, but I'm sure that you can find the right references in his 1936 paper La topologie des espaces représentatifs des groupes de Lie (in Part I, volume 2 of Cartan's Collected Works).
Jan 23, 2013 at 19:12 history answered Allen Knutson CC BY-SA 3.0