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Ideally, I would like a constructive, algorithmic proof of this fact. I have convinced myself that it is true, but my "proof" is not pretty. I would like to know if a more attractive or intuitive proof exists, or if I am in fact wrong.

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Can you explain the setting in a little more detail? (Also, the name is "Chevalley".) –  Jim Humphreys Jun 27 '13 at 0:30
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