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Let G be a semisimple algebraic group. I can see (geometrically) why every dominant weight has to be a non-negative combination of simple roots (and if it strictly dominant then it has to be a positive combination). I assume that the coefficients are allowed to be in $\mathbb{Q}$.

Does anybody have an algebraic proof?

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To show the non-negativity. Write a dominant character as the sum of negative linear combination of some of the simple roots and non-negative of the others. Now, taking scalar product with the negative part, gives the deusired result.

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