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Can someone help me with this exercise? (ex 2 page 141, "Introduction to Lie Algebras and Representation Theory - Humphreys)

Use Weyl's dimension formula https://en.wikipedia.org/wiki/Weyl_character_formula#Weyl_dimension_formula to show that an irreducible finite dimensional L-module of smallest possible dimension has highest weight $\lambda_i$ for some $1 \leq i \leq l$.

Thanks!

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  • $\begingroup$ Sorry! I forgot to say that $\lambda_i$ is a fundamental dominant weight $\endgroup$ – Zahi000 Jan 11 '18 at 16:08
  • $\begingroup$ Well I just discovered that there's a mistake in the book: the representation doesn't have to be necessarily faithful. Anyway the problem remains... I thought that the smallest dimension has to be 1, because the dominant weight is actually a highest weight of some representation and if one of them is $\pi$ then with weyl's formula we have $dim\pi=1$... $\endgroup$ – Zahi000 Jan 12 '18 at 14:48
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    $\begingroup$ The exercise is for simple Lie algebras, I think. So the kernel of the representation would be trivial, being a proper ideal, i.e., we have faithfulness. $\endgroup$ – Dietrich Burde Jan 13 '18 at 14:46
  • $\begingroup$ I'm sorry again, I'm new here! L is a semisimple Lie algebra (not a simple one) :) $\endgroup$ – Zahi000 Jan 14 '18 at 17:17

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