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Given a semisimple complex Lie algebra $\frak{g}$ of rank $r$, with Chevally generators $E_i,F_i,K_i$. Let $V$ be a finite dimensional representation of $\mathfrak{g}$ such that each weight space of $V$ is $1$-dimensional. Let $(i_1,\dots,i_k)$ be an ordered set of elements of $\{1,\dots,r\}$ (allowing repeats), and let $\{j_1,\dots,j_k\}$ be some permutation of $\{1,\dots,r\}$. For $v$ a highest weight of $V$, the elements $$ F_{i_1} F_{i_2} \cdots F_{i_k}(v),\quad \text { and } \quad F_{j_1} F_{j_2} \cdots F_{j_k}(v), $$ must have the same weight. Thus by our assumption they must differ by a scalar multiple. Will this scalar multiple always be an element of $\mathbb{Q}$?

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  • $\begingroup$ There is an obvious typo, where you mean to write "For $v$ a highest weight vector of $V$ ...." And in the display, one composition symbol should be suppressed. $\endgroup$ – Jim Humphreys Sep 11 at 2:00
  • $\begingroup$ thanks a lot, it has been fixed. $\endgroup$ – Pierre Dubois Sep 11 at 5:36
  • $\begingroup$ No, your edit consisted in adding another typo. I've fixed. $\endgroup$ – YCor Sep 11 at 6:07
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    $\begingroup$ By a rational is much weaker than by an integer up to inversion. Then it's certainly true, since the Lie algebra can be defined over $\mathbf{Q}$ with the given Cartan subalgebra as split Cartan subalgebra, and the representation is then split too. $\endgroup$ – YCor Sep 11 at 6:16
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    $\begingroup$ This is true with integer replaced by rational since all irreducible representations of a Chevalley LIe algebra over the rationals are defined over the rationals (JIm Humphreys book on semisimple lie algebras has a construction, I think). $\endgroup$ – Venkataramana Sep 11 at 6:19
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Let me flesh out the answer given in the comments. Any simple Lie algebra can be defined over the field of rational numbers. Then its irreducible finite dimensional representations can be constructed as quotients of Verma modules. That is $$ V_\lambda = \mathfrak{U}(\mathfrak{g}_\mathbb{Q}) \otimes_{ \mathfrak{U}(\mathfrak{b}_\mathbb{Q})} \mathbb{Q}_\lambda / \text{maximal submodule}, $$ where $\mathbb{Q}_\lambda$ is the one dimensional representation on which the split Cartan subalgebra of $\mathfrak{g}_\mathbb{Q}$ acts by character $\lambda.$

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