Theory of Representation – Semi-Simple Lie Algebra Modules with Dimensional Weight Spaces at $ 1

Given a semi-simple complex Lie algebra $ frak {g} $ of rank $ r $, with Chevally generators $ E_i, F_i, K_i $. Let $ V $ to be a finite dimensional representation of $ frak {g} $ such as each weight space of $ V $ is $ 1 $-dimensional. Let $ (i_1, dots, i_k) $ to be an ordered set of elements of $ {1, dots, r } $ (allowing rehearsals), and let $ {j_1, points, j_k } $ to be a permutation of $ {1, dots, r } $. For $ v $ a higher weight of $ v $, the elements
$$
F_ {i_1} circ F_ {i_2} cdots circ F_ {i_k} (v), ~~ text {and} ~~~ circ F_ {j_1} circ F_ {j_2} cdots F_ {j_k} (v)
$$

must have the same weight. Thus, by our hypothesis, they must differ by a scalar multiple. Will this scalar multiple always be an integer?