# 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?