2 problems

Weight-space and dimension conjecture. The weight spaces of simple $A$-modules are one dimensional. Consequently, the dimension of a simple module is the multinomial coefficient

Let $W$ be a Weyl group, let $R^+$ be its positive roots ordered by $\alpha\prec\beta$ when $\beta-\alpha$ is a positive sum of simple roots, and let $J(R^+,\prec)$ be the set of u…