Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 1 / b / ii / Solution

Take the simple roots $L_1-L_2$ and $L_2-L_3$. A weight represented by $(k_1,k_2,k_3)$ is in the Closed dominant Weyl chamber exactly when $k_1\ge k_2\ge k_3$. Among nonnegative triples of total four, the possibilities are $(4,0,0),(3,1,0),(2,2,0),(2,1,1)$. Using $L_1+L_2+L_3=0$, their explicit weight space bases areEach listed tensor has the required summed weight, and the monomial basis from part (i) shows linear independence and exhausts all possible tensors of that weight. Thus these are all the requested bases.