Exercise 15, § 4

Determine the minuscule weights ($§ 1$, Exerc. 24) for each type of reduced, irreducible root system. (One finds the fundamental weights $\omega_1, \ldots, \omega_l$ for $A_l$, the weight $\omega_l$ for $B_l$, the weight $\omega_1$ for $C_l$, the weights $\omega_1, \omega_{l-1}, \omega_l$ for $D_l$, the weights $\omega_1$ and $\omega_6$ for $E_6$, the weight $\omega_7$ for $E_7$, and none for $E_8, F_4$ and $G_2$.)