Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights?
Are these the correct Dynkin labels for the weights of $V$: $$\left(1,0,0,0\right),\left(-1,1,0,0\right),\left(0,-1,1,0\right),\left(0,0,-1,1\right),\left(0,0,0,-1\right)$$
Pick a basis $v_1, \ldots, v_5$ for $V$, where $g \in SU(5)$ acts by matrices. Then the maximal torus $T$ of diagonal matrices in $SU(5)$ acts on $V$ as well, and this representation is completely reducible, and decomposes as $\mathbb{C}v_1 \oplus \cdots \oplus \mathbb{C}v_5$. The action of $(t_1, \ldots, t_5)$ on $v_1$ is $(t_1, \ldots, t_5) \cdot v_1 = t_1 v_1$, which we could call the character $(1, 0, 0, 0, 0)$. All the other characters are similar, for example the action on $v_3$ is by the character $(0, 0, 1, 0, 0)$. Note also that since $t_1 \cdots t_5 = 1 \in T$, these characters are only given up to shift by $(1, 1, 1, 1, 1)$.
Now, for $\bigwedge^2 V$, you can do exactly the same thing. It has a basis $v_1 \wedge v_2, v_1 \wedge v_3, \ldots, v_4 \wedge v_5$, and the action of $T$ on $v_1 \wedge v_2$ is by $(t_1, \ldots, t_5) \cdot (v_1 \wedge v_2) = t_1 v_1 \wedge t_2 v_2$, which is the character $(1, 1, 0, 0, 0)$. You can work out the others.