I have a question regarding the highest weight of $\Lambda^k V$. Suppose I have an irreducible $\mathfrak{g}$-module, $V$, where $\mathfrak{g}$ is a Lie Algebra with highest weight $[a_1, \cdots, a_n]$. I would like to know what the highest weight of $\Lambda^k V$ will be in terms of the highest weight of $V$ and what is the intuition and reasoning for it.
For $S^kV$, it is clear to me that $[ka_1, \cdots, ka_n]$ is the highest weight since if $v$ is the highest weight vector of $V$ then the highest weight vector of $S^kV$ will be $v^{\otimes k}$ and so the weight will be multiplied by $k$. Any insight will be greatly appreciated!