Highest weight of standard exterior power representation of $\mathfrak sl(n,\mathbb C)$

Question

Consider the standard representation of $\mathfrak sl(n,\mathbb C)$ on $\mathbb{C}^n$. Let $1 \leq i < n$. Consider the quotient representation of $\mathfrak sl(n,\mathbb C)$ on the exterior power $\bigwedge^i(\mathbb{C}^n)$. Consider the Cartan subalgebra $\mathfrak h$ of $\mathfrak sl(n,\mathbb C)$ consisting of diagonal, traceless matrices. Denote by $L_j$ the linear form $\mathfrak h \rightarrow \mathbb{C}$ that picks out the $j$-th diagonal entry. Fulton and Harris claim that $\bigwedge^i(\mathbb{C}^n)$ has heighest weight $L_1 + ...+ L_i$. I am struggling to verify that claim. I am even unable to show that $L_1 + ...+ L_i$ is a weight. Any suggestions?

Answer 1

Write $e_j$ for the column vector in $\mathbf{C}^n$ with a $1$ in position $j$ and zeros elsewhere. The action of a diagonal matrix $x$ on $e_j$ is then given by $$x \cdot e_j=L_j(x) e_j.$$ Now let $J=\{1 \leq j_1<j_2<\cdots<j_k \leq n\} \subseteq \{1,2,\dots,n\}$ be a subset of size $k$. The action of $x$ on the exterior product of the vectors $e_j$ for $j \in J$ is given by $$x \cdot e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_k}=\sum_{\ell=1}^k e_{j_1} \wedge \cdots \wedge x \cdot e_{j_\ell} \wedge \cdots \wedge e_{j_k}=\left(\sum_{\ell=1}^k L_\ell(x) \right) e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_k}.$$

This shows that the weights of the exterior product $\Lambda^k(\mathbf{C}^n)$ are precisely the sums $$L_J:=\sum_{j \in J} L_j$$ where $J$ ranges over subsets of $\{1,2,\dots,n\}$ of size $k$.