I want to show that $\bigwedge^i \mathbb{C}^{n+1}$ is a simple representation for $\mathfrak{sl}(n+1,\mathbb{C})$ for each $1\le i \le n+1$ but I'm already stuck at determining the weight vectors.
So let $\mathfrak{h}\subset\mathfrak{sl}(n+1,\mathbb{C})$ be the subalgebra of all diagonal matrices (with trace $0$).
How does an element $v_1\wedge\dots\wedge v_i$ look like if it satisfies $h(v_1\wedge\dots\wedge v_i) = \lambda(h)( v_1\wedge\dots\wedge v_i)$ for some $\lambda\in\mathfrak{h}^*$ and all $h\in \mathfrak{h}$? I cannot come up with any ideas on how to calculate such vectors...
Thank you for you help!