Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Denote $\bigwedge^i\mathfrak{g}$ the $i$-th exterior power of $\mathfrak{g}$. Define: $$\mathrm{ch}(V)=\sum_{\lambda\in\mathfrak{h}^*}(\dim V_\lambda) e(\lambda),$$
where $V_\lambda$ is the weight space corresponding to weight $\lambda$.
I would like to know what is $\mathrm{ch}\left(\bigwedge^i\mathfrak{g}\right)$ for $0\le i\le n$? Is: $$\sum_{i=0}^{n}(-1)^{i}\mathrm{ch}\left(\bigwedge^i\mathfrak{g}\right),$$ invertible?