I have to proof the fact that $H^*(\mathbb{G}(k,n))$ is, as an abelian group, free with basis the partitions that index the Schubert varieties, but I'm having trouble doing it myself.
Denote $\sigma_{\lambda}=[X_\lambda]$ the Schubert class associated to a partition $\lambda \subset m\times n$. For all partitions $\lambda$ contained in an $m\times n$ rectangle, the Schubert class $\sigma_{\lambda}$ is an element of $H^{2|\lambda|}(\mathbb{G}(m,m+n))$, and we have the decomposition
$H^*(\mathbb{G}(m,m+n))=\bigoplus_{\lambda \subset m\times n} \mathbb{Z}\sigma_\lambda$
Thanks in advance
Edit: The question linked does not have an answer and the notes contain a dead link. Moreover the question asks for an example while I need a more general proof or at least an outline for it.
Edit 2: How can I use the fact that $\Omega_\lambda \simeq \mathbb{C}^{mn-|\lambda|}$ and $\mathbb{G}(m,n)$ is the disjoint union of Schubert cells?
