Let $Gr=Gr(m, V)$ be a Grassmannian of $m$-dimensional vector subspaces in the $n$-dimensional vector space $V$. There is a Plücker embedding $p_1: Gr \hookrightarrow \mathbb P(\Lambda^m V)$ mapping an $m$-dimensional vector subspace $U \subset V$ to the point $p_1(U)=\mathbb P(\Lambda^m U)$. It gives a series of line bundles $p_1^* \mathcal O(k)$. One can prove that they are the only line bundles on $Gr$, thus $Pic(Gr)=\mathbb Z$.
But there is another way to see $Gr$: as $Gr(n-m, V^*)$, replacing $U \subset V$ by $\ker(V^* \to U^*) \subset V^*)$. It gives another Plücker embedding $p_2: Gr \hookrightarrow \mathbb P(\Lambda^{n-m} V^*)$ and another series $p_2^* \mathcal O(l)$ of line bundles. But Picard group is the same.
So what $p_2^* \mathcal O(l)$ corresponds to $p_1^* \mathcal O(k)$?
$\ell = k$.
The two projective spaces are also isomorphic (given a choice of volume form on $V$) and the diagram commutes.