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Consider the Grassmann variety $\mathbb{G}(k,n)$ and its Chow ring $A$. It is known that the classes of Schubert cycles form a $\mathbb{Z}$ basis of $A$. Is it known which of these Schubert cycles can be realized as a Chern class of a vector bundle? I know that this is true for the special Schubert classes which generate $A$ as a ring but not as an abelian group.

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