# Every Schubert cycle a Chern class?

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.