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I learned that any characteristic class of rank-$k$ complex vector bundles on paracompact spaces is determined bijectively by a cohomology class in $H^*(Gr_k^\infty(\mathbb C))$, the cohomology ring of k-th infinite Grassmannian.

So the $n$-th Chern classes should be able to defined just by an element of $H^{2n}(Gr_k^\infty(\mathbb C))$, which is a linear combination of certain Schubert cycles (or Young diagrams, equivalently). Is there a direct combinatorial way to define what the Chern classes are?

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