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Let $X=Gr(r,V), Y=Gr(r+1,W)$ where $V,W$ are complex vector spaces with $\dim V > r$ and $\dim W > \dim V$. Let $\phi:X\rightarrow Y$ be some embedding of varieties. This induces a morphism on cohomology $\phi^{*}:H^{*}(Y)\rightarrow H^{*}(X)$. It is well known that the classes in $H^{*}(X)$ (respectively $H^{*}(Y)$) given by Young diagrams form a $\mathbb{Z}$-basis for $H^{*}(X)$ (respectively $H^{*}(Y)$).

Moreover $\phi^{*}(\sigma_\lambda)=\displaystyle\sum_{\lambda'}c_\lambda^{\lambda'}\sigma_{\lambda'}$ (with the obvious notation) for some nonnegative integers $c_\lambda^{\lambda'}$. I was wondering if there is a criterion for the non-vanishing of these coefficients. I know that for full-flag varieties this is not known but I was hoping that at least for Grassmanians the situation gets better.

I am quite new to Schubert calculus so any reference is more than welcomed.

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This is a weird case where I think the question actually asked is interesting, and that if I try to answer it I won't answer the question intended, which is less interesting. Here goes...

The Schubert basis in cohomology of any $G/P$ is its own dual basis (under the Poincar\'e pairing), albeit reordered. Hence the coefficients you ask about are determined by $\phi_*([X]) \in H_*(Y)$. This is a combination of Schubert classes on $Y$ of the right dimension. So since you don't say what $\phi$ is, the question becomes, which combinations of Schubert classes describe a cycle that deforms to a irreducible subvariety of $Y$ isomorphic to $X$.

There's been a lot of work (by Bryant and others) on deformations of cycles in the Grassmannian; I don't know whether any of it is about the deformations being isomorphic to smaller Grassmannians.

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I agree with Allen that (a) the question is ambiguously phrased, and (b) the most interesting interpretation, likely, is not the one intended by the OP. Having said that, my own interpretation is that the OP wants to know how to compute the pullback of the Schubert classes for a morphism between Grassmannians. The short answer is that Giambelli allows to express all Schubert classes as polynomials in the special Schubert classes. So the OP only needs to compute the pullbacks of the special Schubert classes.

Of course that raises the question, what are the possible values for the pullbacks of the special Schubert classes, or, equivalently, what are the K-theory classes of globally generated, locally free sheaves of rank $r+1$. By analogy with Hartshorne's conjecture, I expect the K-theory class is either a sum of $r+1$ (globally generated) invertible sheaves or else the sum of one (globally generated) invertible sheaf and a (globally generated) twist of the universal rank $r$ locally free sheaf by an invertible sheaf.

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If your $\phi$ is torus-equivariant, you can use equivariant Schubert calculus and Bott localization theorem. Often they make things simpler, because everyting reduces to torus-invariant subvarieties.

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