# branching schubert calculus

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.

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 $X$.
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.
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.