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.