The $d=r$ inequality simply amounts to saying that $$\sum_{j=1}^s {\mathop{\mathrm{codim}}}\Omega_{\lambda^i} \leq \dim Gr(r,n),$$ i.e. this covers the case where the product vanishes for dimensional reasons. If we assume that $S^{\lambda^1} \cdots S^{\lambda^s}$ is of top degree then this inequality is always satisfied.

Schubert calculus in the tangent space
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General statements for $G/P$
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Let $F^1, \ldots, F^s$ be flags on ${\mathbb{C}}^n$. To determine whether a product $S^{\sigma^1} \cdots S^{\sigma^s} \in H^*(Gr(r,n))$ vanishes, it is sufficient to consider whether the Schubert varieties $\bar{\Omega}^{F^i}_{\sigma}$ have a point of intersection when the flags $F^i$ are sufficiently generic. This is due to the Kleiman-Bertini theorem [@Kl], which tells us that if $F^i$ are generic, these Schubert varieties will intersect transversely (in positively oriented points). Moreover, this point of intersection can be assumed to be inside the open cell $\Omega^{F^i}_{\sigma}$

Thus we can take the following approach. Choose the flags $F^i$ such that $V \in G/P$ is an intersection point of the Schubert varieties $\Omega^{F^i}_{\sigma}$. Subject to this restriction, the flags $F^i$ should be as generic as possible. We call generic flags with this restriction [**almost generic**]{} for $V$. The question then becomes whether or not the Schubert varieties with respect to almost generic 