& 0 \\
0 & 0 & * & * \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 
\end{matrix}
\right)\end{aligned}$$ The intersection is codimension $18$; however the expected codimension is ${\mathop{\mathrm{codim}}}X_\lambda + {\mathop{\mathrm{codim}}}X_\mu = 13+6 =19$. Thus $S^\lambda S^\mu = 0 \in H^*(Gr(r,n))$.

### $X_\lambda$ as a set of homomorphisms {#sec:grassmannianhoms}

Let $\phi \in X_\lambda \subset {\mathrm{Hom}}(V,Q)$. This represents a direction in which we can perturb $V$ and remain in the tangent space to the Schubert variety. The kernel, $\ker \phi$ represents the maximal subspace of $V$ which is preserved by this perturbation. One of the key ideas which we take from Belkale’s proof is to examine $\ker \phi$ for a generic $\phi \in \bigcap_{i=1}^s X_{\lambda^i}$. For now, however, let us restrict our attention to a single $X_\lambda$.

Let us recall that the construction of $X_\lambda$ was actually a tangent space of a Schubert variety relative to some generic flag $F$ on ${\mathbb{C}}^n$. Thus $V$ carries an induced flag $F_V$, and $Q$ carries an induced flag $F_Q$. The action of $GL(V) \times GL(Q)$ on ${\mathrm{Hom}}(V,Q)$ corresponds to changing these induced flags. $F_V$ and $F_Q$ carry all the relevant information (for our purposes) 