flags intersect transversely. If they intersect transversely, then Schubert varieties with respect to fully generic flags must have a point of intersection; however if they do not, then the point of intersection is artificial, and fully generic flags will not give any point of intersection.

More importantly, this calculation can be done on the level of tangent spaces. We have the following basic result which appears in $\cite{B, P1}$.

\[lem:tangent1\] $S^{\sigma^1} \cdots S^{\sigma^s} = 0 \in H^*(G/P)$ if and only if the intersection of subspaces $$\bigcap_{i=1}^s T_V \Omega^{F^i}_{\sigma^i} \subset T_V G/P$$ is non-transverse for $F^i$ almost generic.

Our $V$ can be any point of $G/P$, so let us take $V=V^0$. Then $P$ acts transitively on the flags $F^i$ such that $V^0 \in \Omega^{F^i}_{\sigma^i}$. Thus to calculate $T_{V^0} \Omega^{F^i}_{\sigma^i}$, for generic $F^i$, it suffices to compute it for a special $F^i$ and consider the action of a generic element of $P$ on the tangent space $T_{V^0} G/P = {\mathfrak{g}}/{\mathfrak{p}}$.

There is always at least one coordinate flag $\hat{F}(\sigma^i)$ such that $V^0 \in \Omega^{\hat{F}(\sigma^i)}_{\sigma^i}$; we will use one of these. The standard flag on ${\mathbb{C}}^n$ is $${F^{\mathrm{std}}}= {\{0\}}\subsetneq \langle x_1 \rangle
\subsetneq \langle x_1, x_2 \rangle \subsetneq \cdots 
\subsetneq \langle x_1, x_2, 