\ldots, x_{n-1} \rangle$$ The Weyl group $S_n$ acts transitively on the set of coordinate flags of ${\mathbb{C}}^n$, thus $\hat{F}^i = w_i \cdot {F^{\mathrm{std}}}$ for some $w_i \in S_n$. Of all possible choices for the $\hat{F}(\sigma^i)$, we will always choose $\hat{F}(\sigma^i) = w_i \cdot {F^{\mathrm{std}}}$ with $w_i$ minimal in the Bruhat order.

Denote the tangent space to the Schubert variety $\Omega^{\hat{F}(\sigma)}_{\sigma}$ at $V_0$ by $$\hat{Z}_\sigma = T_{V^0} \Omega^{\hat{F}(\sigma)}_{\sigma} 
\subset {\mathfrak{g}}/{\mathfrak{p}}.$$ Let $Z_\sigma$ denote a generic $P$-translate of $\hat{Z}_\sigma$. This notation will be convenient, as we will often need to consider intesections $\bigcap_{i=1}^s p_i \hat{Z}_{\sigma_i}$ where $p_i \in P$ are generic; we can now write this simply as $\bigcap_{i=1}^s Z_{\sigma_i}$. For the special cases of Grassmannians and two-step flag manifolds, we will use the letters $X$ or $Y$ respectively instead of $Z$.

Lemma \[lem:tangent1\] can be stated equivalently as follows:

\[lem:tangent2\] $S^{\sigma^1} \cdots S^{\sigma^s} = 0 \in H^*(G/P)$ if and only if the intersection of subspaces $$\bigcap_{i=1}^s Z_{\sigma^i} \subset {\mathfrak{g}}/{\mathfrak{p}}$$ is non-transverse.

Although we have eliminated the flags $F^i$ from this statement, we will sometimes wish to think of $Z_{\sigma^i}$ as being determined by generic flags, rather than as generic translates of $\hat{Z}_{\sigma^i}$.

Whenever we use the notation $Z_{\sigma^i}$, we tacitly assume 