(unless otherwise specified) that underlying flags $F^i$ are almost generic. If the $F^i$ are almost generic, we say that $Z_{\sigma^1}, \ldots, Z_{\sigma^s}$ are in [**general position**]{}. This is, of course, equivalent to the statement that $Z_{\sigma^i}$ is a *generic* $P$-translate of $\hat{Z}_{\sigma^i}$.

To calculate a particular $\hat{Z}_{\sigma}$, we use the following fact.

\[prop:calcZ\] $\hat{Z}_{\sigma}$ is the image of ${\mathfrak{b}}(\hat{F}(\sigma))$ under ${\mathfrak{g}}\to {\mathfrak{g}}/{\mathfrak{p}}$.

As $\Omega^{\hat{F}(\sigma)}_{\sigma}$ is a $B(\hat{F}(\sigma))$ orbit, the tangent space is generated by ${\mathfrak{b}}(\hat{F}(\sigma))$.

Grassmannians {#sec:grassmannian}
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### The tangent space to a Grassmannian Schubert variety

To distinguish the Grassmannian as a special case, we use the notation $\hat{X}_{\lambda}$ for $\hat{Z}_\sigma$, and $X_\lambda$ for the generic translate $X_\sigma$, where $\lambda$ is the partition corresponding to the $01$-string $\sigma$. We will take $V^0$ to be the coordinate subspace $\langle x_{n-r+1}, \ldots, x_n \rangle \in Gr(r,n)$.

We will now identify the subspace $\hat{X}_\lambda$. Put $V = V^0$, and $Q = {\mathbb{C}}^n/V$. Now ${\mathfrak{g}}/{\mathfrak{p}}$ can be naturally identified with ${\mathrm{Hom}}(V, Q)$, so we can view $\hat{X}_\lambda$ as a set of homomorphisms $\phi:V \to Q$. Both $V$ and $Q$ inherit full flags from the standard flag. $V$ inherits the flag $${F^{\mathrm{std}}}_V = {\{0\}}\subsetneq \langle x_{n-r+1} \rangle
\subsetneq \langle x_{n-r+1}, x_{n-r+2} \rangle \subsetneq \cdots
