string $\sigma(uv)$. This is constructed as follows: we delete from $\sigma = \sigma_1 \ldots \sigma_n$ every $\sigma_l \notin \{u,v\}$. What remains will be a string consisting only of ‘$u$’s and ‘$v$’s. To convert this into a $01$-string, we change every ‘$u$’ to a [[‘$0$’]{} ]{}, and every ‘$v$’ to a [[‘$1$’]{} ]{}.

If $\sigma=01312230132$, then $\sigma(13)$ is produced as follows: $$\sigma = 0\mathbf{131}22\mathbf{3}0\mathbf{13}2
\mapsto \mathbf{131313} \mapsto 010101 = \sigma(13).$$

Each $\sigma(uv)$ corresponds to a Schubert cell on a different Grassmannian. In section \[sec:twostep\] we investigate the significance of these $\sigma(uv)$ on a two-step flag manifold.

In a completely different spirit, we define strings $\sigma[j] = \sigma[j]_1 \ldots \sigma[j]_n$, for $1 \leq j \leq k$, defined as follows. $$\sigma[j]_l = 
\begin{cases}
0,& \text{if $\sigma_l \leq k-j$} \\
1,& \text{if $\sigma_l > k-j$.}
\end{cases}$$

These have a very natural geometric significance. The $k$-step flag variety has $k$ different projections onto Grassmannians $Gr(d_j, {\mathbb{C}}^n)$. The image of the Schubert cell $\Omega_\sigma$ is projected onto $Gr(d_j, {\mathbb{C}}^n)$ is the Schubert cell $\Omega_{\sigma[j]}$.

If $\sigma=2103210$, then $$\begin{aligned}
\sigma[1]&=0001000 \\
\sigma[2]&=1001100 \\
\sigma[3]&=1101110\end{aligned}$$

### Notation for Grassmannians

When $P$ is a maximal parabolic, $G/P \cong Gr(r,n)$ is a Grassmannian, and we can equivalently index the Schubert 