$\Omega^F_\sigma$ is a $B(F)$-orbit, called the [**Schubert cell**]{} of $\sigma$ with respect to the flag $F$. When the flag $F$ is understood, or irrelevant, we shall also denote this $\Omega_\sigma$. The [**Schubert variety**]{} of $\sigma$, with respect to the flag $F$, is its closure $\bar{\Omega}^F_\sigma$, which represents the Schubert class $S^\sigma \in H^*(G/P)$.

If the $\sigma_l$ are weakly decreasing, then $\Omega_\sigma$ consists of a single point which is a subflag of $F$. If the $\sigma_i$ are weakly increasing, then $\Omega_\sigma$ is a dense open subvariety of $G/P$, and $S^\sigma = 1 \in H^*(G/P)$.

In general we can easily calculate the dimension of this orbit: $$\dim \Omega_\sigma = \#\{l<l'\ |\ \sigma_l < \sigma_{l'}\}.$$

If $V \in G/P$, then the [**Schubert position**]{} of $V$ with respect to the flag $F$ is the unique $\sigma$ such that $V \in \Omega^F_\sigma$. When we have multiple flags $F^1, \ldots, F^s$ on ${\mathbb{C}}^n$, the Schubert position of $V$ will be the $s$-tuple $(\sigma^1, \ldots \sigma^s)$, where $\sigma^i$ is the Schubert position of $V$ with respect to $F^i$.

### $01$-strings from $\sigma$

From $\sigma$ we construct [*$01$-strings*]{} in two different ways. First, for each pair $(u,v)$, with $0 \leq u < v \leq k$, we define a 