some of the others merely contain old ideas which have been dressed up in a new context. We will try to indicate whenever possible which ideas and results have been borrowed.

Partial flag manifolds
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### Schubert varieties and Schubert positions

Let $G = GL(n)$, and let $P \subset G$ be a parabolic subgroup, the stabiliser subgroup of some $k$-step flag $$V^0 = {\{0\}}=V^0_0 \subsetneq V^0_1 \subsetneq \cdots 
\subsetneq V^0_k \subsetneq V^0_{k+1} = {\mathbb{C}}^n.$$ We will assume for later convenience that the $V^0_i$ are coordinate subspaces. Let $d_j = \dim V_j$. Then $G/P$ is a partial flag manifold (of type $0<d_1<\cdots<d_k<n$).

Let $$F = {\{0\}}=F_0 \subsetneq F_1 \subsetneq \cdots \subsetneq F_{n-1} 
\subsetneq F_n = {\mathbb{C}}^n.$$ be a full flag on $C^n$. Let $B(F) \subset G$ denote its stabiliser.

Let $\sigma = \sigma_1 \ldots \sigma_n$ be a string of length $n$, where $\sigma_l \in \{0, \ldots, k\}$, and the number of $j$’s in this string is $d_{j+1}-d_j$. It is a basic fact that $G/P$ has finitely many $B(F)$-orbits, and that these orbits are indexed by the set of possible $\sigma$ as follows.

Define $$\Omega^F_\sigma = \{V \in G/P\ \big|\ 
\sum_{j=1}^k \dim V_j \cap F_l - \dim V_j \cap F_{l-1} = \sigma_l\}.$$ 