flag variety

Let $k$ be a field, and let $V$ be a vector space over $k$ of dimension $n$ and choose an increasing sequence $\mathbf{i}=(i_1,\ldots,i_m)$ , with $1\leq i_1<\cdots <i_m\leq n$ . Then the (partial) flag variety $\mc{F}\ell(V,\mathbf{i})$ associated to this data is the set of all flags $\{0\}\leq V_1\subset\cdots\subset V_{n}$ with $\dim V_j=i_j$ . This has a natural embedding into the product of Grassmannians $G(V,i_1)\times\cdots G(V,i_m)$ , and its image here is closed, making $\mc{F}\ell(V,\mathbf{i})$ into a projective variety over $k$ . If $k=\C$ these are often called flag manifolds.

The group $\mathrm{Sl}(V)$ acts transtively on $\mc{F}\ell(V,\mathbf{i})$ , and the stabilizer of a point is a parabolic subgroup. Thus, as a homogeneous space, $\mc{F}\ell(V,\mathbf{i})\cong\mathrm{Sl}(V)/P$ where $P$ is a parabolic subgroup of $\mathrm{Sl}(V)$ . In particular, the complete flag variety is isomorphic to $\mathrm{Sl}(V)/B$ , where $B$ is the Borel subgroup.