# 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. Then the (partial) flag variety $\mathcal{F}\ell(V,\mathbf{i})$ 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 $\mathcal{F}\ell(V,\mathbf{i})$ into a projective variety over $k$. If $k=\mathbb{C}$ these are often called flag manifolds.

The group $\mathrm{Sl}(V)$ acts transtively on $\mathcal{F}\ell(V,\mathbf{i})$, and the stabilizer of a point is a parabolic subgroup. Thus, as a homogeneous space, $\mathcal{F}\ell(V,\mathbf{i})\cong\mathrm{Sl}(V)/P$ 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.

Title flag variety FlagVariety 2013-03-22 13:27:56 2013-03-22 13:27:56 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 14M15 flag manifold complete flag variety partial flag variety