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 $\mathcal{F}\ell(V,\mathbf{i})$ 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
Canonical name	FlagVariety
Date of creation	2013-03-22 13:27:56
Last modified on	2013-03-22 13:27:56
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	6
Author	bwebste (988)
Entry type	Definition
Classification	msc 14M15
Synonym	flag manifold
Defines	complete flag variety
Defines	partial flag variety