# flag variety

Let $k$ be a field, and let $V$ be a vector space^{} over $k$ of dimension^{} $n$ and
choose an increasing sequence $\mathbf{i}=({i}_{1},\mathrm{\dots},{i}_{m})$, with
$$. Then the (partial) flag variety
$\mathcal{F}\mathrm{\ell}(V,\mathbf{i})$ associated to this data is the set of all
flags $\{0\}\le {V}_{1}\subset \mathrm{\cdots}\subset {V}_{n}$ with $dim{V}_{j}={i}_{j}$. This has a
natural embedding into the product of Grassmannians
$G(V,{i}_{1})\times \mathrm{\cdots}G(V,{i}_{m})$, and its image here is closed, making
$\mathcal{F}\mathrm{\ell}(V,\mathbf{i})$ into a projective variety over $k$. If $k=\u2102$
these are often called flag manifolds.

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