flag variety

Let $k$ be a field, and let $V$ be a vector space over $k$ of dimension $n$ andchoose an increasing sequence $𝐢=(i_1,\mathrm{\dots },i_m)$, with. Then the (partial) flag variety$ℱ\mathrm{\ell }(V,𝐢)$ associated to this data is the set of allflags $\{0\}\le V_1\subset \mathrm{\cdots }\subset V_n$ with $dim{V}_j=i_j$. This has anatural embedding into the product of Grassmannians$G(V,i_1)\times \mathrm{\cdots }G(V,i_m)$, and its image here is closed, making$ℱ\mathrm{\ell }(V,𝐢)$ into a projective variety over $k$. If $k=ℂ$these are often called flag manifolds.

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