User:Maximilian Janisch/latexlist/Algebraic Groups/Flag

of type $2$ in an $12$-dimensional vector space $V$
A collection of linear subspaces $V _ { 1 } , \ldots , V _ { k }$ of $V$ of corresponding dimensions $n _ { 1 } , \ldots , n _ { k }$, such that $V _ { 1 } \subset \ldots \subset V _ { k }$ (here $\nu = ( n 1 , \ldots , n _ { k } )$, $1 \leq k \leq n - 1$; $0 < n _ { 1 } < \ldots < n _ { k } < n$). A flag of type $\nu _ { 0 } = ( 1 , \ldots , n - 1 )$ is called a complete flag or a full flag. Any two flags of the same type can be mapped to each other by some linear transformation of $V$, that is, the set $F _ { \nu } ( V )$ of all flags of type $2$ in $V$ is a homogeneous space of the general linear group $GL ( V )$. The unimodular group $SL ( V )$ also acts transitively on $F _ { \nu } ( V )$. Here the stationary subgroup $H _ { F }$ of $H ^ { \prime }$ in $GL ( V )$ (and also in $SL ( V )$) is a parabolic subgroup of $GL ( V )$ (respectively, of $SL ( V )$). If $H ^ { \prime }$ is a complete flag in $V$, defined by subspaces $V _ { 1 } \subset \ldots \subset V _ { n - 1 }$, then $H _ { F }$ is a complete triangular subgroup of $GL ( V )$ (respectively, of $SL ( V )$) relative to a basis $e _ { 1 } , \ldots , e _ { x }$ of $V$ such that $e _ { i } \in V$, $i = 1 , \dots , n$. In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For $k = 1$, a flag of type $( n _ { 1 } )$ is simply an $n 1$-dimensional linear subspace of $V$ and $F ( x _ { 1 } ) ( V )$ is the Grassmann manifold $G _ { n , n _ { 1 } } = Gr _ { n _ { 1 } } ( V )$. In particular, $F ( 1 ) ( V )$ is the projective space associated with the vector space $V$. Every flag variety $F _ { \nu } ( V )$ can be canonically equipped with the structure of a projective algebraic variety (see [1]). If $V$ is a real or complex vector space, then all the varieties $F _ { \nu } ( V )$ are compact. Cellular decompositions and cohomology rings of the $F _ { \nu } ( V )$ are known (see [3], and also Bruhat decomposition).