# Flag

of type $\nu$ in an $n$ - dimensional vector space $\nu$
A collection of linear subspaces $V$ of $V _{1} \dots V _{k}$ of corresponding dimensions $V$ , such that $n _{1} \dots n _{k}$ ( here $V _{1} \subset \dots \subset V _{k}$ , $\nu = (n _{1} \dots n _{k} )$ ; $1 \leq k \leq n - 1 ;\quad 0 < n _{1} < \dots < n _{k} < n$ ). A flag of type $\nu _{0} = (1 \dots n - 1 )$ 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 $\nu$ in $V$ is a homogeneous space of the general linear group $\mathop{\rm GL}\nolimits (V)$ . The unimodular group $\mathop{\rm SL}\nolimits (V)$ also acts transitively on $F _ \nu (V)$ . Here the stationary subgroup $H _{F}$ of $F$ in $\mathop{\rm GL}\nolimits (V)$ ( and also in $\mathop{\rm SL}\nolimits (V)$ ) is a parabolic subgroup of $\mathop{\rm GL}\nolimits (V)$ ( respectively, of $\mathop{\rm SL}\nolimits (V)$ ). If $F$ is a complete flag in $V$ , defined by subspaces $V _{1} \subset \dots \subset V _ {n - 1}$ , then $H _{F}$ is a complete triangular subgroup of $\mathop{\rm GL}\nolimits (V)$ ( respectively, of $\mathop{\rm SL}\nolimits (V)$ ) relative to a basis $e _{1} \dots e _{n}$ of $V$ such that $e _{i} \in V _{i}$ , $i = 1 \dots n$ . 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 $F _ {(n _{1} )} (V)$ is the Grassmann manifold $G _ {n, n _{1}} = \mathop{\rm Gr}\nolimits _ {n _{1}} (V)$ . 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 ). 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 , and also Bruhat decomposition).