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).

For references see Flag structure.