of type in an -dimensional vector space
A collection of linear subspaces of of corresponding dimensions , such that (here , ; ). A flag of type 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 , that is, the set of all flags of type in is a homogeneous space of the general linear group . The unimodular group also acts transitively on . Here the stationary subgroup of in (and also in ) is a parabolic subgroup of (respectively, of ). If is a complete flag in , defined by subspaces , then is a complete triangular subgroup of (respectively, of ) relative to a basis of such that , . In general, quotient spaces of linear algebraic groups by parabolic subgroups are sometimes called flag varieties. For , a flag of type is simply an -dimensional linear subspace of and is the Grassmann manifold . In particular, is the projective space associated with the vector space . Every flag variety can be canonically equipped with the structure of a projective algebraic variety (see ). If is a real or complex vector space, then all the varieties are compact. Cellular decompositions and cohomology rings of the are known (see , and also Bruhat decomposition).
For references see Flag structure.
Flag. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flag&oldid=19141