A flag in a vector space or a projective space is a ‘nest’ of subspaces, one of each dimension from to . Given a field , the space of all flags in an -dimensional -vector space has the structure of a projective variety over , the flag variety. If is the field of real or complex numbers, then it has a structure of a smooth manifold. It can be considered as the homogeneous space where is the subgroup of lower (or upper if you like) triangular matrices. By the Gram–Schmidt orthogonalization procedure, one shows the isomorphism as real manifolds where is the subgroup of the diagonal -matrices.
More generally, the generalized flag variety is the complex projective variety obtained as the coset space where is a compact Lie group, its maximal torus, the complexification of , which is a complex semisimple group, and is the Borel subgroup. It has a structure of a compact Kaehler manifold. It is a special case of the bigger family of coset spaces of semisimple groups modulo parabolics which includes for example grassmanian?s. There are quantum, noncommutative and infinite-dimensional generalizations. Flag varieties have rich combinatorial and geometric structure and play an important role in representation theory and integrable systems.
D. Monk, The geometry of flag manifolds, Proceedings of the London Mathematical Society 1959 s3-9(2):253–286; doi:10.1112/plms/s3-9.2.253
Generalized flag variety at wikipedia
M. Brion, Lectures on the geometry of flag varieties, pdf
A. Borel, Linear algebraic groups, Springer
See also: building