# nLab flag variety

A flag in a vector space or a projective space is a ‘nest’ of subspaces, one of each dimension from $0$ to $n-1$. Given a field $k$, the space of all flags in an $n$-dimensional $k$-vector space has the structure of a projective variety over $k$, the flag variety. If $k$ 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 $\mathrm{SL}\left(n,k\right)/B$ where $B$ 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 $\mathrm{SL}\left(n,ℂ\right)/B\cong \mathrm{SU}\left(n\right)/T$ where $T$ is the subgroup of the diagonal $n×n$-matrices.

More generally, the generalized flag variety is the complex projective variety obtained as the coset space $G/T\cong {G}^{ℂ}/B$ where $G$ is a compact Lie group, $T$ its maximal torus, ${G}^{ℂ}$ the complexification of $G$, which is a complex semisimple group, and $B\subset {G}^{ℂ}$ 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