Flag Manifold

For any Sequence of Integers $0 < n_1 < \ldots < n_k$, there is a flag manifold of type ($n_1$, ..., $n_k$) which is the collection of ordered pairs of vector Subspaces of $\Bbb{R}^{n_k}$ ($V_1$, ..., $V_k$) with $\dim(V_i)=n_i$ and $V_i$ a Subspace of $V_{i+1}$. There are also Complex flag manifolds with Complex subspaces of $\Bbb{C}^{n_k}$ instead of Real Subspaces of a Real $n_k$-space. These flag manifolds admit the structure of Manifolds in a natural way and are used in the theory of Lie Groups.

See also Grassmann Manifold

References

Lu, J.-H. and Weinstein, A. ``Poisson Lie Groups, Dressing Transformations, and the Bruhat Decomposition.'' J. Diff. Geom. 31, 501-526, 1990.