---
tag: BB0TSMW93P
kind: concept
name: Flag manifold
msc: [14M15, "22E46", 53C30]
specializes: [85TWR6E75P]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_flag_variety
---

## Summary

A flag manifold is a quotient $G/P$ of a connected complex semisimple Lie group $G$ by a parabolic subgroup $P$.
The basic example is the space of complete flags $V_1 \subset V_2 \subset \cdots \subset V_{n-1}$ of subspaces of $\C^n$ with $\dim V_i = i$.
Projective spaces and Grassmannians are flag manifolds too.
Flag manifolds are compact complex manifolds on which $G$ acts, and they carry the geometry behind the Borel–Weil theorem and Schubert calculus.

## Explanations

### L1 human

A flag in $\C^3$ is a line through the origin together with a plane that contains it, like a flagpole and a flag.
The set of all such flags is a smooth, compact space of complex dimension $3$.
Invertible matrices move flags around, and any flag can be moved to any other.
So the space of flags is a group divided by the subgroup that fixes one flag, the upper triangular matrices.
Other flag manifolds keep only some of the subspaces, as the projective plane keeps only the line.