---
tag: NHRWCQQKTH
kind: concept
name: Flag variety
msc: [14M15, 14M17]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_flag_variety
same_as:
  wikidata: Q938283
  wikipedia: Generalized flag variety
---

## Summary

The flag variety of a vector space $V$ parametrizes chains of subspaces $V_1 \subset V_2 \subset \cdots$ of prescribed dimensions.
It is a smooth projective [[homogeneous space|85TWR6E75P]] for $\mathrm{GL}(V)$; [[projective spaces|ZT9A6JK8QF]] and [[Grassmannians|CYPA882RB9]] are special cases.
In general, a flag variety is a quotient $G / P$ of a reductive group by a parabolic subgroup.
Flag varieties are where representation theory, combinatorics and geometry meet, through [[Schubert varieties|Z014Q8KPDD]] and the Borel–Weil theorem.

## Explanations

### L1 human

A flag in three-dimensional space is a line through the origin together with a plane through the origin that contains the line.
Think of a flagpole and a flag attached to it.
The set of all such flags is a smooth, compact space of dimension $3$, the full flag variety of $\C^3$.
In general a flag is a nested chain of subspaces, and the flag variety is the space of all of them.