---
created: 2022-04-03T21:20
updated: 2023-03-31T16:01
title: "flag variety"
aliases: [flag variety, Bruhat decomposition, Bruhat order, flag varieties]
---

---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Hodge structure](Unsorted/pure%20Hodge%20structure.md)
	- [Borel–Weil–Bott theorem](Borel–Weil–Bott%20theorem)

---

# flag variety

For $G$ a [reductive](Unsorted/Borel.md) algebraic group and $B\leq G$ a Borel, the **flag variety** of $G$ is the quotient variety $G/B$. 
This is a moduli of subgroups of $B$, where the bijection is $gB\mapsto gBg\inv$.


Example: for $G=\GL_n(k)$, $G/B \cong \mathrm{Flag}_n(k)$, the variety of full flags in $k^n$.

- For $V\in \mods{\CC}$, since $\GL(V)\actson \Frame(V)$ transitively (the set of bases), for any fixed $W\in \Gr_k(V)$ one has $$\Gr_k(W) = {G\over P} \da {\GL(V) \over \Stab_{\GL(V)} W}$$
	- Note $\T_W \Gr_k(W) = \Hom(W, V/W)$.
	- For any fixed flag $\vector d$, there is an embedding $\Gr_{\vector d}(V) \injects \prod \Gr_{d_i}(V)$ as a closed subspace, which is thus a submanifold.

## Bruhat decomposition/order

If you fix $w\in W \da N_G(T)/T$ and a lift $\tilde w\in N_G(T)$, there is a locally closed subvariety $BwB \da B\tilde w B \subseteq G$. Setting $X_w\da {BwB\over B}$, one obtains the **Bruhat decomposition** $G = \disjoint_{w\in W} BwB = \disjoint_{w\in W} X_w$.
Define the **Bruhat order** on $W$ by $w_1\leq w_2 \iff X_{w_1} \subseteq \cl^\Top_{G/B} X_{w_2}$.

This is the transitive closure of the partial order $w_1 \leq w_2 \iff \ell(w_1) \geq \ell(w_2)$. So $x\to y$ iff $\ell(x) \geq \ell(y)$, and the arrows point toward shorter length words.

If $G = \GL_n(k)$ and $P$ is the stabilizer of a fixed line $L \subseteq \AA^n\slice k$ then $G/P\cong \PP^{n-1}$ and the Bruhat decomposition is the well-known [affine paving](affine%20paving) of projective space.

# Flag bundle

Used in the [splitting principle for Chern classes](Unsorted/Chern%20class.md#Splitting%20principle):
![](attachments/2023-03-31-9.png)
![](attachments/2023-03-31-10.png)
![](attachments/2023-03-31-11.png)

Used to define **Chern roots**:
![](attachments/2023-03-31-12.png)


# Partial flag varieties

![](2023-03-31-67.png)
![](2023-03-31-68.png)