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flag variety

For $G$ a reductive 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 $gB\mapsto gBg^{-1}$.

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

Bruhat decomposition/order

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

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 $G = \operatorname{GL}_n(k)$ and $P$ is the stabilizer of a fixed line $L \subseteq {\mathbf{A}}^n_{/ {k}} $ then $G/P\cong {\mathbf{P}}^{n-1}$ and the Bruhat decomposition is the well-known affine paving of projective space.

Flag bundle

Used in the splitting principle for Chern classes: attachments/Pasted%20image%2020220403212049.png

Used to define Chern roots: attachments/Pasted%20image%2020220403212153.png

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affine paving in Unsorted/flag variety
equivariant K theory

flag variety
dominant weight

There is a special weight \begin{align*} \rho \coloneqq\varpi_{1}+\cdots+\varpi_{\ell} \in \Lambda^{+} = {1\over 2}\sum_{\alpha\in \Phi^+} \alpha ,\end{align*} i.e. the sum of fundamental weights or the half-sum of positive roots. It satisfies \begin{align*} \left\langle\rho, \alpha^{\vee}\right\rangle=1 \qquad\text{ and }\qquad s_{\alpha} \rho=\rho-\alpha \qquad \forall \alpha\in \Delta .\end{align*} It is the smallest regular dominant weight fixed by no nontrivial element of $W$, and the associated line bundle on the flag variety $G/B$ is ample, and is in fact a square root of the canonical bundle.