Vector bundles

Flags

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 11 May 2007

A flag is a sequence of subspaces

0 \subseteq V_{1} \subseteq V_{2} \subseteq \dots \subseteq V_{n} = ℂ^{n} with \dim V_{i} = i .

Our favourite flag is

p = (0 \subseteq ⟨ e_{1} ⟩ \subseteq \dots \subseteq ⟨ e_{1}, \dots, e_{n} ⟩) where e_{i} = (0, \dots, 0, \overset{i th}{1}, 0, \dots, 0)^{t} .

The general linear group $G = {GL}_{n} (ℂ)$ acts on $ℂ^{n}$ and on flags and

B = {(\begin{matrix} * & * \\ ⋱ \\ 0 & * \end{matrix})} is the stabilizer of p

so that

\begin{matrix} G / B & ⟷ & {flags} \\ g B & \mapsto & g p \end{matrix}

and the flag variety

G / B

may be viewed either as a set of cosets or as a set of flags.

The Grassmanian of $k$ -planes in $ℂ^{n}$ is

{Gr}_{k, n} = {(0 \subseteq V_{k} \subseteq ℂ^{n}) ∣ V_{k} is a subspace, \dim V_{k} = k} .

The general linear group

{GL}_{n} (ℂ)

acts on

{Gr}_{k, n}

and

P = {(\begin{matrix} * & * \\ 0 & * \end{matrix})} is the stabilizer of p = (0 \subseteq ⟨ e_{1}, \dots, e_{k} ⟩ \subseteq ℂ^{n}),

our favourite

k

-plane. Hence

\begin{matrix} G / P & ⟷ & {Gr}_{k, n} \\ g P & \mapsto & g p \end{matrix}

and the Grassmanian can be viewed either as the set of cosets in

G / P

or as the set of

k

-dimensional subspaces of

ℂ^{n}

Projective space $ℙ^{n - 1}$ is the space of lines in $ℂ^{n}$ ,

$ℙ^{n - 1} = {Gr}_{1, n} = {0 \subseteq ⟨ v ⟩ \subseteq ℂ^{n} ∣ v \in ℂ^{n}, v \neq 0} = {[v] ∣ v \in ℂ^{n}, v \neq 0},$

where $[v] = ⟨ v ⟩ = span {v}$ , for $v \in ℂ^{n} - {0}$ .

The general linear group

{GL}_{n} (ℂ)

acts on

ℙ^{n - 1}

and

P = {(\begin{matrix} * & * & \dots & * \\ 0 & * & * \end{matrix})} is the stabilizer of p = (0 \subseteq ⟨ e_{1}, \dots, e_{k} ⟩ \subseteq ℂ^{n}),

our favourite

k

-plane. Hence

\begin{matrix} G / P & ⟷ & {Gr}_{k, n} \\ g P & \mapsto & g p \end{matrix}