Flags and Grassmannians

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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Flags

A flag is a sequence of subspaces

$$0⊊V_1\subseteq V_2\subseteq \dots \subseteq V_n={ℂ}^n\text{with}\text{dim}\hspace{0.17em}{V}_i=i\text{.}$$

Our favourite flag is

$$p=(0\subseteq ⟨e_1⟩\subseteq \dots \subseteq ⟨e_1,\dots ,e_n⟩)$$

where $e_i={(0,\dots ,0,\stackrel{i\text{th}}{1},0,\dots ,0)}^t\text{.}$

$G={GL}_n\left(ℂ\right)$ acts on ${ℂ}^n$ and on flags and

$$B=\left\{\left(\begin{array}{cc}*& *\\ 0& *\end{array}\right)\right\}$$

is the stabilizer of $p$ so that

$$\begin{array}{ccc}\left\{\text{flags}\right\}& \iff & \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.\\ gp& ↤& gB\end{array}$$

We handle the flag variety with

Linear algebra Theorem 2

$$G=\underset{w\in W}{⨆}BwB\text{where}W=S_n$$

the group of permutation matrices.

Recall that

$$\begin{array}{c}{ℛ}^{+}=\{{\epsilon }_i-{\epsilon }_j\hspace{0.17em}|\hspace{0.17em}1\le i<j\le n\}\\ {𝔛}_{ij}={𝔛}_{{\epsilon }_i-{\epsilon }_j}=\left\{x_{ij}\left(c\right)\hspace{0.17em}\right|\hspace{0.17em}c\in ℂ\}\text{where}{x}_{ij}\left(c\right)=\begin{array}{ccccc}& & & & j\\ & (& 1& & & & & )\\ i& & \ddots & c\\ & & & \ddots \\ & 0& & & 1\end{array}\\ w{𝔛}_{ij}{w}^{-1}={𝔛}_{w\left(i\right)w\left(j\right)}\\ ℛ\left(w\right)=\{\alpha \in {ℛ}^{+}\hspace{0.17em}|\hspace{0.17em}w\alpha \notin {ℛ}^{+}\}=\left\{(i,j)\hspace{0.17em}\right|\hspace{0.17em}1\le i<j\le u,w\left(i\right)>w\left(j\right)\}\text{.}\\ \ell \left(w\right)=\text{Card}\left(ℛ\left(w\right)\right)\text{is the length of}\hspace{0.17em}w\text{.}\end{array}$$

The simple reflections $s_1,\dots ,s_{n-1}$ are the elements of length $1$ in $S_n$

$$s_i=\begin{array}{cccccccccc}& & & & i& i+1& & & & \\ (& 1& & & & & & & 0& )\\ & \ddots \\ & & 1\\ & & & 0& 1\\ & & & 1& 0\\ & & & & & 1\\ & & & & & & \ddots \\ 0& & & & & & & 1\end{array}\text{and}{x}_i\left(c\right)=x_{i\hspace{0.17em}i+1}\left(c\right)=\begin{array}{ccccc}& & & & i+1\\ & (& 1& & & & & )\\ i& & \ddots & c\\ & & & \ddots \\ & 0& & & 1\end{array}$$

If $w=s_{i_1}\dots s_{i_{\ell }}$ is a reduced word

$$\begin{array}{ccc}BwB& =& B{s}_{i}B\dots B{s}_{i_{\ell }}B\\ & =& \left\{x_{i_1}\left(c_1\right)s_{i_1}\dots x_{i_{\ell }}\left(c_{\ell }\right)s_{i_{\ell }}B\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{\ell }\in ℂ\}\text{.}\end{array}$$

 

$$\begin{array}{ccc}w& =& \end{array}$$

Grassmanians

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

$${\text{Gr}}_{k,n}=\left\{(0\subseteq V_k\subseteq {ℂ}^n)\hspace{0.17em}\right|\hspace{0.17em}{V}_k\hspace{0.17em}\text{is a subspace,}\hspace{0.17em}\text{dim}\hspace{0.17em}{V}_k=k\}\text{.}$$

Our favourite $k\text{-plane}$ is

$$p=(0\subseteq ⟨e_1,\dots ,e_k⟩\subseteq {ℂ}^n)\text{and}P=\left\{\begin{array}{ccccc}k& (& *& *& )\\ & 0& *\end{array}\right\}$$

is the stabilizer of $p$ in $G={GL}_n\left(ℂ\right)\text{.}$ So

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P$}\right.& ⟷& {\text{Gr}}_{k,n}\\ gP& ⟼& gp\end{array}$$

Then

$$P=\underset{w\in W_j}{⨆}BwB\text{where}{W}_J=⟨s_1,\dots ,s_{k-1},s_{k+1},\dots ,s_{n-1}⟩$$

so that

$$W_J=S_k\times S_{n-k}=\left\{\stackrel{\stackrel{k}{⏞}}{\begin{array}{c} \end{array}}\stackrel{\stackrel{n-k}{⏞}}{\begin{array}{c} \end{array}}\right\}\text{.}$$

Let $W^J$ be the set of minimal length coset representatives of cosets in $\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$W_J$}\right.\text{.}$ Then

$$W^J=\left\{\begin{array}{c} i1 i2 ik ⏟kkkkk ⏟kkn-kkk \end{array}\hspace{0.17em}\right|\hspace{0.17em}1\le i_1<\dots <i_k\le n\}$$

is in bijection with

$$\{\lambda \subseteq \left(k^{n-k}\right)\}$$

the set of partitions that fit inside a $k\times (n-k)$ box. A partition is a collection of boxes in a corner

$$\lambda =\begin{array}{c} \end{array}\lambda =(4,4,2,2,1,1)⊢14\text{.}$$

The bijection is

$$\begin{array}{ccc} 4 4 2 2 1 1 10 9 8 7 6 5 4 3 2 1 & ⟼& (2,3,5,6,9,10)\text{.}\end{array}$$ $$14 13 12 11 10 9 8 7 6 5 4 3 2 1$$

So we write

$$W^J=\left\{w_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \subseteq \left(k^{n-k}\right)\}\text{.}$$

Then

$$G=\underset{w\in W^J}{⨆}BwP=\underset{\lambda \subseteq \left(k^{n-k}\right)}{⨆}B{w}_{\lambda }P$$

since

$$G=\underset{w\in W}{⨆}BwB=\underset{w_{\lambda }\in W^J}{⨆}\underset{w\in W_J}{⨆}B{w}_{\lambda }wB=\underset{\underset{w\in W_J}{w_{\lambda }\in W^J}}{⨆}B{w}_{\lambda }B·BwB=\underset{w_{\lambda }\in W^J}{⨆}BwP\text{.}$$

Projective space

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

$$\begin{array}{ccc}{\mathbb{P}}^{n-1}& =& {\text{Gr}}_{1,n}=\{0\subseteq ⟨v⟩\subseteq {ℂ}^n\hspace{0.17em}|\hspace{0.17em}v\in {ℂ}^n,v\ne 0\}\\ & =& \left\{\left[v\right]\hspace{0.17em}\right|\hspace{0.17em}v\in {ℂ}^n,v\ne 0\}\end{array}$$

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

Our favourite point of ${\mathbb{P}}^{n-1}$ is

$$p=\left[(1,0,\dots ,0)\right]=⟨e_1⟩=(0\subseteq ⟨e_1⟩\subseteq {ℂ}^n)$$

which has stabilizer

$$P=\left(\begin{array}{c}*\\ & \multicolumn{2}{c}{*}\\ & & \phantom{*}\end{array}\right)\text{in}G={GL}_n\left(ℂ\right)$$

and

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P$}\right.& ⟷& {\mathbb{P}}^{n-1}\\ gP& ⟼& gp& =[g_{11},\dots ,g_{1n}]\text{if}g=\left(g_{ij}\right)\text{.}\end{array}$$

In this case

$$W_J=⟨s_2,\dots ,s_{n-1}⟩=S_{n-1}=\left\{\begin{array}{c} \end{array}\underset{\underset{k-1}{⏟}}{\begin{array}{c} \end{array}}\right\}$$

and

$$W_J=\left\{\begin{array}{c} i ⏟1 ⏟kkn-1kk \end{array}\hspace{0.17em}\right|\hspace{0.17em}1\le i\le n\}$$

so that

$$P=\underset{w\in W_J}{⨆}BwB\text{and}G=\underset{i=1}{\overset{n}{⨆}}B{w}_{i}P\text{.}$$

Specifically

$$w_i=\begin{array}{c} i \end{array}=s_{i-1}{s}_{i-2}\dots s_2{s}_1$$

and

$$\begin{array}{ccc}B{w}_{i}P& =& \left\{x_{i-1}\left(c_1\right)s_{i-1}{x}_{i-2}\left(c_2\right)s_{i-2}\dots x_1\left(c_{i-1}\right)s_{1}P\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{i-1}\in ℂ\}\\ & =& \left\{x_{i-1,i}\left(c_1\right)x_{i-2,i}\left(c_2\right)\dots x_{1i}\left(c_{i-1}\right)s_{i-1}\dots s_{i}P\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{i-1}\in ℂ\}\\ & =& \left\{\left(\begin{array}{ccccc}1& \ddots & \begin{array}{c}{c}_{i-1}\\ ⋮\\ c_1\end{array}& & 0\\ \\ & 1\\ & & & \ddots \\ 0& & & & 1\end{array}\right)s_{i-1}\dots s_{1}P\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{i-1}\in ℂ\}\text{.}\end{array}$$

Note that

$$\left(\begin{array}{ccccc}1& \ddots & \begin{array}{c}{c}_{i-1}\\ ⋮\\ c_1\end{array}& & 0\\ \\ & 1\\ & & & \ddots \\ 0& & & & 1\end{array}\right)s_{i-1}\dots s_{1}P=\left[\left(\begin{array}{ccccc}1& \ddots & \begin{array}{c}{c}_{i-1}\\ ⋮\\ c_1\end{array}& & 0\\ \\ & 1\\ & & & \ddots \\ 0& & & & 1\end{array}\right)\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)\right]=\left(\begin{array}{c}{c}_{i-1}\\ ⋮\\ c_1\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)=[c_{i-1},\dots ,c_1,1,0,\dots ,0]\text{.}$$

The group

$$T=\left\{\left(\begin{array}{ccc}{t}_1& & 0\\ & \ddots \\ 0& & t_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{t}_i\in {ℂ}^{\times }\}\text{acts on}\hspace{0.17em}{\mathbb{P}}^{n-1}\hspace{0.17em}\text{by}t[c_1,\dots ,c_n]=[t_1{c}_1,\dots ,t_n{c}_n]\text{.}$$

If $[c_1,\dots ,c_n]$ is a fixed point then for all $t_1,\dots ,t_n\in {ℂ}^{\times }$ $(t_1{c}_1,\dots ,t_n{c}_n)=(\gamma c_1,\dots ,\gamma c_n)$ for some $\gamma \in {ℂ}^{\times }\text{.}$

So all but one of the $c_i$ is $0,$ since $\gamma =t_i$ if $c_i\ne 0\text{.}$

So the $T\text{-fixed}$ points are

$$\underset{\phantom{\text{...}}{i}^{\text{th}}}{[0,\dots ,0,1,0,\dots ,0]}⟼s_{i-1}\dots s_{1}P\text{.}$$

Notes and References

This is a typed copy of handwritten notes by Arun Ram.

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