The Fano plane as a flag variety

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

Last update: 2 June 2013

$\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.$ and $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.$ and the Fano Plane

The vector space of column vectors of length $n$

$${𝔽}^n=\left\{\left(\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{c}_i\in 𝔽\}$$

has basis $e_1,e_2,\dots ,e_n$ where $e_i=\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)i\text{th.}$

Let $ℒ$ be the lattice of subspaces of ${𝔽}^n$ partially ordered by inclusion, and

$$\begin{array}{ccc}ℱ𝓁& =& \left\{\text{maximal chains in}\hspace{0.17em}ℒ\right\}\\ & =& \left\{(0\subseteq V_1\subseteq \dots \subseteq V_{n-1}\subseteq {𝔽}^n)\hspace{0.17em}\right|\hspace{0.17em}\text{dim}\hspace{0.17em}{V}_i=i\}\text{.}\end{array}$$

Our favourite flag is

$$0\subseteq E_1\subseteq \dots \subseteq E_{n-1}\subseteq {𝔽}^n$$

where

$$E_i=\left\{\left(\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_i\\ 0\\ ⋮\\ 0\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{c}_j\in 𝔽\}=\text{span}\{e_1,\dots ,e_i\}\text{.}$$

The automorphism group of the vector space ${𝔽}^n$ is

$$G=\text{Aut}\left({𝔽}^n\right)={GL}_n\left(𝔽\right)$$

and we write $g=\left(g_{ij}\right)$ where $g{e}_i=\sum _{j=1}^n{g}_{ji}{e}_j$ so that

$$g{e}_i=\left(\begin{array}{c}1\\ g_i\\ 1\end{array}\right)$$

is the $i\text{th}$ column of $g\text{.}$

The stabilizer of $E_i$ is

$$P_i={\text{Stab}}_G\left(E_i\right)=\begin{array}{cc}\phantom{AAA}\stackrel{\phantom{AAA}i\phantom{AAA}}{⏞}& \stackrel{\phantom{AA}n-i\phantom{AA}}{⏞}\phantom{AA}\\ \multicolumn{2}{c}{\left\{\left(\begin{array}{cc}\begin{array}{ccc}\phantom{*}& \phantom{*}& \phantom{*}\\ \phantom{*}& *& \phantom{*}\\ \phantom{*}& \phantom{*}& \phantom{*}\end{array}& \begin{array}{cc}\phantom{*}& \phantom{*}\\ \multicolumn{2}{c}{0}\\ \phantom{*}& \phantom{*}\end{array}\\ \begin{array}{ccc}\phantom{*}& 0& \phantom{*}\\ \phantom{*}& \phantom{*}\end{array}& *\end{array}\right)\right\}}\end{array}\subseteq {GL}_n\left(𝔽\right)$$

and the stabilizer of $0\subseteq E_1\subseteq E_2\subseteq \dots \subseteq E_{n-1}\subseteq {𝔽}^n$ is

$$B=P_1\cap P_2\cap \dots \cap P_{n-1}=\left\{\left(\begin{array}{ccccc}*& & & & *\\ & *\\ & & \ddots \\ & & & *\\ 0& & & & *\end{array}\right)\right\}\text{.}$$

The maps

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_i$}\right.& ⟶& \left\{\text{subspaces of dimension}\hspace{0.17em}i\hspace{0.17em}\text{in}\hspace{0.17em}{𝔽}^n\right\}\\ g{P}_i& ⟼& g{E}_i\end{array}$$

and

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$B$}\right.& ⟶& ℱ𝓁\\ gB& ⟼& (0\subseteq g{E}_1\subseteq g{E}_2\subseteq \dots \subseteq g{E}_{n-1}\subseteq {𝔽}^n)\end{array}$$

are bijections and

$$g{E}_i=\text{span}\{\left(\begin{array}{c}1\\ g_1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ g_2\\ 1\end{array}\right),\dots ,\left(\begin{array}{c}1\\ g_i\\ 1\end{array}\right),\}$$

is the span of the first $i$ columns of $g\text{.}$

Representations of cosets in $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_i$}\right.$

Let

$$x_i\left(c\right)=\begin{array}{ccccccc}& & & & & i& i+1\\ & (& 1& & & & & & & & )\\ & & \ddots \\ & & & 1\\ i& & & & 1& c\\ i+1& & & & 0& 1\\ & & & & & & 1\\ & & & & & & & \ddots \\ & & & & & & & & 1\end{array}{s}_i=\begin{array}{ccccccc}& & & & & i& i+1\\ & (& 1& & & & & & & & )\\ & & \ddots \\ & & & 1\\ i& & & & 0& 1\\ i+1& & & & 1& 0\\ & & & & & & 1\\ & & & & & & & \ddots \\ & & & & & & & & 1\end{array}$$

The Weyl group of $G={GL}_n\left(ℂ\right)$ is

$$W_0=S_n=⟨s_1,\dots ,s_{n-1}⟩=\{n\times n\hspace{0.17em}\text{permutation matrices}\},$$

the subgroup of ${GL}_n\left(ℂ\right)$ generated by $s_1,\dots ,s_{n-1}\text{.}$

If

$$W_i=W_0\cap P_i=⟨s_1,s_2,\dots ,s_{j-1},s_{j+1},s_{j+2},\dots ,s_{n-1}⟩$$

then coset representatives of the cosets in $\raisebox{1ex}{$W_0$}\!\left/ \!\raisebox{-1ex}{$W_i$}\right.$ are

$$W^i=\left\{\left(\begin{array}{ccccccccc}1& 2& \dots & i-1& i& i+1& i+2& \dots & n\\ {\sigma }_1& {\sigma }_2& \dots & {\sigma }_{i-1}& {\sigma }_i& {\tau }_1& {\tau }_2& \dots & {\tau }_{n-i}\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{c}{\sigma }_1<{\sigma }_2<\dots <{\sigma }_{i-1}<{\sigma }_i\\ {\tau }_1<{\tau }_2<\dots <{\tau }_{n-i}\end{array}\}$$

and

$$\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_i$}\right.=\underset{u\in W^i}{⨆}Bu{P}_i$$

where, if $u=s_{j_1}\dots s_{j_{\ell }}$ is a minimal length expression of $u$ as a product of the generators $s_1,\dots ,s_{n-1}$ of $W_0,$ then

$$Bu{P}_i=\left\{x_{j_1}\left(c_1\right)s_{j_1}\dots x_{j_{\ell }}\left(c_{\ell }\right)s_{j_{\ell }}{P}_i\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{\ell }\in 𝔽\}\text{.}$$

The cases $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.$ and $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.$

$$P_1=\left\{\left(\begin{array}{cc}*& \begin{array}{ccc}\phantom{*}& *& \phantom{*}\end{array}\\ \begin{array}{c}\phantom{*}\\ 0\\ \phantom{*}\end{array}& \begin{array}{ccc}\phantom{*}& \phantom{*}& \phantom{*}\\ \phantom{*}& *& \phantom{*}\\ \phantom{*}& \phantom{*}& \phantom{*}\end{array}\end{array}\right)\right\}\text{and}{P}_2=\left\{\left(\begin{array}{cc}\begin{array}{cc}*& *\\ *& *\end{array}& \begin{array}{ccc}\phantom{*}& *& \phantom{*}\end{array}\\ \begin{array}{c}\phantom{*}\\ 0\\ \phantom{*}\end{array}& \begin{array}{ccc}\phantom{*}& \phantom{*}& \phantom{*}\\ \phantom{*}& *& \phantom{*}\\ \phantom{*}& \phantom{*}& \phantom{*}\end{array}\end{array}\right)\right\}$$ $$\begin{array}{ccc}{W}_1& =& s_1\times s_{n-1}=⟨s_2,\dots ,s_{n-1}⟩\subseteq S_n\text{and}\\ W_2& =& s_2\times s_{n-2}=⟨s_1,s_3,s_4,\dots ,s_{n-1}⟩\subseteq S_n\text{.}\\ W^1& =& \left\{\begin{array}{c} 1 2 … j … n \end{array}\hspace{0.17em}\right|\hspace{0.17em}j\in \{1,2,\dots ,n\}\}\\ & =& \left\{s_{j-1}{s}_{j-2}\dots s_2{s}_1\hspace{0.17em}\right|\hspace{0.17em}j\in \{1,2,\dots ,n\}\}\\ W^2& =& \left\{\begin{array}{c} 1 … i … j … n \end{array}\hspace{0.17em}\right|\hspace{0.17em}i,j\in \{1,2,\dots ,n\},i<j\}\\ & =& \left\{s_{i-1}\dots s_2{s}_1{s}_{j-1}{s}_{j-2}\dots s_3{s}_2\hspace{0.17em}\right|\hspace{0.17em}i,j\in \{1,2,\dots ,n\},i<j\}\text{.}\end{array}$$

Then

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.& ⟶& \{V_1\subseteq {𝔽}^n\hspace{0.17em}|\hspace{0.17em}\text{dim}\hspace{0.17em}{V}_1=1\}\\ g{P}_1& ⟼& \text{span}\left\{\left(\begin{array}{c}1\\ g_1\\ 1\end{array}\right)\right\}\end{array}$$

and

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.& ⟶& \{V_2\subseteq {𝔽}^n\hspace{0.17em}|\hspace{0.17em}\text{dim}\hspace{0.17em}{V}_2=2\}\\ g{P}_2& ⟼& \text{span}\{\left(\begin{array}{c}1\\ g_1\\ 1\end{array}\right),\left(\begin{array}{c}1\\ g_2\\ 1\end{array}\right)\}\text{.}\end{array}$$

Next

$$\begin{array}{ccc}\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.& =& \underset{u\in W^1}{⨆}Bu{P}_1=\underset{j=1}{\overset{n}{⨆}}B{s}_{j-1}{s}_{j-2}\dots s_2{s}_1{P}_1\text{and}\\ \raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.& =& \underset{u\in W^2}{⨆}Bu{P}_2=\underset{\begin{array}{c}i,j\in \{1,2,\dots ,n\}\\ i<j\end{array}}{⨆}B{s}_{i-1}\dots s_2{s}_1{s}_{j-1}\dots s_3{s}_2{P}_2\end{array}$$

with

$$B{s}_{j-1}\dots s_2{s}_1{P}_1=\left\{x_{j-1}\left(c_{j-1}\right)s_{j-1}\dots x_2\left(c_2\right)s_2{x}_1\left(c_1\right)s_1{P}_1\hspace{0.17em}\right|\hspace{0.17em}{c}_1,\dots ,c_{j-1}\in 𝔽\}$$

and

$$B{s}_{i-1}\dots s_2{s}_1{s}_{j-1}\dots s_3{s}_2{P}_2=\left\{\begin{array}{c}{x}_{i-1}\left(c_{i-1}\right)s_{i-1}\dots x_2\left(c_2\right)s_2{x}_1\left(c_1\right)s_1{x}_{j-1}\left(d_{j-1}\right)s_{j-1}\dots x_2\left(d_2\right)s_2{P}_2\\ \text{with}{c}_1,c_2,\dots ,c_{i-1},d_2,d_3,\dots ,d_{j-1}\in 𝔽\end{array}\right\}\text{.}$$

Note that

$$x_{j-1}\left(c_{j-1}\right)s_{j-1}\dots x_2\left(c_2\right)s_2{x}_1\left(c_1\right)s_1=\left(\begin{array}{cc}\begin{array}{cc}{c}_1& 1\\ c_2& 0& 1\\ ⋮& ⋮& 0& \ddots \\ ⋮& ⋮& ⋮& \ddots & 1\\ c_{j-1}& 0& 0& \dots & 0& 1\\ 1& 0& 0& \dots & 0& 0\end{array}& 0\\ 0& \begin{array}{c}1\\ & \ddots \\ & & 1\end{array}\end{array}\right)$$

and

$$x_{i-1}\left(c_{i-1}\right)s_{i-1}\dots x_2\left(c_2\right)s_2{x}_1\left(c_1\right)s_1{x}_{j-1}\left(d_{j-1}\right)s_{j-1}\dots x_2\left(d_2\right)s_2=\left(\begin{array}{cc}\begin{array}{ccc}{c}_1& d_2& 1\\ c_2& d_3& 0& 1\\ ⋮& ⋮& ⋮& 0& \ddots \\ ⋮& ⋮& ⋮& ⋮& \ddots & 1\\ c_{i-1}& d_i& 0& 0& \dots & 0& 1\\ 1& 0& 0& 0& \dots & 0& 0& 0\\ 0& d_{i+1}& 0& 0& \dots & 0& 0& 1\\ ⋮& ⋮& ⋮& ⋮& & & & 0& \ddots \\ ⋮& ⋮& ⋮& ⋮& & & & ⋮& \ddots & 1\\ 0& d_{j-1}& 0& 0& \dots & 0& 0& 0& \dots & 0& 1\\ 0& 1& 0& 0& \dots & 0& 0& 0& \dots & 0& 0\end{array}& 0\\ 0& \begin{array}{c}1\\ & \ddots \\ & & 1\end{array}\end{array}\right)\text{.}$$

The case $G={GL}_3\left({𝔽}_2\right)$

$$P_1=\left\{\left(\begin{array}{ccc}*& *& *\\ 0& *& *\\ 0& *& *\end{array}\right)\right\}\text{and}{P}_2=\left\{\left(\begin{array}{ccc}*& *& *\\ *& *& *\\ 0& 0& *\end{array}\right)\right\}$$

and cosets in $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.$ have representatives

$$\left(\begin{array}{c}1\\ & 1\\ & & 1\end{array}\right),\left(\begin{array}{ccc}{c}_1& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}{c}_1& 1& 0\\ c_2& 0& 1\\ 1& 0& 0\end{array}\right)\text{with}{c}_1,c_2\in {𝔽}_2$$

so that $\text{Card}\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.\right)=1+2+4=7\text{.}$

Cosets in $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.$ have representatives

so that $\text{Card}\left(\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.\right)=1+2+4=7\text{.}$

Then the lattice $ℒ$

$$𝔽23 ( 10 01 00 ) ( 10 00 01 ) ( 10 01 01 ) ( 00 10 01 ) ( 10 10 01 ) ( 01 10 01 ) ( 11 10 01 ) (100) (010) (110) (001) (101) (011) (111) (000)$$

have representatives of $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_1$}\right.$ on level 1 and representatives of $\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$P_2$}\right.$ on level 2.

Another way to encode this poset is via the following picture of the Fano plane

$$(001) (011) (111) (101) (010) (110) (100)$$

so that the inclusion of points in lines matches the poset $ℒ\text{.}$

Notes and References

This is a typed copy of handwritten notes by Arun Ram on 12/11/2012.

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