## The Fano plane as a flag variety

Last update: 2 June 2013

## $G}{{P}_{1}}$ and $G}{{P}_{2}}$ and the Fano Plane

The vector space of column vectors of length $n$

$𝔽n= { ( c1c2⋮cn ) | ci∈𝔽 }$

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

$ℱ𝓁 = {maximal chains in ℒ} = { (0⊆V1⊆…⊆Vn-1⊆𝔽n) | dim Vi=i } .$

Our favourite flag is

$0⊆E1⊆…⊆ En-1⊆𝔽n$

where

$Ei= { (c1c2⋮ci0⋮0) | cj∈𝔽 } =span{e1,…,ei}.$

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

$G=Aut(𝔽n)= GLn(𝔽)$

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

$gei=(1gi1)$

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

The stabilizer of ${E}_{i}$ is

$Pi=StabG(Ei)= AAA ⏞ AAA i AAA ⏞ AA n-i AA AA { ( *** *** *** ** 0 ** * 0 * ** * ) } ⊆GLn(𝔽)$

and the stabilizer of $0\subseteq {E}_{1}\subseteq {E}_{2}\subseteq \dots \subseteq {E}_{n-1}\subseteq {𝔽}^{n}$ is

$B=P1∩P2∩…∩ Pn-1= { ( ** * ⋱ * 0* ) } .$

The maps

$GPi⟶ {subspaces of dimension i in 𝔽n} gPi⟼gEi$

and

$GB⟶ℱ𝓁 gB⟼(0⊆gE1⊆gE2⊆…⊆gEn-1⊆𝔽n)$

are bijections and

$gEi=span { (1g11), (1g21), …, (1gi1), }$

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

## Representations of cosets in $G}{{P}_{i}}$

Let

$xi(c)= i i+1 ( 1 ) ⋱ 1 i 1 c i+1 0 1 1 ⋱ 1 si= i i+1 ( 1 ) ⋱ 1 i 0 1 i+1 1 0 1 ⋱ 1$

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

$W0=Sn= ⟨s1,…,sn-1⟩ ={n×n permutation matrices},$

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

If

$Wi=W0∩Pi= ⟨ s1,s2,…, sj-1, sj+1, sj+2,…, sn-1 ⟩$

then coset representatives of the cosets in ${W}_{0}}{{W}_{i}}$ are

$Wi= { ( 12…i-1i i+1i+2…n σ1σ2… σi-1σi τ1τ2… τn-i ) | σ1<σ2<…< σi-1<σi τ1<τ2<…< τn-i }$

and

$GPi= ⨆u∈Wi BuPi$

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

$BuPi= { xj1(c1) sj1… xjℓ(cℓ) sjℓPi | c1,…, cℓ∈𝔽 } .$

## The cases $G}{{P}_{1}}$ and $G}{{P}_{2}}$

$P1= { ( * *** * 0 * *** *** *** ) } andP2= { ( ** ** *** * 0 * *** *** *** ) }$ $W1 = s1×sn-1= ⟨s2,…,sn-1⟩ ⊆Snand W2 = s2×sn-2= ⟨ s1,s3,s4, …,sn-1 ⟩ ⊆Sn. W1 = { 1 2 \dots j \dots n | j∈ {1,2,…,n} } = { sj-1 sj-2 …s2s1 | j∈{1,2,…,n} } W2 = { 1 \dots i \dots j \dots n | i,j∈ {1,2,…,n} ,i

Then

$GP1 ⟶ { V1⊆𝔽n | dim V1=1 } gP1⟼span{(1g11)}$

and

$GP2 ⟶ { V2⊆𝔽n | dim V2=2 } gP2⟼ span { (1g11), (1g21) } .$

Next

$GP1 = ⨆u∈W1 BuP1= ⨆j=1nB sj-1sj-2 …s2s1P1and GP2 = ⨆u∈W2 BuP2= ⨆i,j∈{1,2,…,n}i

with

$Bsj-1…s2s1 P1= { xj-1 (cj-1) sj-1… x2(c2)s2 x1(c1)s1 P1 | c1 ,…,cj-1∈𝔽 }$

and

$Bsi-1…s2s1 sj-1…s3s2 P2= { xi-1 (ci-1) si-1 … x2(c2)s2 x1(c1)s1 xj-1 (dj-1) sj-1 … x2(d2)s2 P2 withc1, c2,…, ci-1, d2,d3,… ,dj-1∈𝔽 } .$

Note that

$xj-1 (cj-1) sj-1 … x2(c2)s2 x1(c1)s1= ( c11 c201 ⋮ ⋮ 0 ⋱ ⋮ ⋮ ⋮ ⋱ 1 cj-1 00… 01 100… 00 0 0 1 ⋱ 1 )$

and

$xi-1 (ci-1) si-1 … x2(c2)s2 x1(c1)s1 xj-1 (dj-1) sj-1 … x2(d2)s2= ( c1 d2 1 c2 d3 01 ⋮ ⋮ ⋮ 0 ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ 1 ci-1 di 00… 01 1000 … 000 0 di+1 0 0 … 0 0 1 ⋮ ⋮ ⋮ ⋮ 0⋱ ⋮ ⋮ ⋮ ⋮ ⋮⋱1 0 dj-1 00… 000 …01 0100 …000 … 00 0 0 1 ⋱ 1 ) .$

## The case $G={GL}_{3}\left({𝔽}_{2}\right)$

$P1= { ( *** 0** 0** ) } and P2= { ( *** *** 00* ) }$

and cosets in $G}{{P}_{1}}$ have representatives

$( 1 1 1 ) , ( c110 100 001 ) , ( c110 c201 100 ) withc1, c2∈𝔽2$

so that $\text{Card}\left(G}{{P}_{1}}\right)=1+2+4=7\text{.}$

Cosets in $G}{{P}_{2}}$ have representatives

$( 1 1 1 ) , ( 100 0d21 010 ) , ( c1d21 100 010 ) withc1, d2∈𝔽2$

so that $\text{Card}\left(G}{{P}_{2}}\right)=1+2+4=7\text{.}$

Then the lattice $ℒ$

${𝔽}_{2}^{3} \left(\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right) \left(\begin{array}{cc}1& 0\\ 0& 0\\ 0& 1\end{array}\right) \left(\begin{array}{cc}1& 0\\ 0& 1\\ 0& 1\end{array}\right) \left(\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right) \left(\begin{array}{cc}1& 0\\ 1& 0\\ 0& 1\end{array}\right) \left(\begin{array}{cc}0& 1\\ 1& 0\\ 0& 1\end{array}\right) \left(\begin{array}{cc}1& 1\\ 1& 0\\ 0& 1\end{array}\right) \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right) \left(\begin{array}{c}0\\ 1\\ 0\end{array}\right) \left(\begin{array}{c}1\\ 1\\ 0\end{array}\right) \left(\begin{array}{c}0\\ 0\\ 1\end{array}\right) \left(\begin{array}{c}1\\ 0\\ 1\end{array}\right) \left(\begin{array}{c}0\\ 1\\ 1\end{array}\right) \left(\begin{array}{c}1\\ 1\\ 1\end{array}\right) \left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$

have representatives of $G}{{P}_{1}}$ on level 1 and representatives of $G}{{P}_{2}}$ 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.