## Flags and Grassmannians

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## Flags

A flag is a sequence of subspaces

$0⊊V1⊆V2⊆… ⊆Vn=ℂnwith dim Vi=i.$

Our favourite flag is

$p= ( 0⊆⟨e1⟩ ⊆…⊆⟨e1,…,en⟩ )$

where ${e}_{i}={\left(0,\dots ,0,\stackrel{i\text{th}}{1},0,\dots ,0\right)}^{t}\text{.}$

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

$B= {(**0*)}$

is the stabilizer of $p$ so that

${flags}⇔GB gp↤gB$

We handle the flag variety with

Linear algebra Theorem 2

$G=⨆w∈WBwB whereW=Sn$

the group of permutation matrices.

Recall that

$ℛ+= { εi-εj | 1≤i w(j) } . ℓ(w)=Card(ℛ(w)) is the length of w.$

The simple reflections ${s}_{1},\dots ,{s}_{n-1}$ are the elements of length $1$ in ${S}_{n}$

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

If $w={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}$ is a reduced word

$BwB = BsiB…BsiℓB = { xi1(c1) si1…xiℓ (cℓ) siℓB | c1,…,cℓ∈ℂ } .$

$w =$

## Grassmanians

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

$Grk,n= { (0⊆Vk⊆ℂn) | Vk is a subspace, dim Vk=k } .$

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

$p=(0⊆⟨e1,…,ek⟩⊆ℂn) andP= { k ( * * ) 0 * }$

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

$GP ⟷ Grk,n gP ⟼ gp$

Then

$P=⨆w∈WjBwB whereWJ= ⟨ s1,…,sk-1, sk+1,…,sn-1 ⟩$

so that

$WJ=Sk×Sn-k = { ⏞k ⏞n-k } .$

Let ${W}^{J}$ be the set of minimal length coset representatives of cosets in $W}{{W}_{J}}\text{.}$ Then

$WJ= { i1 i2 ik ⏟kkkkk ⏟kkn-kkk | 1≤i1<…< ik≤n }$

is in bijection with

${λ⊆(kn-k)}$

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

$λ= λ=(4,4,2,2,1,1) ⊢14.$

The bijection is

$4 4 2 2 1 1 10 9 8 7 6 5 4 3 2 1 ⟼ (2,3,5,6,9,10).$ $14 13 12 11 10 9 8 7 6 5 4 3 2 1$

So we write

$WJ= {wλ | λ⊆(kn-k)} .$

Then

$G=⨆w∈WJ BwP= ⨆λ⊆(kn-k) BwλP$

since

$G=⨆w∈WBwB= ⨆wλ∈WJ ⨆w∈WJB wλwB= ⨆wλ∈WJw∈WJ BwλB·BwB= ⨆wλ∈WJ BwP.$

## Projective space

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

$ℙn-1 = Gr1,n= { 0⊆⟨v⟩ ⊆ℂn | v∈ℂn,v≠0 } = { [v] | v ∈ℂn,v≠0 }$

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

Our favourite point of ${ℙ}^{n-1}$ is

$p=[(1,0,…,0)] =⟨e1⟩= (0⊆⟨e1⟩⊆ℂn)$

which has stabilizer

$P= ( * * * ) inG=GLn(ℂ)$

and

$GP ⟷ ℙn-1 gP ⟼ gp =[g11,…,g1n] ifg=(gij).$

In this case

$WJ= ⟨s2,…,sn-1⟩ =Sn-1= { ⏟k-1 }$

and

$WJ= { i ⏟1 ⏟kkn-1kk | 1≤i≤n }$

so that

$P=⨆w∈WJ BwBandG= ⨆i=1n BwiP.$

Specifically

$wi= i = si-1 si-2…s2s1$

and

$BwiP = { xi-1(c1)si-1 xi-2(c2)si-2 … x1(ci-1)s1P | c1,…, ci-1∈ℂ } = { xi-1,i(c1) xi-2,i(c2) … x1i(ci-1) si-1…siP | c1,…, ci-1∈ℂ } = { ( 1 ⋱ ci-1 ⋮ c1 0 1 ⋱ 01 ) si-1…s1P | c1,…, ci-1∈ℂ } .$

Note that

$( 1 ⋱ ci-1 ⋮ c1 0 1 ⋱ 01 ) si-1…s1P= [ ( 1 ⋱ ci-1 ⋮ c1 0 1 ⋱ 01 ) ( 0 ⋮ 0 1 0 ⋮ 0 ) ] = ( ci-1 ⋮ c1 1 0 ⋮ 0 ) = [ ci-1,…,c1, 1,0,…,0 ] .$

The group

$T= { ( t1 0 ⋱ 0 tn ) | ti∈ℂ× } acts on ℙn-1 by t[c1,…,cn]= [t1c1,…,tncn].$

If $\left[{c}_{1},\dots ,{c}_{n}\right]$ is a fixed point then for all ${t}_{1},\dots ,{t}_{n}\in {ℂ}^{×}$ $\left({t}_{1}{c}_{1},\dots ,{t}_{n}{c}_{n}\right)=\left(\gamma {c}_{1},\dots ,\gamma {c}_{n}\right)$ for some $\gamma \in {ℂ}^{×}\text{.}$

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

$[0,…,0,1,0,…,0] ...ith ⟼ si-1…s1P.$

## Notes and References

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