## Bott Towers

Last update: 19 September 2012

## Bott Towers

Let $\ell \in {ℤ}_{>0}$ and

$letC= {cij} 1≤i

The Bott tower $Y$ is ${\left({ℂ}^{2}-\left\{\left(0,0\right)\right\}\right)}^{\ell }=\left({ℂ}^{2}-\left\{\left(0,0\right)\right\}\right)×\dots ×\left({ℂ}^{2}-\left\{\left(0,0\right)\right\}\right)$ with the relation

$[ z1,w1,… ,zℓ,wℓ ] = [ z1,w1,… ,zi-1, wi-1zit ,wit,zi+1 ,wi+1 tci,i+1,… ,zℓ,wℓ tci,ℓ,$

for $i=1,2,\dots ,\ell$ and $t\in {ℂ}^{*}\text{.}$ For $S\subseteq \left\{1,\dots ,\ell \right\}$ let

$YS= { [ z1,w1,… ,zℓ,wℓ ] ∣wi=0 ifi∉Sand wi∈ℂ×is i∈S } .$

Then

$Y= ⨆S⊆{1,…,N} YS,YS‾= ⨆T⊆SYT, andYS≅ ℂCard(S).$

The torus $D={\left({ℂ}^{×}\right)}^{\ell }$ acts on $Y$ by

$(t1,tℓ) [ z1,w1,… ,zℓ,wℓ ] = [ z1,t1w1,… ,zℓ,tℓwℓ ] ,$

and the $D$-fixed points in $Y$ are

$[S]= [σ1,…,σℓ] withσi= { (1,0) , ifi∉S , (0,1) , ifi∈S.$

Let ${P}_{1},\dots ,{P}_{n}$ be the minimal parabolic subgroups ${P}_{i}⊉B$ of $G\text{.}$ Let $\ell \in {ℤ}_{>0}$ and let $\left({i}_{1},\dots ,{i}_{\ell }\right)$ be a sequence in $\left\{1,\dots ,n\right\}$ of length $\ell \text{.}$ The Bott-Samelson variety is

$Γ=Pi1×B Pi2×B… ×BPiℓ/B,$

so that $\Gamma$ is ${P}_{{i}_{1}}×\dots {P}_{{i}_{\ell }}$ with the relations

$[g1,…,gℓb] =[g1,…,gℓ] and [ g1,…,gi-1, gib,gi+1,… ,gℓ ] = [ g1,…,gi-1, gi,bgi+1,… ,gℓ ] ,$

for $b\in B$ and $i=1,2,\dots ,N-1\text{.}$ The torus $T$ acts on $\Gamma$ by

$t[g1,…,gℓb]= [tg1,…,gℓb], fort∈T, [g1,…,gℓb] ∈Γ.$

Since each component $\left[{z}_{i},{w}_{i}\right]$ of a point $Y$ lives in ${ℙ}^{1}$ and each component ${x}_{{i}_{j}}\left({c}_{j}\right){s}_{j}$ of a point of $\Gamma$ lives in ${ℙ}^{1},$

$Γ≅Ywherecjk = ⟨ αik, αij∨ ⟩ , for1≤j

One must be quote careful at this step as this isomorphism is not an isomorphism in the algebraic category (THANKS TO DAVE ANDERSON FOR FLAGGING THIS). See [Wi1, Remark 2.10].

Furthermore the $T$ action on $\Gamma$ is the restriction of the $D$ action on $Y$ to $T$ via the homomorphism

$T ⟶ D t ⟼ ( Xαi1(t) ,…, Xαiℓ(t) ) .$

There is a $T$–equivariant morphism

$Γi1,…,iℓ =Pi1×B Pi2×B…×B Piℓ/B ⟶ γi1,…,iℓ Xw↪G/B ( xi1(c1) si1,…,xiℓ (cℓ)siℓB ) ⟼ xi1(c1) si1,…,xiℓ (cℓ)siℓB$

where $w={s}_{{i}_{1}}\dots {s}_{{i}_{\ell }}\text{.}$

The "Borel picture" for $Y$ and $\Gamma$ are

$HD*(Y) ≅ ℂ [ y1,…,yℓ, x1,…,xℓ ] ⟨ xj2=yjxj -c1jx1 xj-…-cj-1,j xj-1xj,for j=1,…,ℓ ⟩ , HT*(Γ) ≅ ℂ [ α1,…,αn, x1,…,xℓ ] ⟨ xj2=αij xj- ⟨ αi1, αij∨ ⟩ x1xi-…- ⟨ αij-1, αij∨ ⟩ xj-1xj,for j=1,…,ℓ ⟩ .$

The "moment graph picture" is given by Willems, who computes the map $\Phi =\underset{S\subseteq \left\{1,\dots ,\ell \right\}}{⨁}{\iota }_{S}^{*}$ and finds

$ℂ [ y1,…,yℓ, x1,…,xℓ ] ⟨ xj2=yjxj -c1jx1 xj-…-cj-1,j xj-1xj,for j=1,…,ℓ ⟩ ⟶Φ ⨁S⊆{1,…,ℓ} ιS* ℂ [ y1,…,yℓ ]$

is given by $\Phi {\left({x}_{j}\right)}_{S}=0$ if $j\notin S$ and

$Φ(xj)S=yj+ ∑m∈ℤ>0 (-1)m+1 ∑ {i1,…,im,j} ⊆S ci1i2… cimjyi1, ifj∈S.$

Willems also computes the map

$γ*:HT(G/B) →HT(Γ) and findsγ* ([Xw])= ∑ S=(i1,…,ir) ⊆{1,…,ℓ} r=ℓ(w),w= si1…sir xi1…xir.$

The favourite basis of ${H}_{D}\left(Y\right)$ is

$σεD= ∏π+(ε) σ^iD=xi1 …xiℓ,if ε={i1,…,iℓ} .$

FIX THIS LAST SENTENCE SO THAT IT MAKES SOME SENSE!!!

### Example: Bott towers and Bott-Samelson varieties

The Bott tower $Y$ for the data $C=\left\{\begin{array}{cc}{c}_{12},& {c}_{13}\\ & {c}_{23}\end{array}\right\}$ has moment graph

$\left\{123\right\} \left\{12\right\} \left\{13\right\} \left\{23\right\} \left\{1\right\} \left\{2\right\} \left\{3\right\} \varnothing {\epsilon }_{1} {\epsilon }_{1} {\epsilon }_{1} {\epsilon }_{1} {\epsilon }_{2} {\epsilon }_{2} {\epsilon }_{2}-{c}_{12}{\epsilon }_{1} {\epsilon }_{2}-{c}_{12}{\epsilon }_{1} {\epsilon }_{3} {\epsilon }_{3}-{c}_{23}{\epsilon }_{2} {\epsilon }_{3}-{c}_{13}{\epsilon }_{2} {\epsilon }_{3}-{c}_{13}{\epsilon }_{1}-{c}_{23}{\epsilon }_{2}+{c}_{12}{c}_{23}{\epsilon }_{1}$

and the favourite basis for ${h}_{D}\left(Y\right)$ consists of the sections

$Φ(x1)= yε1 yε1 yε1 0 yε1 0 0 0$ $Φ(x2)= yε2-c12ε1 yε2-c12ε1 0 yε2 0 yε2 0 0$ $Φ(x3)= y ε2-c13 ε1-c23 ε2+c12 c23ε1 0 yε3-c13ε1 yε3-c23ε2 0 0 yε3 0$ $Φ(x1x2)= yε1 yε2-c12ε1 yε1 yε2-c12ε1 0 0 0 0 0 0$ $Φ(x2x3)= yε2-c12ε1 y ε2-c13ε1 -c23ε2+ c12c23ε1 0 0 yε2 yε3-c23ε2 0 0 0 0$ $Φ(x1x3)= yε1 y ε2-c13ε1 -c23ε2+ c12c23ε1 0 yε1 yε3-c13ε1 0 0 0 0 0$ $Φ(x1x2x3)= yε1 yε2-c12ε1 y ε2-c13ε1 -c23ε2+ c12c23ε1 0 0 0 0 0 0 0$

which provide the relations

$x12 = yε1x1, x22 = yε2x2+ yε2-c12ε1 -yε2 yε1 x1x2, x32 = yε3x3+ yε3-c23ε2 -yε3 yε2 x2x3+ yε3-c13ε1 -yε3 yε1 x1x3 + yε2 yε2-c12ε1 + yε2-c12ε1 yε3-c23ε2 - yε2-c12ε1 yε3 - y ε2-c13ε1- c23ε2+ c12c23ε1 yε2 yε1 yε2 yε2-c12ε1 x1x2x3.$

In particular, the "Borel picture" in cohomology is

$HD(Y)= ℂ [ y1,y2,y3, x1,x2,x3 ] ⟨ x12 = y1x1, x22 = y2x2-c12 x1x2, x32 = y3x3-c23 x2x3-c13 x1x3 ⟩$

Now consider the Bott-Samelson variety $\Gamma ={P}_{1}{×}_{B}{P}_{2}{×}_{B}{P}_{1}/B$ for $G=S{L}_{3}$ for which

$C= { c12=-1, c13=2 c23=-1 } since ⟨α1,α2∨⟩ =-1, ⟨α1,α1∨⟩ =2, ⟨α2,α1∨⟩ =-1.$

The "Borel picture" in cohomology is

$HT(Γ)= ℂ [ α1,α2,α3, x1,x2,x3 ] ⟨ x12 = α1x1, x22 = α2x2+ x1x2, x32 = α3x3+ x2x3+2 x1x3 ⟩$

since ${y}_{1}={y}_{-{\alpha }_{1}},{y}_{2}={y}_{-{\alpha }_{2}},{y}_{3}={y}_{-{\alpha }_{3}}\text{.}$ In terms of moment graphs, the inclusion ${\gamma }^{*}:\phantom{\rule{0.2em}{0ex}}{H}_{T}\left(G/B\right)⟶{H}_{T}\left(\Gamma \right)$ is given by

${f}_{1} {f}_{{s}_{2}} {f}_{{s}_{1}} {f}_{{s}_{1}{s}_{2}} {f}_{{s}_{1}{s}_{2}} {f}_{{s}_{1}{s}_{2}{s}_{1}} {y}_{-{\alpha }_{2}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} {f}_{1} {f}_{{s}_{2}} {w}_{0}{f}_{{s}_{1}{s}_{2}{s}_{1}} {f}_{{s}_{1}} {f}_{{s}_{1}{s}_{2}} {f}_{{s}_{2}{s}_{1}} {f}_{{s}_{1}{s}_{2}} {f}_{{s}_{1}{s}_{2}{s}_{1}} {y}_{1} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{1}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{2}} {y}_{-{\alpha }_{1}} {y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)} {y}_{-{\alpha }_{1}} {y}_{-{s}_{2}{\alpha }_{1}} {y}_{-{s}_{1}{\alpha }_{1}} {y}_{-{\alpha }_{2}}$

Then

$γ*: HT(G/B) ⟶ HT(Γ) [Xs1s2s1] ⟼ 1 [Xs1s2] ⟼ x1+x3 [Xs2s1] ⟼ x2 [Xs1] ⟼ x1x2 [Xs2] ⟼ x2x3 [X1] ⟼ x1x2x3$

and the composite

$HT(G/B) ⟶γ*HT (Γ)⟶Φ ⨁S⊆{1,2,3} HT(pt)= ⨁S⊆{1,2,3} ℂ[yα1,yα2]$

has

$Φ(x1)= y-α1 y-α1 y-α1 0 y-α1 0 0 0$ $Φ(x2)= y-(α1+α2) y-(α1+α2) 0 y-α2 0 y-α2 0 0 = y-s1α2 y-s1α2 0 y-α2 0 y-α2 0 0$ $Φ(x3)= y-α2 0 yα1 y-(α1+α2) 0 0 y-α1 0 = y-s1s2α2 0 y-s1α1 y-s2α1 0 0 y-α1 0$

giving

$Φ ( γ*[Xs1s2] ) =Φ(x1+x3)= y-(α1+α2) y-α1 0 y-(α1+α2) y-α1 0 y-α1 0$ $Φ ( γ*[Xs2s1] ) =Φ(x2)= y-(α1+α2) y-(α1+α2) 0 y-α2 0 y-α2 0 0$ $Φ ( γ*[Xs1] ) =Φ(x1x2)= y-α1y-(α1+α2) y-α1y-(α1+α2) 0 0 0 0 0 0$ $Φ ( γ*[Xs2] ) =Φ(x2x3)= y-α2y-(α1+α2) 0 0 y-α2y-(α1+α2) 0 0 0 0$ $Φ ( γ*[X1] ) =Φ(x1x2x3)= y-α1y-α2y-(α1+α2) 0 0 0 0 0 0 0$