Bott Towers

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

Last update: 19 September 2012

Bott Towers

Let >0 and

letC= {cij} 1i<j ,withcij .

The Bott tower Y is ( 2- {(0,0)} ) = ( 2- {(0,0)} ) ×× ( 2- {(0,0)} ) with the relation

[ z1,w1, ,z,w ] = [ z1,w1, ,zi-1, wi-1zit ,wit,zi+1 ,wi+1 tci,i+1, ,z,w tci,,

for i=1,2,, and t*. For S{1,,} let

YS= { [ z1,w1, ,z,w ] wi=0 ifiSand wi×is iS } .

Then

Y= S{1,,N} YS,YS= TSYT, andYS Card(S).

The torus D=(×) acts on Y by

(t1,t) [ z1,w1, ,z,w ] = [ z1,t1w1, ,z,tw ] ,

and the D-fixed points in Y are

[S]= [σ1,,σ] withσi= { (1,0) , ifiS , (0,1) , ifiS.

Let P1,,Pn be the minimal parabolic subgroups PiB of G. Let >0 and let (i1,,i) be a sequence in {1,,n} of length . The Bott-Samelson variety is

Γ=Pi1×B Pi2×B ×BPi/B,

so that Γ is Pi1×Pi with the relations

[g1,,gb] =[g1,,g] and [ g1,,gi-1, gib,gi+1, ,g ] = [ g1,,gi-1, gi,bgi+1, ,g ] ,

for bB and i=1,2,,N-1. The torus T acts on Γ by

t[g1,,gb]= [tg1,,gb], fortT, [g1,,gb] Γ.

Since each component [zi,wi] of a point Y lives in 1 and each component xij(cj)sj of a point of Γ lives in 1,

ΓYwherecjk = αik, αij , for1j<k.

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 Γ is the restriction of the D action on Y to T via the homomorphism

T D t ( Xαi1(t) ,, Xαi(t) ) .

There is a T–equivariant morphism

Γi1,,i =Pi1×B Pi2×B×B Pi/B γi1,,i XwG/B ( xi1(c1) si1,,xi (c)siB ) xi1(c1) si1,,xi (c)siB

where w=si1 si.

The "Borel picture" for Y and Γ are

HD*(Y) [ y1,,y, x1,,x ] xj2=yjxj -c1jx1 xj--cj-1,j xj-1xj,for j=1,, , HT*(Γ) [ α1,,αn, x1,,x ] xj2=αij xj- αi1, αij x1xi-- αij-1, αij xj-1xj,for j=1,, .

The "moment graph picture" is given by Willems, who computes the map Φ=S{1,,} ιS* and finds

[ y1,,y, x1,,x ] xj2=yjxj -c1jx1 xj--cj-1,j xj-1xj,for j=1,, Φ S{1,,} ιS* [ y1,,y ]

is given by Φ(xj)S=0 if jS and

Φ(xj)S=yj+ m>0 (-1)m+1 {i1,,im,j} S ci1i2 cimjyi1, ifjS.

Willems also computes the map

γ*:HT(G/B) HT(Γ) and findsγ* ([Xw])= S=(i1,,ir) {1,,} r=(w),w= si1sir xi1xir.

The favourite basis of HD(Y) is

σεD= π+(ε) σ^iD=xi1 xi,if ε={i1,,i} .

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= { c12, c13 c23 } has moment graph

{123} {12} {13} {23} {1} {2} {3} ε1 ε1 ε1 ε1 ε2 ε2 ε2-c12ε1 ε2-c12ε1 ε3 ε3-c23ε2 ε3-c13ε2 ε3-c13ε1-c23ε2+c12c23ε1

and the favourite basis for hD(Y) 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)= [ y1,y2,y3, x1,x2,x3 ] x12 = y1x1, x22 = y2x2-c12 x1x2, x32 = y3x3-c23 x2x3-c13 x1x3

Now consider the Bott-Samelson variety Γ=P1×BP2 ×BP1/B for G=SL3 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)= [ α1,α2,α3, x1,x2,x3 ] x12 = α1x1, x22 = α2x2+ x1x2, x32 = α3x3+ x2x3+2 x1x3

since y1=y-α1, y2=y-α2, y3=y-α3. In terms of moment graphs, the inclusion γ*:HT (G/B)HT (Γ) is given by

f1 fs2 fs1 fs1s2 fs1s2 fs1s2s1 y-α2 y-(α1+α2) y-(α1+α2) y-α1 y-α1 y-(α1+α2) y-α2 y-α2 y-α1 f1 fs2 w0fs1s2s1 fs1 fs1s2 fs2s1 fs1s2 fs1s2s1 y1 y-α1 y-α1 y-α1 y-α2 y-α2 y-α1 y-(α1+α2) y-α1 y-s2α1 y-s1α1 y-α2

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} [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

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