Schubert Products A1 and A2

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

Last update: 2 September 2012

Examples

Type SL2

The nil affine Hecke algebra has [Zpt] =y-α b1 so that [Zpt] 2 =y-α2 b1 =y-α [Zpt] and A1 [Zpt] = (1+ts1) 1x-α [Zpt] = (1+ts1) 1y-α y-α1b1 =b1+bs2 =Φ(11). Writing these classes in terms of sections on the moment graph [Z1] =11, and [Zpt] =y-α 0 making it easy to see that [Zpt]2 =y-α [Zpt] and [Z1] =A1[Zpt] = (1+ts1) 1x-α[Zpt] = (1+ts1) ( y-α y-α 0 ) = (1+ts1) (10) = (10) + (01) = (11) =1.

Type SL3

Here we will use

W0= s1, s2 si2=1, s1s2s1 =s2 s1s2 with s1α1 =-α1, s1α2 =α1+ α2, s2α1 =α1+ α2, s2α2 =-α2. In this case A1 =(1+ts1) 1x-α1 and A2= (1+ts2) 1x-α2.

Moment graph pictures:

b1 bs1 bs2 bs1s2 bs2s1 bw0 bw basis   y-α1 yα1 y-(α1+α2) yα1+α2 y-α2 yα2 x-α1   y-α2 y-(α1+α2) yα2 y-α2 yα1+α2 yα1 x-α2

Since [Z1] =A1 [Zpt], [Z2] =A2 [Zpt], [Z12] =A1 [Z2], , the first few Bott-Samelson classes are y-(α1+α2) y-α2y-α1 00 00 0 [Zpt] y-(α1+α2) y-α2 y-(α1+α2) y-α2 0 00 0 [Z1] y-(α1+α2) y-α1 0 y-(α1+α2) y-α1 00 0 [Z2] y-(α1+α2) y-(α1+α2) y-α1 y-α1 0 0 [Z12] y-(α1+α2) y-α2 y-(α1+α2) 0 y-α2 0 [Z21] Δ121 Δ121 1 11 1 [Z121] Δ212 1 Δ212 11 1 [Z212] where Δ121= y-(α1+α2) y-α1 + y-α2 yα1 and Δ212= y-(α1+α2) y-α2 + y-α1 yα2 .

In this case, all the Schubert varieties are smooth so that [Xpt] =[Zpt], [Xs1] =[Z1], [Xs2] =[Z2], [Xs1s2] = [Z12], [Xs2s1] = [Z21] and 1=Xs1s2s1= 1 11 11 1 .

Though [Xs1s2s1] [Z121] and [Xs1s2s1] = [Xs2s1s2] [Z212] , the formulas [Z121] =1+ Δ121-1 y-(α1+α2) y-α2 [Xs1] and [Z212] =1+ Δ212-1 y-(α1+α2) y-α1 [Xs2] are reflected in [CPZ, 17.3 first equation] and [HK, §5.2]. Note that in HT, y-αi =-αi and Δ121= -(α1+α2) -α2 + -α1α2= -α1-α2+α1 -α2 =1, and, in KT, y-αi= 1-e-αi and Δ212= 1- e-(α1+α2) 1- e-α2 + 1-e-α1 1-eα2 = 1- e-(α1+α2) -(1-e-α1) e-α2 1-e-α2 =1, so that, in HT and Kt, one does have [Xs1s2s1] = [Z121] =[Z212] =1 .

Since 0= y-α1 +α1, Δ121-1 y-(α1+α2) y-α2 = y-(α1+α2) yα1+y-α2 y-α1- y-α1yα1 yα1y-α1 y-(α1+α2) y-α2 = y-(α1+α2) yα1-y-α1 yα1-y-α2 yα1+y-α2 yα1+y-α2 y-α1 yα1y-α1 y-(α1+α2) y-α2 = q( y-α1, y-α2 ) y-α1 y-α2 yα1 - y-α2 yα1 y-α1 q(yα1, y-α1 ) yα1 y-α1 = q( y-α1, y-α2 ) y-α2- y-α2 q( yα1, y-α1 ) y-(α1+α2) y-α2 = 1y-(α1+α2) ( a11-a11+ a12 ( y-α1 - y-α1 ) +a21 ( y-α2 - yα1 ) + ) Alternatively, Δ121-1 y-(α1+α2) y-α2 = 1 y-α1 y-α2 + 1 y-s1α1 y-s1α2 - 1 y-s1α2 y-α2 = 1y-α2 ( 1y-α1 - 1y-s1α2 ) + 1 y-s1α1 y-s1α2 =SOMETHINNG IS OFF HERE 1y-α2 ( y-s2α1- y-α1 y-α1 y-s2α1 ) + 1 y-s1α1 y-s1α2 = 1 y-α2 y-s2α1 ( B2y-α1 ) + 1 yα1 y-s1α2 = 1 y-α1 y-s2α1 ( B2y-α1 ) - 1+q( yα1, y-α1 ) y-α1 y-α1 y-s1α2

Pieri-Chevalley formulas:  Using fAi=Ai (sif)+ Bif gives

xλZpt = yλZpt, xλZ1 = xλA1Zpt= ( A1xs1λ+ (B1xλ) ) Zpt= ys1λZ1+ (B1yλ) Zpt, xλZ12 = xλA1A2Zpt= ( A1xs1λ+ (B1xλ) ) A2Zpt, = ( A1A2 xs2s1λ+A1 (B2xs1λ)+ A2 ( s2B1xλ ) + (B2B1xλ) ) Zpt, = ys2s1λ Z12+ ( B2ys1λ ) Z1+ ( s2B1yλ ) Z2+ ( B2B1yλ ) Zpt,

Then

xλ=xλ·1=xλ ( Z121- Δ121-1 y-s1α2 y-α2 Z1 )

and

xλZ121 = ys1s2s1λ Z121 + ( B1s2s1Yλ Z21 ) + ( s1B2s1Yλ ) Z12 + ( B1B2s1Yλ ) Z1 + ( s1B2B1Yλ ) Z1 + ( B1s2B1Yλ ) Z2 + ( B1B2B1Yλ ) Zpt - Δ121-1 Y-s1α2 Y-α2 Ys1λZ1 - Δ121-1 Y-s1α2 Y-α2 (B1Yλ)Zpt

which is not very pleasing.

A more pleasing derivation of the last Pieri-Chevalley formular for Type A2 is

xλ= yλ ys1λ ys2λ ys2s1λ ys1s2λ ys1s2s1λ

Hence

xλ- ys1s2s1λ = yλ- ys1s2s1λ ys1λ- ys1s2s1λ ys2λ- ys1s2s1λ ys2s1λ- ys1s2s1λ ys1s2λ- ys1s2s1λ 0

Then

xλ- ys1s2s1λ - ( ys2s1λ- ys1s2s1λ ) 1y-α1 Z12- ( ys1s2λ- ys1s2s1λ ) 1y-α2 Z21 xλ- ys1s2s1λ - ( B1ys2s1λ ) Z12- ( B2ys1s2λ ) Z21 = γe γ1 γ2 0 0 0

where

γ1 = ys1λ- ys1s2s1λ- ( ys2s1λ- ys1s2s1λ ) y-(α1+α2) y-α1 - ( ys1s2λ- ys1s2s1λ ) = ys1λ- ys1s2λ- ( B1ys2s1λ ) y-(α1+α2) , γ2 = ys2λ- ys2s1λ- ( B2ys1s2λ ) y-(α1+α2) ,

and

γe = yλ- ys1s2s1λ - ( ys2s1λ- ys1s2s1λ ) y-(α1+α2) y-α1 - ( ys1s2λ- ys1s2s1λ ) y-(α1+α2) y-α2 = yλ- ys1s2s1λ - ( B1ys2s1λ ) y-(α1+α2) - ( B2ys1s2λ ) y-(α1+α2)

Then

xλ- ys1s2s1λ - ( B1ys2s1λ ) Z12- ( B2ys1s2λ ) Z21- γ1 y-(α1+α2) y-α2 Z1- γ2 y-(α1+α2) y-α1 Z2 = γe-γ1-γ2 0 0 0 0 0

where

γe-γ1-γ2 = yλ- ys1s2s1λ - ( B1ys2s1λ ) y-(α1+α2) - ( B2ys1s2λ ) y-(α1+α2) - ( ys1λ- ys1s2λ- ( B1ys2s1λ ) y-(α1+α2) ) - ( ys2λ- ys2s1λ- ( B2ys1s2λ ) y-(α1+α2) ) = yλ- ys1s2s1λ- ys1λ+ ys1s2λ- ys2λ+ ys2s1λ

and

γ1 y-(α1+α2) y-α2 = ( ys1λ- ys1s2λ- ( ys2s1λ- ys1s2s1λ ) y-(α1+α2) y-α1 ) y-(α1+α2) y-α2 = ( ys1λ- ys1s2λ ) y-α1- ( ys2s1λ- ys2s1s2λ ) y-(α1+α2) y-(α1+α2) y-α2 y-α1

In summary,

xλ = ys1s2s1λ + ( B1ys2s1λ ) Z12+ + ( B2ys1s2λ ) Z21+ γ1 y-(α1+α2) y-α2 Z1+ γ2 y-(α1+α2) y-α1 Z2 + ys1s2s1λ -ys1s2λ- ys2s1λ+ ys1λ+ ys2λ-yλ y-α1y-α2 y-(α1+α2) Zpt

is a decomposition of xλ into Schubert classes.

Schubert products:  The moment graph sections provide fairly quick computations of the products [Zpt]2 = y-α1 y-α2 y-(α1+α2) [Zpt], [Zpt] [Z1] = y-α2 y-(α1+α2) [Zpt], [Zpt] [Z2] = y-α1 y-(α1+α2) [Zpt], [Zpt] [Z12] = y-(α1+α2) [Zpt], [Zpt] [Z21] = y-(α1+α2) [Zpt],

 

[Z1]2 = y-α2 y-(α1+α2) [Z1], [Z1] [Z2] = y-(α1+α2) [Zpt], [Z1] [Z12] = y-(α1+α2) [Z1] , [Z1] [Z21] = y-α2 [Z1] + y-(α1+α2) -y-α2 y-α1 [Zpt] = y-α2 [Z1] +(q( y-α1, y-α2 ) y-α2+1 ) [Zpt],

 

[Z2]2 = y-α1 y-(α1+α2) [Z1] , [Z2] [Z12] = y-α1 [Z2] + y-(α1+α2) -y-α1 y-α2 [Zpt] = y-α1 [Z2] + ( q( y-α1, y-α2 ) y-α1 +1 ) [Zpt], [Z2] [Z21] = y-(α1+α2) [Z2],

 

[Z12]2 = y-α1 [Z12] + y-(α1+α2) -y-α1 y-α2 [Z1] = y-α1 [Z12] + (q( y-α1, y-α2 ) y-α1 +1 ) [Z1] , [Z21] 2 = y-α2 [Z21] + y-(α1+α2) -D-α2 y-α1 [Z2] = y-α2 [Z21] + (q( y-α1, y-α2 ) y-α2 +1 ) [Z2], [Z12] [Z21] = [Z1] +[Z2] + y-(α1+α2) -y-α1- y-α2 y-α1 y-α2 [Zpt] = [Z1] +[Z2] +q( y-α1, y-α2 ) [Zpt],

An example of a check of one of these products is: y-α2 [Z21] + y-(α1+α2) -y-α2 y-α1 [Z2] = y-(α1+α2) y-α2 y-α22 y-(α1+α2) y-α2 0 y-α22 0 + ( y-(α1+α2) -y-α2 ) y-(α1+α2) 0 ( y-(α1+α2) -y-α2 ) y-(α1+α2) 0 0 0 = y-(α1+α2)2 y-α22 y-(α1+α2)2 0 y-α22 0 = [Z21] 2.

Using q(yλ, yμ) = { 0 , inHT, -1, inKT,

provides a quick check that these formulas agree with the computations for equivariant cohomology and equivariant K-theory which are appear in [GR, §5]. More specifically, in [GR] we have [1]2=-α10 α01α11 [1], [1][s1]= α01α11 [1], [1][s2]= α10α11 [1], [1][s1s2]= -α11[1], [1][s2s1]= α11[1], [s1]2= α01α11 [s1], [s1][s2] =-α11[1], [s1][s1s2] =y01[1]- α01[s1], [s1][s1s2] =-α11[s1], [s2]2=α10 α11[s2], [s2][s1s2] =-α11[s2], [s2][s2s1] =y10[1]- α10[s2], [s1s2]2= y01[s2]- α01 [s1s2], [s1s2] [s2s1]= {-[1]}+ [s1]+[s2], [s2s1]2= y10[s1]- α10 [s2s1],

Notes and References

These notes are part of an ongoing project with Nora Ganter studying generalised cohomologies of flag varieties.

References

[CPZ] B. Calmès, V. Petrov, K. Zainoulline, Invariants, torsion indeices and oriented cohomology of complete flags, arxiv:0905.1341

[Ga] N. Ganter, The elliptic Weyl character formula, arXiv:1206.0528.

[GR] S. Griffeth and A. Ram, Affine Hecke algebras and the Schubert calculus, European J. Combinatorics 25 (2004) 1263-1283.

[HK] J. Hornbostel and V. Kiritchenko, Schubert caculus foralgebraic cobordism, arxiv:0903.3936v3.

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