Schubert classes and products in rank 2

Last update: 16 September 2012

Schubert classes and products in rank 2

In rank 2, ${W}_{0}$ is a dihredral group generated by ${s}_{1}$ and ${s}_{2}$ with ${s}_{i}^{2}=1$ and

$s1α2 = jα1+α2, s2α1 = α1+α2, with j= { 1 , in TypeA2 , 2 , in TypeB2 , 3 , in TypeG2 , and b1 bs1 bs2 bs1s2 bs2s1 bs1s2s1 bs2s1s2 bs1s2s1s2 bs2s1s2s1 bs1s2s1s2s1 bs2s1s2s1s2 ⋮ ⋮ bwbasis y-α1 yα1 y-s2α1 ys2α1 y-s1s2α1 ys1s2α1 y-s2s1s2α1 ys2s1s2α1 y-s1s2s1s2α1 ys1s2s1s2α1 y-s2s1s2s1s2α1 ⋮ ⋮ x-α1 y-α2 y-s1α2 yα2 y-s2s1α2 ys1α2 y-s1s2s1α2 ys2s1α2 y-s2s1s2s1α2 ys1s2s1α2 y-s1s2s1s2s1α2 ys2s1s2s1α2 ⋮ ⋮ x-α2$

Let

$YR- = ∏α∈R+ y-α, and Δ121 = yR- y-α1 y-α2 y-s1α2 ( B1y-α2 +p ( yα1, y-α1 ) y-α2 ) , Δ212 = yR- y-α2 y-α1 y-s2α1 ( B2y-α1 +p ( yα2, y-α2 ) y-α1 ) ,$

The Schubert and Bott–Samelson cycles for rank 2 and length $\le 1$ are given

$yR- 00 00 00 00 ⋮⋮ [X1]= [Zpt] yR-y-α1 yR-y-α10 00 00 00 ⋮⋮ [Xs1]= [Z1] yR-y-α2 0yR-y-α2 00 00 00 ⋮⋮ [Xs2]= [Z2]$

The remaining Schubert and Bott–Samelson cycles for rank 2 and length $\le 3$ are given in Figure 3.

where

$Δ212 = yR- ( 1 y-α2 y-α1 y-α2 + 1 y-s2α2 y-s2α1 y-α2 ) = yR- y-α2 ( 1 y-α2 y-α1 - 1-p ( yα2, y-α2 ) y-α2 y-α2 y-s2α1 ) = yR- y2-α2 ( y-s2α1- y-α1+p ( yα2, y-α2 ) y-α1 y-α2 y-α1 y-s2α1 ) = yR- y-α2 y-α1 y-s2α1 ( B2y-α1+p ( yα2, y-α2 ) y-α1 ) ,$

and, in ${H}_{T},$ ${B}_{2}{y}_{-{\alpha }_{1}}+p\left({y}_{{\alpha }_{2}},{y}_{-{\alpha }_{2}}\right){y}_{-{\alpha }_{1}}=\frac{-\left({\alpha }_{1}+{\alpha }_{2}\right)+{\alpha }_{1}}{-{\alpha }_{2}}+0=1,$ and, in ${K}_{T},$ ${B}_{2}{y}_{-{\alpha }_{1}}+p\left({y}_{{\alpha }_{2}},{y}_{-{\alpha }_{2}}\right){y}_{-{\alpha }_{1}}=\frac{1-{e}^{-\left({\alpha }_{1}+{\alpha }_{2}\right)}-\left(1-{e}^{-{\alpha }_{1}}\right)}{1-{e}^{-{\alpha }_{2}}}+1-{e}^{-{\alpha }_{1}}=1.$ Then

$[Z121] = [Xs1s2s1] + Δ121-N y-α1 [Xs1],$

where

$Δ121-N y-α1 = yR- y2-α1 y-α2 y-s1α2 ( B1y-α2+p ( yα1, y-α1 ) y-α2-1 ) ,$

$yR- y-α1 y-α2 yR- y-α1 y-α2 yR- y-α2 y-s2α1 yR- y-α2 y-s2α1 0 00 00 ⋮⋮ [Xs1s2]= [Z12] yR- y-α1 y-α2 yR- y-α1 y-s1α2 yR- y-α1 y-α2 0 yR- y-α1 y-s1α2 00 00 ⋮⋮ [Xs2s1]= [Z21] NyR- y-α1 y-α2 y-s1α2 NyR- y-α1 y-α2 y-s1α2 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-s1α2 y-s1s2α1 yR- y-α1 y-s1α2 y-s1s2α1 0 00 ⋮⋮ [Xs1s2s1] yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-α2 y-s1α2 yR- y-α1 y-α2 y-s2α1 yR- y-α2 y-s1α1 y-s2s1α2 yR- y-α1 y-α2 y-s1α2 0 yR- y-α2 y-s2α1 y-s2s1α2 00 ⋮⋮ [Xs2s1s2] Δ121 Δ121 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-s1α2 y-s1s2α1 yR- y-α1 y-s1α2 y-s1s2α1 0 00 ⋮⋮ [Z121] Δ212 yR- y-α1 y-α2 y-s1α2 Δ212 yR- y-α2 y-s1α1 y-s2s1α2 yR- y-α1 y-α2 y-s1α2 0 yR- y-α2 y-s2α1 y-s2s1α2 00 ⋮⋮ [Z212] Figure 2: Schubert and Bott–Samelson cycles for rank 2 and length≤3.$

CAN WE EXPAND THIS RIGHT HAND SIDE WITH (4.4). IN ${K}_{T},$

$N = B1y-α2+p ( yα1, y-α1 ) y-α2= y-s1α2 -y-α2 y-α1 +p ( yα1, y-α1 ) y-α2 = { 1 , in TypeA2 , 1+e(-α1+α2) , in TypeB2 , 1+e-(α1+α2) +e-(2α1+α2) , in TypeG2 ,$

Pieri–Chevalley formulas: Using $f{A}_{i}={A}_{i}\left({s}_{i}f\right)+{B}_{i}f$ gives

$xλ[X1] = yλ[X1], xλ[Xs1] = xλA1 [Zpt]= ( A1xs1λ+ (B1xλ) ) [Zpt]= ys1λ [Xs1]+ (B1yλ) [X1], xλ[Xs2] = ys2λ [Xs2]+ (B2yλ) [X2], xλ[Xs1s2] = xλA1A2 [Zpt]= ( A1xs1λ+ (B1xλ) ) A2[Zpt], = ( A1A2 xs2s1λ +A1 (B2xs1λ) +A2 (s2B1xλ)+ (B2B1xλ) ) [Zpt], = ys2s1λ [Xs1s2]+ (B2ys1λ) [Xs1]+ (s2B1yλ) [Xs2]+ (B2B1yλ) [Xs1], xλ[Xs2s1] = ys1s2λ [Xs2s1]+ (B1ys2λ) [Xs2]+ (s1B2yλ) [Xs1]+ (B1B2yλ) [Xs1],$

and, in Type ${A}_{2},$ (CAN WE GET THE REST OF THESE FOR TYPE ${B}_{2}$??)

$xλ [Xs1s2s1] = xλ·1 = ys1s2s1λ+ (B1ys2s1λ) [Xs1s2]+ (B2ys1s2λ) [Xs2s1] + γ1 y-(α1+α2) y-α2 [Xs1]+ γ2 y-(α1+α2) y-α1 [Xs2] ys1s2s1λ -ys1s2λ- ys2s1λ+ ys1λ+ ys2λ-yλ y-α1 y-α2 y-(α1+α2) [X1],$

where ${\alpha }_{1}+{\alpha }_{2}={s}_{1}{\alpha }_{2}={s}_{2}{\alpha }_{1},$ ${s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2}$ and

$γ1= ys1λ- ys1s2λ- ( B1 ys2s1λ ) y-(α1+α2), γ2= ys2λ- ys2s1λ- ( B2 ys1s2λ ) y-(α1+α2).$

Schubert products in rank 2

Let

$f=∑w∈W0 fwbw.$

Then

$f[Xs1] = f1[Xs1], f[Xs1] = fs1 [Xs1]+ (f1-fs1) [Xs1]1 [X1]1 [X1]= fs1 [Xs1]+ ( f1-fs1 y-α1 ) [X1], f[Xs1s2] = fs1s2 [Xs1s2]+ ( fs1- fs1s2 ) [Xs1s2] s1 [Xs1] s1 [Xs1]+ ( fs2- fs1s2 ) [Xs1s2]s2 [Xs2]s2 [Xs2] + ( ( f1- fs1s2 ) [Xs1s2]1 [X1]1 - ( fs1- fs1s2 ) [Xs1s2]s1 [Xs1]s1 [Xs1]1 [X1]1 - ( fs2- fs1s2 ) [Xs1s2]s2 [Xs1]s2 [Xs2]1 [X1]1 ) [X1], = fs1s2 [Xs1s2] + fs1- fs1s2 y-α2 [Xs1]+ fs2- fs1s2 y-s2α1 [Xs2] + ( f1- fs1s2 y-α1 y-α2 - fs1- fs1s2 y-α1 y-α2 - fs2- fs1s2 y-s2α1 y-α2 ) [X1], = fs1s2 [Xs1s2] + fs1- fs1s2 y-α2 [Xs1] + fs2- fs1s2 y-s2αa [Xs2] + ( f1-fs1 y-α1 y-α2 - fs2- fs1s2 y-s2α1 y-α2 ) [X1] , = fs1s2 [Xs1s2] + fs1- fs1s2 y-α2 [Xs1] + fs2- fs1s2 y-s2α1 [Xs2] +1y-α2 ( (f1-fs1) y-α1 - ( fs2- fs1s2 ) y-s2α1 ) [X1], f[Xs1s2s1] = fs1s2s1 [Xs1s2s1] + ( fs1s2- fs1s2s1 ) [Xs1s2s1] s1s2 [Xs1s2] s1s2 [Xs1s2] + ( fs2s1- fs1s2s1 ) [Xs1s2s1] s2s1 [Xs2s1] s2s1 [Xs1s2] + ( ( fs1- fs1s2s1 ) [Xs1s2s1] s1 [Xs1] s1 - ( fs1s2- fs1s2s1 ) [Xs1s2s1] s1s2 [Xs1s2] s1s2 [Xs1s2] s1 [Xs1] s1 - ( fs2s1- fs1s2s1 ) [Xs1s2s1] s2s1 [Xs2s1] s2s1 [Xs2s1] s1 [Xs1] s1 ) [Xs1] + ( ( fs2- fs1s2s1 ) [Xs1s2s1] s2 [Xs2] s2 - ( fs1s2- fs1s2s1 ) [Xs1s2s1] s1s2 [Xs1s2] s1s2 [Xs1s2] s2 [Xs2] s2 - ( fs2s1- fs1s2s1 ) [Xs1s2s1] s2s1 [Xs2s1] s2s1 [Xs2s1] s2 [Xs2] s2 ) [Xs2] +(STUFF) [X1] = fs1s2s1 [Xs1s2s1] + ( fs1s2- fs1s2s1 ) y-α1 [Xs1s2] + ( fs1s2- fs1s2s1 ) y-s1s2α1 [Xs2s1] + ( N ( fs1- fs1s2s1 ) y-α2 y-s1α2 - ( fs1s2- fs1s2s1 ) y-α1 y-α2 - ( fs2s1- fs1s2s1 ) y-s1s2α1 y-s1α2 ) [Xs1] + ( ( fs2- fs1s2s1 ) y-α1 y-s2α1 - ( fs1s2- fs1s2s1 ) y-α1 y-s2α1 - ( fs2s1- fs1s2s1 ) y-α1 y-s1s2α1 ) [Xs2] +(STUFF)[X1] = fs1s2s1 [Xs1s2s1] + ( fs1s2- fs1s2s1 ) y-α1 [Xs1s2] + ( fs1s2- fs1s2s1 ) y-s1s2α1 [Xs2s1] + ( N ( fs1- fs1s2s1 ) y-α2 y-s1α2 - ( fs1s2- fs1s2s1 ) y-α1 y-α2 - ( fs2s1- fs1s2s1 ) y-s1s2α1 y-s1α2 ) [Xs1] + ( ( fs2- fs1s2 ) y-α1 y-s2α1 - ( fs2s1- fs1s2s1 ) y-α1 y-s1s2α1 ) [Xs2] +(STUFF)[X1]$

where STUFF is obtained by multiplying both sides by $\left[{X}_{1}\right]$ and solving.

These formulas allow for quick computation of Schubert products in rank 2. The formulas up to length 3 are below. It is straightforward to check that these generalise the corresponding computations for equivalent cohomology and equivariant K–theory which were given in [GR, §5]. Since $\left[{X}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}}\right]=\left[{X}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}}\right]=1$ in Type ${B}_{2},$ these calculations completely determine all Schubert products generalised equivariant Schubert products for Types ${A}_{2}$ and ${B}_{2}\text{.}$ It is interesting to note that the terms in the products above are naturally indexed by chains in the Bruhat order (compare to, for example, see REFERENCE????).

The Schubert products are

$[X1]2= yR-[X1], [X1][Xs1]= yR-y-α1 [X1], [X1][Xs2]= yR-y-α2 [X1],$ $[X1] [Xs1s2]= yR- y-α1 y-α2 [X1], [X1] [Xs2s1]= yR- y-α2 y-α1 [X1],$ $[X1] [Xs1s2s1]= NyR- y-α1 y-α2 y-s1α2 [X1], [X1] [Xs2s1s2]= yR- y-α2 y-α1 y-s2α1 [X1],$

The Schubert products are

$[Xs1]2= yR- y-α1 [Xs1], [Xs1] [Xs1s2]= yR- y-α1 y-α2 [Xs1], [Xs1] [Xs1s2s1] = NyR- y-α1 y-α2 y-s1α2 [Xs1],$

$[Xs1] [Xs2] = yR- y-α1 y-α2 [X1], [Xs1] [Xs2s1] = yR- y-α1 y-s1α2 [Xs1]+ yR- y-α2 y-α1 y-s1α2 ( y-s1α2 -y-α2 y-α1 ) [X1], [Xs1] [Xs2s1s2] = yR- y-α2 y-α1 y-s1α2 [Xs1]+ yR- y-α1 y-α2 y-s1α2 y-s2α1 ( y-s1α2- y-s2α1 y-α1 ) [X1],$

The next Schubert products are

$[Xs2]2= yR- y-α2 [Xs2], [Xs2] [Xs2s1]= yR- y-α2 y-α1 [Xs2], [Xs2] [Xs2s1s2]= yR- y-α2 y-α1 y-s2α1 [Xs2],$

$[Xs2] [Xs1s2] = yR- y-α2 y-s2α1 [Xs2]+ yR- y-α1 y-α2 y-s2α1 ( y-s2α1 -y-α1 y-α2 ) [X1], [Xs2] [Xs1s2s1] = yR- y-α1 y-α2 y-s2α1 [Xs2]+ yR- y-α1 y-α22 ( N y-s1α2 - 1y-s2α1 ) [X1],$

The next Schubert products are

$[Xs1s2]2 = yR- y-α2 y-s2α1 [Xs1s2]+ yR- y-α2 y-α1 y-s2α1 ( y-s2α1- y-α1 y-α2 ) [Xs1], [Xs1s2] [Xs2s1] = yR- y-α1 y-α2 y-s1α2 [Xs1]+ yR- y-α1 y-α2 y-s2α1 [Xs2] + yR- y-α1 y-α2 y-s1α2 y-s2α1 ( ( y-s2α1 -y-α1 y-α2 ) ( y-s1α2 -y-α2 y-α1 ) -1 ) [X1], [Xs1s2] [Xs1s2s1] = yR- y-α1 y-α2 y-s2α1 [Xs1s2]+ yR- y-α1 y-α22 ( N y-s1α2 - 1 y-s2α1 ) [Xs1] [Xs1s2] [Xs2s1s2] = yR- y-α2 y-s2α1 y-s2s1α2 [Xs1s2]+ yR- y-α22 ( 1 y-α1 y-s1α2 - 1 y-s2α1 y-s2s1α2 ) [Xs1] + yR- y-s2α12 y-α2 ( 1 y-α1 - 1 y-s2s1α2 ) [Xs2] + yR- y-α22 ( 1 y-α12 y-s2α1 - 1 y-s2α12 y-α1 - 1 y-α12 y-s1α2 - 1 y-s2α12 y-s2s1α2 ) [X1],$

The next Schubert products are

$[Xs2s1]2 = yR- y-α1 y-s1α2 [Xs2s1] + yR- y-α1 y-α2 y-s1α2 ( y-s1α2 -y-α2 y-α1 ) [Xs2], [Xs2s1] [Xs1s2s1] = yR- y-α1 y-s1α2 y-s1s2α1 [Xs2s1]+ yR- y-α1 y-s1α22 ( 1y-α2- 1y-s1s2α1 ) [Xs1] + yR- y-α12 ( 1 y-s2α1 y-α2 - 1 y-s1s2α1 y-s1α2 ) [Xs2]+ (STUFF) [Xs1], [Xs2s1] [Xs2s1s2] = yR- y-α2 y-α1 y-s1α2 [Xs2s1]+ yR- y-α2 y-α12 ( 1 y-s2α1 - 1 y-s1α2 ) [Xs2],$

The next Schubert products are

$[Xs1s2s1]2 = yR- y-α1 y-s1α2 y-s1s2α1 [Xs1s2s1] + yR- y-α12 ( 1 y-α2 y-s2α1 - 1 y-s1α2 y-s1s2α1 ) [Xs1s2] + yR- y-α1 y-α2 ( N2 y-α2 y-s1α22 - N y-s1α22 y-s1s2α1 - 1 y-α1 y-α2 y-s2α1 + 1 y-α1 y-s1α2 y-s1s2α1 ) [X1], [Xs1s2s1] [Xs2s1s2] = yR- y-α1 y-α2 y-s2α1 y-s2s1α2 [Xs1s2]+ yR- y-α1 y-α2 y-s1α2 y-s1s2α1 [Xs2s1] + yR- y-α1 y-α2 ( N y-α2 y-s1α22 - 1 y-α2 y-s2α1 y-s2s1α2 - 1 y-s1α22 y-s1s2α1 ) [Xs1] + yR- y-α1 y-α2 ( 1 y-α1 y-s2α12 - 1 y-α1 y-s1α2 y-s1s2α1 - 1 y-s2α12 y-s2s1α2 ) [Xs2] [Xs2s1s2]2 = yR- y-α2 y-s2α1 y-s2s1α2 [Xs2s1s2] + yR- y-α22 ( 1 y-α1 y-s1α2 - 1 y-s2α1 y-s2s1α2 ) [Xs2s1] + yR- y-α1 y-α2 ( 1 y-α1 y-s2α12 - 1 y-s2α12 y-s2s1α2 - 1 y-α1 y-α2 y-s1α2 + 1 y-α2 y-s2α1 y-s2s1α2 ) [X1],$