## Schubert classes and products in rank 2

Last update: 17 February 2013

## Schubert classes and products in rank 2

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

$s1α1 = -α1, s1α2 = jα1+α2, s2α1 = α1+α2, s2α2 = -α2, with j= { 1, in Type A2, 2, in Type B2, 3, in Type G2, and b1 bs1 bs2 bs1s2 bs2s1 bs1s2s1 bs2s1s2 bs1s2s1s2 bs2s1s2s1 bs1s2s1s2s1 bs2s1s2s1s2 ⋮⋮ bw basis$ $y-α1 yα1 y-s2α1 ys2α1 y-s1s2α1 ys1s2α1 y-s2s1s2α1 ys2s1s2α1 y-s1s2s1s2α1 ys1s2s1s2α1 y-s2s1s2s1s2α1 ⋮⋮ x-α1 y-α2 y-s1α2 yα2 y-s2s1α2 ys1α2 y-s1s2s1α2 ys2s1α2 y-s2s1s2s1α2 ys1s2s1α2 y-s1s2s1s2s1α2 ys2s1s2s1α2 ⋮⋮ x-α2$

Let

$yR- = ∏α∈R+ y-α, (7.1) Δ121 = yR- ( 1 y-α2 y-α1 y-α2 + 1 y-s2α2 y-s2α1 y-α2 ) = yR- y-α1 y-α2 y-s1α2 ( y-s1α2 -y-α2 y-α1 +p ( yα1, y-α1 ) y-α2 ) Δ212 = yR- y-α2 y-α1 y-s2α1 ( y-s2α1 -y-α1 y-α2 +p ( yα2, y-α2 ) y-α1 ) ,and (7.2) N= = 1+ ( 1-p ( y-α2, y-jα1 ) y-α2 ) ( ∑k=1j-1 ( 1-p ( y-α1, y-kα1 ) y-kα1 ) ) . (7.3)$

We note that, for ordinary cohomology ${H}_{T}$ and K-theory ${K}_{T},$

$N= { 1+(j-1), in HT, 1+e-α2 ( e-α1+…+ e-(j-1)α1 ) , in KT, andΔ121= NyR- y-α1 y-α2 y-s1α2 .$

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 2.

### Schubert products in rank 2

Using the explicit moment graph representations of the Schubert classes, the formulas for products $g\left[{X}_{w}\right]$ given at the end of Section 6 allow for quick computations of the products of Schubert classes in rank 2 for Weyl group elements up to length 3. It is straightforward to check that these generalise the corresponding computations for equivariant cohomology and equivariant K-theory which were given in [GRa0405333, §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 generalized equivariant Schubert products for Types ${A}_{2}$ and ${B}_{2}\text{.}$

The Schubert products for low dimensional Schubert varieties are as follows.

$[X1]2= yR- [X1], [X1] [Xs1]= yR- y-α1 [X1], [X1] [Xs2]= yR- y-α2 [X1], [X1] [Xs1s2] = yR- y-α1 y-α2 [X1], [X1] [Xs2s1]= yR- y-α2 y-α1 [X1], [X1] [Xs1s2s1] = NyR- y-α1 y-α2 y-s1α2 [X1], [X1] [Xs2s1s2] = yR- y-α2 y-α1 y-s2α1 [X1], [Xs1]2= yR- y-α1 [Xs1], [Xs1] [Xs1s2]= yR- y-α1 y-α2 [Xs1], [Xs1] [Xs1s2s1] = NyR- y-α1 y-α2 y-s1α2 [Xs1],$ $[Xs1] [Xs2] = yR- y-α1 y-α2 [X1], [Xs1] [Xs2s1] = yR- y-α1 y-s1α2 [Xs1]+ yR- y-α2 y-α1 y-s1α2 ( y-s1α2 -y-α2 y-α1 ) [X1], [Xs1] [Xs2s1s2] = yR- y-α2 y-α1 y-s1α2 [Xs1]+ yR- y-α1 y-α2 y-s1α2 y-s2α1 ( y-s1α2 -y-s2α1 y-α1 ) [X1],$ $[Xs2]2= yR- y-α2 [Xs2], [Xs2] [Xs2s1]= yR- y-α2 y-α1 [Xs2], [Xs2] [Xs2s1s2]= yR- y-α2 y-α1 y-s2α1 [Xs2],$ $[Xs2] [Xs1s2] = yR- y-α2 y-s2α1 [Xs2]+ yR- y-α1 y-α2 y-s2α1 ( y-s2α1- y-α1 y-α2 ) [X1], [Xs2] [Xs1s2s1] = yR- y-α1 y-α2 y-s2α1 [Xs2]+ yR- y-α1 y-α2 y-s1α2 y-s2α1 ( Ny-s2α1- y-α1 y-s1α2 ) [X1], [Xs1s2]2 = yR- y-α2 y-s2α1 [Xs1s2]+ yR- y-α2 y-α1 y-s2α1 ( y-s2α1- y-α1 y-α2 ) [Xs1], [Xs1s2] [Xs2s1] = yR- y-α1 y-α2 y-s1α2 [Xs1]+ yR- y-α1 y-α2 y-s2α1 [Xs2] + yR- y-α1 y-α2 y-s1α2 y-s2α1 ( ( y-s2α1- y-α1 y-α2 ) ( y-s1α2- y-α2 y-α1 ) -1 ) [X1], [Xs1s2] [Xs1s2s1] = yR- y-α1 y-α2 y-s2α1 [Xs1s2]+ yR- y-α1 y-α2 y-s1α2 y-s2α1 ( Ny-s2α1 -y-s1α2 y-α2 ) [Xs1] [Xs1s2] [Xs2s1s2] = yR- y-α2 y-s2α1 y-s2s1α1 [Xs1s2] + yR- y-α1 y-α2 y-s2α2 y-s1α2 y-s2s1α2 ( y-s2α1 y-s2s1α2 -y-α1 y-s1α2 y-α2 ) [Xs1] + yR- y-α1 y-α2 y-s2α1 y-s2s1α2 ( y-s2s1α2 -y-α1 y-s2α1 ) [Xs2] + yR- y-α22 ( 1 y-α12 y-s2α1 - 1 y-s2α12 y-α1 - 1 y-α12 y-s1α2 + 1 y-s2α12 y-s2s1α2 ) [X1], [Xs2s1]2 = yR- y-α1 y-s1α2 [Xs2s1]+ yR- y-α1 y-α2 y-s1α2 ( y-s1α2- y-α2 y-α1 ) [Xs2], [Xs2s1] [Xs1s2s1] = yR- y-α1 y-s1α2 y-s1s2α1 [Xs2s1]+ yR- y-α1 y-s1α22 ( Ny-α2- 1y-s1s2α1 ) [Xs1], + yR- y-α12 ( 1 y-s2α1 y-α2 - 1 y-s1s2α1 y-s1α2 ) [Xs2] + yR- y-α12 ( N y-α22 y-s1α2 - N y-α2 y-α12 - 1 y-α22 y-s2α1 + 1 y-s1α22 y-s1s2α1 ) [X1], [Xs2s1] [Xs2s1s2] = yR- y-α2 y-α1 y-s1α2 [Xs2s1]+ yR- y-α2 y-α12 ( 1y-s2α1- 1y-s1α2 ) [Xs2], [Xs1s2s1]2 = yR- y-α1 y-s1α2 y-s1s2α1 [Xs1s2s1]+ yR- y-α12 ( 1 y-α2 y-s2α1 - 1 y-s1α2 y-s1s2α1 ) [Xs1s2] + yR- y-α1 y-α2 ( N2 y-α2 y-s1α22 - N y-s1α22 y-s1s2α1 - 1 y-α1 y-α2 y-s2α1 + 1 y-α1 y-s1α2 y-s1s2α1 ) [X1], [Xs1s2s1] [Xs2s1s2] = yR- y-α1 y-α2 y-s2α1 y-s2s1α2 [Xs2s1]+ yR- y-α1 y-α2 y-s1α2 y-s1s2α1 [Xs2s1] + yR- y-α1 y-α2 ( N y-α2 y-s1α22 - 1 y-α2 y-s2α1 y-s2s1α2 - 1 y-s1α22 y-s1s2α1 ) [Xs1] + yR- y-α1 y-α2 ( 1 y-α1 y-s2α12 - 1 y-α1 y-s1α2 y-s1s2α1 - 1 y-s2α12 y-s2s1α2 ) [Xs2], [Xs2s1s2]2 = yR- y-α2 y-s2α1` y-s2s1α2 [Xs2s1s2]+ yR- y-α22 ( 1 y-α1 y-s1α2 - 1 y-s2α1 y-s2s1α2 ) [Xs2s1] + yR- y-α1 y-α2 ( 1 y-α1 y-s2α12 - 1 y-s2α12 y-s2s1α2 - 1 y-α1 y-α2 y-s1α2 + 1 y-α2 y-s2α1 y-s2s1α2 ) [X1].$ $yR- y-α1 y-α2 yR- y-α1 y-α2 yR- y-α2 y-s2α1 yR- y-α2 y-s2α1 0 00 00 ⋮⋮ [Xs1s2] =[Z12] yR- y-α1 y-α2 yR- y-α1 y-s1α2 yR- y-α1 y-α2 0 yR- y-α1 y-s1α2 00 00 ⋮⋮ [Xs2s1] =[Z21] NyR- y-α1 y-α2 y-s1α2 NyR- y-α1 y-α2 y-s1α2 NyR- y-α1 y-α2 y-s2α1 NyR- y-α1 y-α2 y-s2α1 NyR- y-α1 y-s1α2 y-s1s2α1 NyR- y-α1 y-s1α2 y-s1s2α1 0 00 ⋮⋮ [Xs1s2s1] yR- y-α1 y-α2 y-s2α1 NyR- y-α1 y-α2 y-s1α2 NyR- y-α1 y-α2 y-s2α1 NyR- y-α2 y-s2α1 y-s2s1α2 NyR- y-α1 y-α2 y-s1α2 0 NyR- y-α2 y-s2α1 y-s2s1α2 00 ⋮⋮ [Xs2s1s2] Δ121 Δ121 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-α2 y-s2α1 yR- y-α1 y-s1α2 y-s1s2α1 yR- y-α1 y-s1α2 y-s1s2α1 0 0 0 ⋮⋮ [Z121] Δ212 yR- y-α1 y-α2 y-s1α2 Δ212 yR- y-α2 y-s2α1 y-s2s1α2 yR- y-α1 y-α2 y-s1α2 0 yR- y-α1 y-s2α1 y-s2s1α2 0 0 ⋮⋮ [Z212]$ $Figure 2: Schubert and Bott-Samelson cycles for rank 2 and length ≤ 3.$

## Notes and References

This is an excerpt from a paper entitled Generalized Schubert Calculus authored by Nora Ganter and Arun Ram. It was dedicated to C.S. Seshadri on the occasion of his 90th birthday.