## Schubert Products ${A}_{1}$ and ${A}_{2}$

Last update: 2 September 2012

## Examples

### Type $S{L}_{2}$

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

### Type $S{L}_{3}$

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:

Since $\left[{Z}_{1}\right]={A}_{1}\left[{Z}_{\text{pt}}\right],\phantom{\rule{0.2em}{0ex}}\left[{Z}_{2}\right]={A}_{2}\left[{Z}_{\text{pt}}\right],\phantom{\rule{0.2em}{0ex}}\left[{Z}_{12}\right]={A}_{1}\left[{Z}_{2}\right],\phantom{\rule{0.2em}{0ex}}\dots ,$ 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 $\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]\ne \left[{Z}_{121}\right]$ and $\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]=\left[{X}_{{s}_{2}{s}_{1}{s}_{2}}\right]\ne \left[{Z}_{212}\right]$, 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 ${H}_{T}$, ${y}_{-{\alpha }_{i}}=-{\alpha }_{i}$ and $Δ121= -(α1+α2) -α2 + -α1α2= -α1-α2+α1 -α2 =1,$ and, in ${K}_{T}$, ${y}_{-{\alpha }_{i}}=1-{e}^{-{\alpha }_{i}}$ 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 ${H}_{T}$ and ${K}_{t}$, one does have $\left[{X}_{{s}_{1}{s}_{2}{s}_{1}}\right]=\left[{Z}_{121}\right]=\left[{Z}_{212}\right]=1$.

Since $0={y}_{-{\alpha }_{1}+{\alpha }_{1}}$, $Δ121-1 y-(α1+α2) y-α2 = y-(α1+α2) yα1+y-α2 y-α1- y-α1yα1 yα1y-α1 y-(α1+α2) y-α2 = y-(α1+α2) yα1-y-α1 yα1-y-α2 yα1+y-α2 yα1+y-α2 y-α1 yα1y-α1 y-(α1+α2) y-α2 = q( y-α1, y-α2 ) y-α1 y-α2 yα1 - y-α2 yα1 y-α1 q(yα1, y-α1 ) yα1 y-α1 = q( y-α1, y-α2 ) y-α2- y-α2 q( yα1, y-α1 ) y-(α1+α2) y-α2 = 1y-(α1+α2) ( a11-a11+ a12 ( y-α1 - y-α1 ) +a21 ( y-α2 - yα1 ) +… )$ 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 $f{A}_{i}={A}_{i}\left({s}_{i}f\right)+{B}_{i}f$ gives

$\begin{array}{ccc}{x}_{\lambda }{Z}_{\text{pt}}& =& {y}_{\lambda }{Z}_{\text{pt}},\\ {x}_{\lambda }{Z}_{1}& =& {x}_{\lambda }{A}_{1}{Z}_{\text{pt}}=\left({A}_{1}{x}_{{s}_{1}\lambda }+\left({B}_{1}{x}_{\lambda }\right)\right){Z}_{\text{pt}}={y}_{{s}_{1}\lambda }{Z}_{1}+\left({B}_{1}{y}_{\lambda }\right){Z}_{\text{pt}},\\ {x}_{\lambda }{Z}_{12}& =& {x}_{\lambda }{A}_{1}{A}_{2}{Z}_{\text{pt}}=\left({A}_{1}{x}_{{s}_{1}\lambda }+\left({B}_{1}{x}_{\lambda }\right)\right){A}_{2}{Z}_{\text{pt}},\\ & =& \left({A}_{1}{A}_{2}{x}_{{s}_{2}{s}_{1}\lambda }+{A}_{1}\left({B}_{2}{x}_{{s}_{1}\lambda }\right)+{A}_{2}\left({s}_{2}{B}_{1}{x}_{\lambda }\right)+\left({B}_{2}{B}_{1}{x}_{\lambda }\right)\right){Z}_{\text{pt}},\\ & =& {y}_{{s}_{2}{s}_{1}\lambda }{Z}_{12}+\left({B}_{2}{y}_{{s}_{1}\lambda }\right){Z}_{1}+\left({s}_{2}{B}_{1}{y}_{\lambda }\right){Z}_{2}+\left({B}_{2}{B}_{1}{y}_{\lambda }\right){Z}_{\text{pt}},\end{array}$

Then

${x}_{\lambda }={x}_{\lambda }·1={x}_{\lambda }\left({Z}_{121}-\frac{{\Delta }_{121}-1}{{y}_{-{s}_{1}{\alpha }_{2}}{y}_{-{\alpha }_{2}}}{Z}_{1}\right)$

and

$\begin{array}{ccc}{x}_{\lambda }{Z}_{121}& =& {y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }{Z}_{121}+\left({B}_{1}{s}_{2}{s}_{1}{Y}_{\lambda }{Z}_{21}\right)+\left({s}_{1}{B}_{2}{s}_{1}{Y}_{\lambda }\right){Z}_{12}+\left({B}_{1}{B}_{2}{s}_{1}{Y}_{\lambda }\right){Z}_{1}+\left({s}_{1}{B}_{2}{B}_{1}{Y}_{\lambda }\right){Z}_{1}+\left({B}_{1}{s}_{2}{B}_{1}{Y}_{\lambda }\right){Z}_{2}\\ & & +\left({B}_{1}{B}_{2}{B}_{1}{Y}_{\lambda }\right){Z}_{\text{pt}}-\frac{{\Delta }_{121}-1}{{Y}_{-{s}_{1}{\alpha }_{2}}{Y}_{-{\alpha }_{2}}}{Y}_{{s}_{1}\lambda }{Z}_{1}-\frac{{\Delta }_{121}-1}{{Y}_{-{s}_{1}{\alpha }_{2}}{Y}_{-{\alpha }_{2}}}\left({B}_{1}{Y}_{\lambda }\right){Z}_{\text{pt}}\end{array}$

which is not very pleasing.

A more pleasing derivation of the last Pieri-Chevalley formular for Type ${A}_{2}$ is

${x}_{\lambda }\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\begin{array}{ccc}& {y}_{\lambda }& \\ {y}_{{s}_{1}\lambda }& & {y}_{{s}_{2}\lambda }\\ {y}_{{s}_{2}{s}_{1}\lambda }& & {y}_{{s}_{1}{s}_{2}\lambda }\\ & {y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }& \end{array}$

Hence

${x}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\begin{array}{ccc}& {y}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }& \\ {y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }& & {y}_{{s}_{2}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\\ {y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }& & {y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\\ & 0& \end{array}$

Then

$\begin{array}{c}{x}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{1}{{y}_{-{\alpha }_{1}}}{Z}_{12}-\left({y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{1}{{y}_{-{\alpha }_{2}}}{Z}_{21}\\ {x}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){Z}_{12}-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){Z}_{21}\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\begin{array}{ccc}& {\gamma }_{e}& \\ {\gamma }_{1}& & {\gamma }_{2}\\ 0& & 0\\ & 0& \end{array}\end{array}$

where

$\begin{array}{ccc}{\gamma }_{1}& =& {y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-{\alpha }_{1}}}-\left({y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\\ & =& {y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)},\\ {\gamma }_{2}& =& {y}_{{s}_{2}\lambda }-{y}_{{s}_{2}{s}_{1}\lambda }-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)},\end{array}$

and

$\begin{array}{ccc}{\gamma }_{e}& =& {y}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-{\alpha }_{1}}}-\left({y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-{\alpha }_{2}}}\\ & =& {y}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\end{array}$

Then

$\begin{array}{cc}\multicolumn{2}{c}{{x}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){Z}_{12}-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){Z}_{21}-\frac{{\gamma }_{1}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{2}}}{Z}_{1}-\frac{{\gamma }_{2}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{1}}}{Z}_{2}}\\ =& \begin{array}{ccc}& {\gamma }_{e}-{\gamma }_{1}-{\gamma }_{2}& \\ 0& & 0\\ 0& & 0\\ & 0& \end{array}\end{array}$

where

$\begin{array}{ccc}{\gamma }_{e}-{\gamma }_{1}-{\gamma }_{2}& =& {y}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\\ & & -\left({y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}\lambda }-\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\right)-\left({y}_{{s}_{2}\lambda }-{y}_{{s}_{2}{s}_{1}\lambda }-\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\right)\\ & =& {y}_{\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}\lambda }+{y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{2}\lambda }+{y}_{{s}_{2}{s}_{1}\lambda }\end{array}$

and

$\begin{array}{ccc}\frac{{\gamma }_{1}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{2}}}& =& \frac{\left({y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}\lambda }-\left({y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }\right)\frac{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-{\alpha }_{1}}}\right)}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{2}}}\\ & =& \frac{\left({y}_{{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}\lambda }\right){y}_{-{\alpha }_{1}}-\left({y}_{{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{2}{s}_{1}{s}_{2}\lambda }\right){y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{2}}{y}_{-{\alpha }_{1}}}\end{array}$

In summary,

$\begin{array}{ccc}{x}_{\lambda }& =& {y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }+\left({B}_{1}{y}_{{s}_{2}{s}_{1}\lambda }\right){Z}_{12}++\left({B}_{2}{y}_{{s}_{1}{s}_{2}\lambda }\right){Z}_{21}+\frac{{\gamma }_{1}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{2}}}{Z}_{1}+\frac{{\gamma }_{2}}{{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}{y}_{-{\alpha }_{1}}}{Z}_{2}\\ & & +\frac{{y}_{{s}_{1}{s}_{2}{s}_{1}\lambda }-{y}_{{s}_{1}{s}_{2}\lambda }-{y}_{{s}_{2}{s}_{1}\lambda }+{y}_{{s}_{1}\lambda }+{y}_{{s}_{2}\lambda }-{y}_{\lambda }}{{y}_{-{\alpha }_{1}}{y}_{-{\alpha }_{2}}{y}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}}{Z}_{\text{pt}}\end{array}$

is a decomposition of ${x}_{\lambda }$ 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.