Schubert Products $A_1$ and $A_2$

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 $S{L}_2$

The nil affine Hecke algebra has $$\left[Z_{\text{pt}}\right]=y_{-\alpha }{b}_1\text{so that}{\left[Z_{\text{pt}}\right]}^2=y_{-\alpha }^2{b}_1=y_{-\alpha }\left[Z_{\text{pt}}\right]$$ and $$A_1\left[Z_{\text{pt}}\right]=(1+t_{s_1})\frac{1}{x_{-\alpha }}\left[Z_{\text{pt}}\right]=(1+t_{s_1})\frac{1}{y_{-\alpha }}{y}_{-{\alpha }_1}{b}_1=b_1+b_{s_2}=\Phi (1\otimes 1).$$ Writing these classes in terms of sections on the moment graph $$\left[Z_1\right]=11,\text{and}\left[Z_{\text{pt}}\right]=y_{-\alpha }0$$ making it easy to see that ${\left[Z_{\text{pt}}\right]}^2=y_{-\alpha }\left[Z_{\text{pt}}\right]$ and $$\begin{array}{ccc}\left[Z_1\right]=A_1\left[Z_{\text{pt}}\right]& =& (1+t_{s_1})\frac{1}{x_{-\alpha }}\left[Z_{\text{pt}}\right]=(1+t_{s_1})\left(\frac{y_{-\alpha }}{y_{-\alpha }}0\right)\\ & =& (1+t_{s_1})\left(10\right)=\left(10\right)+\left(01\right)=\left(11\right)=1\text{.}\end{array}$$

Type $S{L}_3$

Here we will use

$$W_0=⟨s_1,s_2\mid s_i^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩\text{with}\begin{array}{cc}{s}_1{\alpha }_1=-{\alpha }_1,& s_1{\alpha }_2={\alpha }_1+{\alpha }_2,\\ s_2{\alpha }_1={\alpha }_1+{\alpha }_2,& s_2{\alpha }_2=-{\alpha }_2.\end{array}$$ In this case $$A_1=(1+t_{s_1})\frac{1}{x_{-{\alpha }_1}}\text{and}{A}_2=(1+t_{s_2})\frac{1}{x_{-{\alpha }_2}}.$$

Moment graph pictures:

$$\begin{array}{ccccc}\begin{array}{ccc}& b_1& \\ b_{s_1}& & b_{s_2}\\ b_{s_1{s}_2}& & b_{s_2{s}_1}\\ & b_{w_0}& \\ \\ & b_w\text{basis}& \end{array}& & \begin{array}{ccc}& y_{-{\alpha }_1}& \\ y_{{\alpha }_1}& & y_{-({\alpha }_1+{\alpha }_2)}\\ y_{{\alpha }_1+{\alpha }_2}& & y_{-{\alpha }_2}\\ & y_{{\alpha }_2}& \\ \\ & x_{-{\alpha }_1}& \end{array}& & \begin{array}{ccc}& y_{-{\alpha }_2}& \\ y_{-({\alpha }_1+{\alpha }_2)}& & y_{{\alpha }_2}\\ y_{-{\alpha }_2}& & y_{{\alpha }_1+{\alpha }_2}\\ & y_{{\alpha }_1}\\ \\ & x_{-{\alpha }_2}& \end{array}\end{array}$$

Since $\left[Z_1\right]=A_1\left[Z_{\text{pt}}\right],\left[Z_2\right]=A_2\left[Z_{\text{pt}}\right],\left[Z_{12}\right]=A_1\left[Z_2\right],\dots ,$ the first few Bott-Samelson classes are $$\begin{array}{ccc}& \begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}{y}_{-{\alpha }_1}& \\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & \left[Z_{\text{pt}}\right]& \end{array}& \\ \begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}& \\ y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}& & 0\\ 0& & 0\\ & 0& \\ \\ & \left[Z_1\right]& \end{array}& & \begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}& \\ 0& & y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}\\ 0& & 0\\ & 0& \\ \\ & \left[Z_2\right]& \end{array}\\ \begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-({\alpha }_1+{\alpha }_2)}& & y_{-{\alpha }_1}\\ y_{-{\alpha }_1}& & 0\\ & 0& \\ \\ & \left[Z_{12}\right]& \end{array}& & \begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}& \\ y_{-{\alpha }_2}& & y_{-({\alpha }_1+{\alpha }_2)}\\ 0& & y_{-{\alpha }_2}\\ & 0& \\ \\ & \left[Z_{21}\right]& \end{array}\\ \begin{array}{ccc}& {\Delta }_{121}& \\ {\Delta }_{121}& & 1\\ 1& & 1\\ & 1& \\ \\ & \left[Z_{121}\right]& \end{array}& & \begin{array}{ccc}& {\Delta }_{212}& \\ 1& & {\Delta }_{212}\\ 1& & 1\\ & 1& \\ \\ & \left[Z_{212}\right]& \end{array}\end{array}$$ where $${\Delta }_{121}=\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}+\frac{y_{-{\alpha }_2}}{y_{{\alpha }_1}}\text{and}{\Delta }_{212}=\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_2}}+\frac{y_{-{\alpha }_1}}{y_{{\alpha }_2}}.$$

In this case, all the Schubert varieties are smooth so that $$\begin{array}{c}\left[X_{\text{pt}}\right]=\left[Z_{\text{pt}}\right],\\ \left[X_{s_1}\right]=\left[Z_1\right],\\ \left[X_{s_2}\right]=\left[Z_2\right],\end{array}\begin{array}{c}\left[X_{s_1{s}_2}\right]=\left[Z_{12}\right],\\ \left[X_{s_2{s}_1}\right]=\left[Z_{21}\right]\end{array}\text{and}1=X_{s_1{s}_2{s}_1}=\begin{array}{ccc}& 1& \\ 1& & 1\\ 1& & 1\\ & 1& \end{array}.$$

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 $$\left[Z_{121}\right]=1+\frac{{\Delta }_{121}-1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\left[X_{s_1}\right]\text{and}\left[Z_{212}\right]=1+\frac{{\Delta }_{212}-1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}}\left[X_{s_2}\right]$$ are reflected in [CPZ, 17.3 first equation] and [HK, §5.2]. Note that in $H_T$, $y_{-{\alpha }_i}=-{\alpha }_i$ and $${\Delta }_{121}=\frac{-({\alpha }_1+{\alpha }_2)}{-{\alpha }_2}+\frac{-{\alpha }_1}{{\alpha }_2}=\frac{-{\alpha }_1-{\alpha }_2+{\alpha }_1}{-{\alpha }_2}=1,$$ and, in $K_T$, $y_{-{\alpha }_i}=1-e^{-{\alpha }_i}$ and $${\Delta }_{212}=\frac{1-e^{-({\alpha }_1+{\alpha }_2)}}{1-e^{-{\alpha }_2}}+\frac{1-e^{-{\alpha }_1}}{1-e^{{\alpha }_2}}=\frac{1-e^{-({\alpha }_1+{\alpha }_2)}-(1-e^{-{\alpha }_1})e^{-{\alpha }_2}}{1-e^{-{\alpha }_2}}=1,$$ 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 $0=y_{-{\alpha }_1+{\alpha }_1}$, $$\begin{array}{ccc}\frac{{\Delta }_{121}-1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}& =& \frac{y_{-({\alpha }_1+{\alpha }_2)}{y}_{{\alpha }_1}+y_{-{\alpha }_2}{y}_{-{\alpha }_1}-y_{-{\alpha }_1}{y}_{{\alpha }_1}}{y_{{\alpha }_1}{y}_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\\ & =& \frac{y_{-({\alpha }_1+{\alpha }_2)}{y}_{{\alpha }_1}-y_{-{\alpha }_1}{y}_{{\alpha }_1}-y_{-{\alpha }_2}{y}_{{\alpha }_1}+y_{-{\alpha }_2}{y}_{{\alpha }_1}+y_{-{\alpha }_2}{y}_{-{\alpha }_1}}{y_{{\alpha }_1}{y}_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\\ & =& \frac{q(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{{\alpha }_1}-y_{-{\alpha }_2}{y}_{{\alpha }_1}{y}_{-{\alpha }_1}q(y_{{\alpha }_1},y_{-{\alpha }_1})}{y_{{\alpha }_1}{y}_{-{\alpha }_1}}\\ & =& \frac{q(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_2}-y_{-{\alpha }_2}q(y_{{\alpha }_1},y_{-{\alpha }_1})}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\\ & =& \frac{1}{y_{-({\alpha }_1+{\alpha }_2)}}(a_{11}-a_{11}+a_{12}(y_{-{\alpha }_1}-y_{-{\alpha }_1})+a_{21}(y_{-{\alpha }_2}-y_{{\alpha }_1})+\dots )\end{array}$$ Alternatively, $$\begin{array}{ccc}\frac{{\Delta }_{121}-1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}& =& \frac{1}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}+\frac{1}{y_{-s_1{\alpha }_1}{y}_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_1{\alpha }_2}{y}_{-{\alpha }_2}}\\ & =& \frac{1}{y_{-{\alpha }_2}}(\frac{1}{y_{-{\alpha }_1}}-\frac{1}{y_{-s_1{\alpha }_2}})+\frac{1}{y_{-s_1{\alpha }_1}{y}_{-s_1{\alpha }_2}}\\ & =\mathrm{SOMETHINNG\; IS\; OFF\; HERE}& \frac{1}{y_{-{\alpha }_2}}\left(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\right)+\frac{1}{y_{-s_1{\alpha }_1}{y}_{-s_1{\alpha }_2}}\\ & =& \frac{1}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(B_2{y}_{-{\alpha }_1}\right)+\frac{1}{y_{{\alpha }_1}{y}_{-s_1{\alpha }_2}}\\ & =& \frac{1}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left(B_2{y}_{-{\alpha }_1}\right)-\frac{1+q(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_1}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\end{array}$$

Pieri-Chevalley formulas:  Using $f{A}_i=A_i\left(s_{i}f\right)+B_{i}f$ gives

$\begin{array}{ccc}{x}_{\lambda }{Z}_{\text{pt}}& =& y_{\lambda }{Z}_{\text{pt}},\\ x_{\lambda }{Z}_1& =& x_{\lambda }{A}_1{Z}_{\text{pt}}=(A_1{x}_{s_1\lambda }+\left(B_1{x}_{\lambda }\right))Z_{\text{pt}}=y_{s_1\lambda }{Z}_1+\left(B_1{y}_{\lambda }\right)Z_{\text{pt}},\\ x_{\lambda }{Z}_{12}& =& x_{\lambda }{A}_1{A}_2{Z}_{\text{pt}}=(A_1{x}_{s_1\lambda }+\left(B_1{x}_{\lambda }\right))A_2{Z}_{\text{pt}},\\ & =& (A_1{A}_2{x}_{s_2{s}_1\lambda }+A_1\left(B_2{x}_{s_1\lambda }\right)+A_2\left(s_2{B}_1{x}_{\lambda }\right)+\left(B_2{B}_1{x}_{\lambda }\right))Z_{\text{pt}},\\ & =& y_{s_2{s}_1\lambda }{Z}_{12}+\left(B_2{y}_{s_1\lambda }\right)Z_1+\left(s_2{B}_1{y}_{\lambda }\right)Z_2+\left(B_2{B}_1{y}_{\lambda }\right)Z_{\text{pt}},\end{array}$

Then

${\displaystyle x_{\lambda }=x_{\lambda }·1=x_{\lambda }(Z_{121}-\frac{{\Delta }_{121}-1}{y_{-s_1{\alpha }_2}{y}_{-{\alpha }_2}}{Z}_1)}$

and

${\displaystyle \begin{array}{ccc}{x}_{\lambda }{Z}_{121}& =& y_{s_1{s}_2{s}_1\lambda }{Z}_{121}+\left(B_1{s}_2{s}_1{Y}_{\lambda }{Z}_{21}\right)+\left(s_1{B}_2{s}_1{Y}_{\lambda }\right)Z_{12}+\left(B_1{B}_2{s}_1{Y}_{\lambda }\right)Z_1+\left(s_1{B}_2{B}_1{Y}_{\lambda }\right)Z_1+\left(B_1{s}_2{B}_1{Y}_{\lambda }\right)Z_2\\ & & +\left(B_1{B}_2{B}_1{Y}_{\lambda }\right)Z_{\text{pt}}-\frac{{\Delta }_{121}-1}{Y_{-s_1{\alpha }_2}{Y}_{-{\alpha }_2}}{Y}_{s_1\lambda }{Z}_1-\frac{{\Delta }_{121}-1}{Y_{-s_1{\alpha }_2}{Y}_{-{\alpha }_2}}\left(B_1{Y}_{\lambda }\right)Z_{\text{pt}}\end{array}}$

which is not very pleasing.

A more pleasing derivation of the last Pieri-Chevalley formular for Type $A_2$ is

$x_{\lambda }=\begin{array}{ccc}& y_{\lambda }& \\ y_{s_1\lambda }& & y_{s_2\lambda }\\ y_{s_2{s}_1\lambda }& & y_{s_1{s}_2\lambda }\\ & y_{s_1{s}_2{s}_1\lambda }& \end{array}$

Hence

$x_{\lambda }-y_{s_1{s}_2{s}_1\lambda }=\begin{array}{ccc}& y_{\lambda }-y_{s_1{s}_2{s}_1\lambda }& \\ y_{s_1\lambda }-y_{s_1{s}_2{s}_1\lambda }& & y_{s_2\lambda }-y_{s_1{s}_2{s}_1\lambda }\\ y_{s_2{s}_1\lambda }-y_{s_1{s}_2{s}_1\lambda }& & y_{s_1{s}_2\lambda }-y_{s_1{s}_2{s}_1\lambda }\\ & 0& \end{array}$

Then

$\begin{array}{c}{x}_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-(y_{s_2{s}_1\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{1}{y_{-{\alpha }_1}}{Z}_{12}-(y_{s_1{s}_2\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{1}{y_{-{\alpha }_2}}{Z}_{21}\\ x_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)Z_{12}-\left(B_2{y}_{s_1{s}_2\lambda }\right)Z_{21}=\begin{array}{ccc}& {\gamma }_e& \\ {\gamma }_1& & {\gamma }_2\\ 0& & 0\\ & 0& \end{array}\end{array}$

where

${\displaystyle \begin{array}{ccc}{\gamma }_1& =& y_{s_1\lambda }-y_{s_1{s}_2{s}_1\lambda }-(y_{s_2{s}_1\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}-(y_{s_1{s}_2\lambda }-y_{s_1{s}_2{s}_1\lambda })\\ & =& y_{s_1\lambda }-y_{s_1{s}_2\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)},\\ {\gamma }_2& =& y_{s_2\lambda }-y_{s_2{s}_1\lambda }-\left(B_2{y}_{s_1{s}_2\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)},\end{array}}$

and

${\displaystyle \begin{array}{ccc}{\gamma }_e& =& y_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-(y_{s_2{s}_1\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}}-(y_{s_1{s}_2\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_2}}\\ & =& y_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)}-\left(B_2{y}_{s_1{s}_2\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)}\end{array}}$

Then

$\begin{array}{cc}\multicolumn{2}{c}{x_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)Z_{12}-\left(B_2{y}_{s_1{s}_2\lambda }\right)Z_{21}-\frac{{\gamma }_1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}{Z}_1-\frac{{\gamma }_2}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}}{Z}_2}\\ =& \begin{array}{ccc}& {\gamma }_e-{\gamma }_1-{\gamma }_2& \\ 0& & 0\\ 0& & 0\\ & 0& \end{array}\end{array}$

where

$\begin{array}{ccc}{\gamma }_e-{\gamma }_1-{\gamma }_2& =& y_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)}-\left(B_2{y}_{s_1{s}_2\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)}\\ & & -(y_{s_1\lambda }-y_{s_1{s}_2\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)})-(y_{s_2\lambda }-y_{s_2{s}_1\lambda }-\left(B_2{y}_{s_1{s}_2\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)})\\ & =& y_{\lambda }-y_{s_1{s}_2{s}_1\lambda }-y_{s_1\lambda }+y_{s_1{s}_2\lambda }-y_{s_2\lambda }+y_{s_2{s}_1\lambda }\end{array}$

and

${\displaystyle \begin{array}{ccc}\frac{{\gamma }_1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}& =& \frac{(y_{s_1\lambda }-y_{s_1{s}_2\lambda }-(y_{s_2{s}_1\lambda }-y_{s_1{s}_2{s}_1\lambda })\frac{y_{-({\alpha }_1+{\alpha }_2)}}{y_{-{\alpha }_1}})}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\\ & =& \frac{(y_{s_1\lambda }-y_{s_1{s}_2\lambda })y_{-{\alpha }_1}-(y_{s_2{s}_1\lambda }-y_{s_2{s}_1{s}_2\lambda })y_{-({\alpha }_1+{\alpha }_2)}}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}{y}_{-{\alpha }_1}}\end{array}}$

In summary,

${\displaystyle \begin{array}{ccc}{x}_{\lambda }& =& y_{s_1{s}_2{s}_1\lambda }+\left(B_1{y}_{s_2{s}_1\lambda }\right)Z_{12}++\left(B_2{y}_{s_1{s}_2\lambda }\right)Z_{21}+\frac{{\gamma }_1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}{Z}_1+\frac{{\gamma }_2}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}}{Z}_2\\ & & +\frac{y_{s_1{s}_2{s}_1\lambda }-y_{s_1{s}_2\lambda }-y_{s_2{s}_1\lambda }+y_{s_1\lambda }+y_{s_2\lambda }-y_{\lambda }}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}}{Z}_{\text{pt}}\end{array}}$

is a decomposition of $x_{\lambda }$ into Schubert classes.

Schubert products:  The moment graph sections provide fairly quick computations of the products $$\begin{array}{ccc}{\left[Z_{\text{pt}}\right]}^2& =& y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\\ \left[Z_{\text{pt}}\right]\left[Z_1\right]& =& y_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\\ \left[Z_{\text{pt}}\right]\left[Z_2\right]& =& y_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\\ \left[Z_{\text{pt}}\right]\left[Z_{12}\right]& =& y_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\\ \left[Z_{\text{pt}}\right]\left[Z_{21}\right]& =& y_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\end{array}$$

 

$$\begin{array}{ccc}{\left[Z_1\right]}^2& =& y_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}\left[Z_1\right],\\ \left[Z_1\right]\left[Z_2\right]& =& y_{-({\alpha }_1+{\alpha }_2)}\left[Z_{\text{pt}}\right],\\ \left[Z_1\right]\left[Z_{12}\right]& =& y_{-({\alpha }_1+{\alpha }_2)}\left[Z_1\right],\\ \left[Z_1\right]\left[Z_{21}\right]& =& y_{-{\alpha }_2}\left[Z_1\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\left[Z_{\text{pt}}\right]=y_{-{\alpha }_2}\left[Z_1\right]+\left(q\right(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_2}+1)\left[Z_{\text{pt}}\right],\end{array}$$

 

$$\begin{array}{ccc}{\left[Z_2\right]}^2& =& y_{-{\alpha }_1}{y}_{-({\alpha }_1+{\alpha }_2)}\left[Z_1\right],\\ \left[Z_2\right]\left[Z_{12}\right]& =& y_{-{\alpha }_1}\left[Z_2\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}\left[Z_{\text{pt}}\right]=y_{-{\alpha }_1}\left[Z_2\right]+\left(q\right(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_1}+1)\left[Z_{\text{pt}}\right],\\ \left[Z_2\right]\left[Z_{21}\right]& =& y_{-({\alpha }_1+{\alpha }_2)}\left[Z_2\right],\end{array}$$

 

$$\begin{array}{ccc}{\left[Z_{12}\right]}^2& =& y_{-{\alpha }_1}\left[Z_{12}\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}\left[Z_1\right]=y_{-{\alpha }_1}\left[Z_{12}\right]+\left(q\right(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_1}+1)\left[Z_1\right],\\ {\left[Z_{21}\right]}^2& =& y_{-{\alpha }_2}\left[Z_{21}\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-D_{-{\alpha }_2}}{y_{-{\alpha }_1}}\left[Z_2\right]=y_{-{\alpha }_2}\left[Z_{21}\right]+\left(q\right(y_{-{\alpha }_1},y_{-{\alpha }_2})y_{-{\alpha }_2}+1)\left[Z_2\right],\\ \left[Z_{12}\right]\left[Z_{21}\right]& =& \left[Z_1\right]+\left[Z_2\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_1}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[Z_{\text{pt}}\right]=\left[Z_1\right]+\left[Z_2\right]+q(y_{-{\alpha }_1},y_{-{\alpha }_2})\left[Z_{\text{pt}}\right],\end{array}$$

An example of a check of one of these products is: $$\begin{array}{ccc}\multicolumn{3}{c}{y_{-{\alpha }_2}\left[Z_{21}\right]+\frac{y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\left[Z_2\right]}\\ =& \multicolumn{2}{c}{\begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}& \\ y_{-{\alpha }_2}^2& & y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}\\ 0& & y_{-{\alpha }_2}^2\\ & 0& \end{array}}\\ \multicolumn{2}{c}{}& +\begin{array}{ccc}& (y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_2})y_{-({\alpha }_1+{\alpha }_2)}& \\ 0& & (y_{-({\alpha }_1+{\alpha }_2)}-y_{-{\alpha }_2})y_{-({\alpha }_1+{\alpha }_2)}\\ 0& & 0\\ & 0& \end{array}\\ =& \multicolumn{2}{c}{\begin{array}{ccc}& y_{-({\alpha }_1+{\alpha }_2)}^2& \\ y_{-{\alpha }_2}^2& & y_{-({\alpha }_1+{\alpha }_2)}^2\\ 0& & y_{-{\alpha }_2}^2\\ & 0& \end{array}={\left[Z_{21}\right]}^2.}\end{array}$$

Using $$q(y_{\lambda },y_{\mu })=\{\begin{array}{ccc}0,& & \text{in}{H}_T,\\ -1,& & \text{in}{K}_T,\end{array}$$

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 $$\begin{array}{c}{\left[1\right]}^2=-{\alpha }_{10}{\alpha }_{01}{\alpha }_{11}\left[1\right],\left[1\right]\left[s_1\right]={\alpha }_{01}{\alpha }_{11}\left[1\right],\left[1\right]\left[s_2\right]={\alpha }_{10}{\alpha }_{11}\left[1\right],\left[1\right]\left[s_1{s}_2\right]=-{\alpha }_{11}\left[1\right],\left[1\right]\left[s_2{s}_1\right]={\alpha }_{11}\left[1\right],\\ {\left[s_1\right]}^2={\alpha }_{01}{\alpha }_{11}\left[s_1\right],\left[s_1\right]\left[s_2\right]=-{\alpha }_{11}\left[1\right],\left[s_1\right]\left[s_1{s}_2\right]=y_{01}\left[1\right]-{\alpha }_{01}\left[s_1\right],\left[s_1\right]\left[s_1{s}_2\right]=-{\alpha }_{11}\left[s_1\right],\\ {\left[s_2\right]}^2={\alpha }_{10}{\alpha }_{11}\left[s_2\right],\left[s_2\right]\left[s_1{s}_2\right]=-{\alpha }_{11}\left[s_2\right],\left[s_2\right]\left[s_2{s}_1\right]=y_{10}\left[1\right]-{\alpha }_{10}\left[s_2\right],\\ {\left[s_1{s}_2\right]}^2=y_{01}\left[s_2\right]-{\alpha }_{01}\left[s_1{s}_2\right],\left[s_1{s}_2\right]\left[s_2{s}_1\right]=\{-\left[1\right]\}+\left[s_1\right]+\left[s_2\right],\\ {\left[s_2{s}_1\right]}^2=y_{10}\left[s_1\right]-{\alpha }_{10}\left[s_2{s}_1\right],\end{array}$$

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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