Schubert classes and products in rank 2

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

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

$$\begin{array}{ccc}\begin{array}{c}\begin{array}{ccc}{s}_1{\alpha }_2& =& j{\alpha }_1+{\alpha }_2,\\ s_2{\alpha }_1& =& {\alpha }_1+{\alpha }_2,\end{array}\\ \\ \text{with}\\ \\ j=\{\begin{array}{cc}1,& \text{in Type}{A}_2,\\ 2,& \text{in Type}{B}_2,\\ 3,& \text{in Type}{G}_2,\end{array}\end{array}& \text{and}& \begin{array}{ccc}& b_1& \\ b_{s_1}& & b_{s_2}\\ b_{s_1{s}_2}& & b_{s_2{s}_1}\\ b_{s_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2}\\ b_{s_1{s}_2{s}_1{s}_2}& & b_{s_2{s}_1{s}_2{s}_1}\\ b_{s_1{s}_2{s}_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2{s}_1{s}_2}\\ ⋮& & ⋮\\ \\ & b_w\text{basis}& \end{array}\\ \begin{array}{ccc}& y_{-{\alpha }_1}& \\ y_{{\alpha }_1}& & y_{-s_2{\alpha }_1}\\ y_{s_2{\alpha }_1}& & y_{-s_1{s}_2{\alpha }_1}\\ y_{s_1{s}_2{\alpha }_1}& & y_{-s_2{s}_1{s}_2{\alpha }_1}\\ y_{s_2{s}_1{s}_2{\alpha }_1}& & y_{-s_1{s}_2{s}_1{s}_2{\alpha }_1}\\ y_{s_1{s}_2{s}_1{s}_2{\alpha }_1}& & y_{-s_2{s}_1{s}_2{s}_1{s}_2{\alpha }_1}\\ ⋮& & ⋮\\ \\ & x_{-{\alpha }_1}& \end{array}& & \begin{array}{ccc}& y_{-{\alpha }_2}& \\ y_{-s_1{\alpha }_2}& & y_{{\alpha }_2}\\ y_{-s_2{s}_1{\alpha }_2}& & y_{s_1{\alpha }_2}\\ y_{-s_1{s}_2{s}_1{\alpha }_2}& & y_{s_2{s}_1{\alpha }_2}\\ y_{-s_2{s}_1{s}_2{s}_1{\alpha }_2}& & y_{s_1{s}_2{s}_1{\alpha }_2}\\ y_{-s_1{s}_2{s}_1{s}_2{s}_1{\alpha }_2}& & y_{s_2{s}_1{s}_2{s}_1{\alpha }_2}\\ ⋮& & ⋮\\ \\ & x_{-{\alpha }_2}& \end{array}\end{array}$$

Let

$$\begin{array}{ccc}\begin{array}{ccc}{Y}_{R^{-}}& =& {\displaystyle \prod _{\alpha \in R^{+}}{y}_{-\alpha },}\end{array}& \text{and}& \begin{array}{ccc}{\Delta }_{121}& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}(B_1{y}_{-{\alpha }_2}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}),\\ {\Delta }_{212}& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}(B_2{y}_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}),\end{array}\end{array}$$

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

$$\begin{array}{ccc}\begin{array}{ccc}& y_{R^{-}}& \\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_1\right]=\left[Z_{\text{pt}}\right]}\end{array}& \begin{array}{ccc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}}& \\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}}& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_1}\right]=\left[Z_1\right]}\end{array}& \begin{array}{ccc}& \frac{y_{R^{-}}}{y_{-{\alpha }_2}}& \\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}}\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_2}\right]=\left[Z_2\right]}\end{array}\end{array}$$

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

where

$$\begin{array}{ccc}{\Delta }_{212}& =& y_{R^{-}}(\frac{1}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-{\alpha }_2}}+\frac{1}{y_{-s_2{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-{\alpha }_2}})\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}}(\frac{1}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}}-\frac{1-p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_2}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}})\\ & =& \frac{y_{R^{-}}}{y_2^{-{\alpha }_2}}\left(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}{y}_{-{\alpha }_2}}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\right)\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}(B_2{y}_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}),\end{array}$$

and, in $H_T,$ $B_2{y}_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}=\frac{-({\alpha }_1+{\alpha }_2)+{\alpha }_1}{-{\alpha }_2}+0=1,$ and, in $K_T,$ $B_2{y}_{-{\alpha }_1}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}=\frac{1-e^{-({\alpha }_1+{\alpha }_2)}-(1-e^{-{\alpha }_1})}{1-e^{-{\alpha }_2}}+1-e^{-{\alpha }_1}=1.$ Then

$$\begin{array}{ccc}\left[Z_{121}\right]& =& \left[X_{s_1{s}_2{s}_1}\right]+\frac{{\Delta }_{121}-N}{y_{-{\alpha }_1}}\left[X_{s_1}\right],\end{array}$$

where

$$\begin{array}{ccc}\frac{{\Delta }_{121}-N}{y_{-{\alpha }_1}}& =& \frac{y_{R^{-}}}{y_2^{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}(B_1{y}_{-{\alpha }_2}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}-1),\end{array}$$

$$\begin{array}{cc}\begin{array}{ccc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}& \\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}& & 0\\ 0& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_1{s}_2}\right]=\left[Z_{12}\right]}\end{array}& \begin{array}{ccc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}& \\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\\ 0& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_2{s}_1}\right]=\left[Z_{21}\right]}\end{array}\\ \begin{array}{ccc}& \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& \\ \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_1{s}_2{s}_1}\right]}\end{array}& \begin{array}{ccc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}& \\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[X_{s_2{s}_1{s}_2}\right]}\end{array}\\ \begin{array}{ccc}& {\Delta }_{121}& \\ {\Delta }_{121}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}& & 0\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[Z_{121}\right]}\end{array}& \begin{array}{ccc}& {\Delta }_{212}& \\ \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}& & {\Delta }_{212}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}& & \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\\ 0& & \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ 0& & 0\\ ⋮& & ⋮\\ \\ \multicolumn{3}{c}{\left[Z_{212}\right]}\end{array}\\ \multicolumn{2}{c}{\text{Figure 2: Schubert and Bott–Samelson cycles for rank 2 and length}\le 3\text{.}}\end{array}$$

CAN WE EXPAND THIS RIGHT HAND SIDE WITH (4.4). IN $K_T,$

$$\begin{array}{ccc}N& =& B_1{y}_{-{\alpha }_2}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}=\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2}\\ & =& \{\begin{array}{ccc}1,& & \text{in Type}{A}_2,\\ 1+e^{(-{\alpha }_1+{\alpha }_2)},& & \text{in Type}{B}_2,\\ 1+e^{-({\alpha }_1+{\alpha }_2)}+e^{-(2{\alpha }_1+{\alpha }_2)},& & \text{in Type}{G}_2,\end{array}\end{array}$$

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

$$\begin{array}{ccc}{x}_{\lambda }\left[X_1\right]& =& y_{\lambda }\left[X_1\right],\\ x_{\lambda }\left[X_{s_1}\right]& =& x_{\lambda }{A}_1\left[Z_{\text{pt}}\right]=(A_1{x}_{s_1\lambda }+\left(B_1{x}_{\lambda }\right))\left[Z_{\text{pt}}\right]=y_{s_1\lambda }\left[X_{s_1}\right]+\left(B_1{y}_{\lambda }\right)\left[X_1\right],\\ x_{\lambda }\left[X_{s_2}\right]& =& y_{s_2\lambda }\left[X_{s_2}\right]+\left(B_2{y}_{\lambda }\right)\left[X_2\right],\\ x_{\lambda }\left[X_{s_1{s}_2}\right]& =& x_{\lambda }{A}_1{A}_2\left[Z_{\text{pt}}\right]=(A_1{x}_{s_1\lambda }+\left(B_1{x}_{\lambda }\right))A_2\left[Z_{\text{pt}}\right],\\ & =& (A_1{A}_2{x}_{s_2{s}_1\lambda }+A_1\left(B_2{x}_{s_1\lambda }\right)+A_2\left(s_2{B}_1{x}_{\lambda }\right)+\left(B_2{B}_1{x}_{\lambda }\right))\left[Z_{\text{pt}}\right],\\ & =& y_{s_2{s}_1\lambda }\left[X_{s_1{s}_2}\right]+\left(B_2{y}_{s_1\lambda }\right)\left[X_{s_1}\right]+\left(s_2{B}_1{y}_{\lambda }\right)\left[X_{s_2}\right]+\left(B_2{B}_1{y}_{\lambda }\right)\left[X_{s_1}\right],\\ x_{\lambda }\left[X_{s_2{s}_1}\right]& =& y_{s_1{s}_2\lambda }\left[X_{s_2{s}_1}\right]+\left(B_1{y}_{s_2\lambda }\right)\left[X_{s_2}\right]+\left(s_1{B}_2{y}_{\lambda }\right)\left[X_{s_1}\right]+\left(B_1{B}_2{y}_{\lambda }\right)\left[X_{s_1}\right],\end{array}$$

and, in Type $A_2,$ (CAN WE GET THE REST OF THESE FOR TYPE $B_2$??)

$$\begin{array}{ccccc}{x}_{\lambda }\left[X_{s_1{s}_2{s}_1}\right]& =& x_{\lambda }·1& =& y_{s_1{s}_2{s}_1\lambda }+\left(B_1{y}_{s_2{s}_1\lambda }\right)\left[X_{s_1{s}_2}\right]+\left(B_2{y}_{s_1{s}_2\lambda }\right)\left[X_{s_2{s}_1}\right]\\ & & & & +\frac{{\gamma }_1}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_2}}\left[X_{s_1}\right]+\frac{{\gamma }_2}{y_{-({\alpha }_1+{\alpha }_2)}{y}_{-{\alpha }_1}}\left[X_{s_2}\right]\\ & & & & \frac{y_{s_1{s}_2{s}_1\lambda }-y_{s_1{s}_2\lambda }-y_{s_2{s}_1\lambda }+y_{s_1\lambda }+y_{s_2\lambda }-y_{\lambda }}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-({\alpha }_1+{\alpha }_2)}}\left[X_1\right],\end{array}$$

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

$${\gamma }_1=y_{s_1\lambda }-y_{s_1{s}_2\lambda }-\left(B_1{y}_{s_2{s}_1\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)},{\gamma }_2=y_{s_2\lambda }-y_{s_2{s}_1\lambda }-\left(B_2{y}_{s_1{s}_2\lambda }\right)y_{-({\alpha }_1+{\alpha }_2)}\text{.}$$

Schubert products in rank 2

Let

$$f=\sum _{w\in W_0}{f}_w{b}_w\text{.}$$

Then

$$\begin{array}{ccc}f\left[X_{s_1}\right]& =& f_1\left[X_{s_1}\right],\\ f\left[X_{s_1}\right]& =& f_{s_1}\left[X_{s_1}\right]+(f_1-f_{s_1})\frac{{\left[X_{s_1}\right]}_1}{{\left[X_1\right]}_1}\left[X_1\right]=f_{s_1}\left[X_{s_1}\right]+\left(\frac{f_1-f_{s_1}}{y_{-{\alpha }_1}}\right)\left[X_1\right],\\ f\left[X_{s_1{s}_2}\right]& =& f_{s_1{s}_2}\left[X_{s_1{s}_2}\right]+(f_{s_1}-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_{s_1}}{{\left[X_{s_1}\right]}_{s_1}}\left[X_{s_1}\right]+(f_{s_2}-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_{s_2}}{{\left[X_{s_2}\right]}_{s_2}}\left[X_{s_2}\right]\\ & & +((f_1-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_1}{{\left[X_1\right]}_1}-(f_{s_1}-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_{s_1}}{{\left[X_{s_1}\right]}_{s_1}}\frac{{\left[X_{s_1}\right]}_1}{{\left[X_1\right]}_1}-(f_{s_2}-f_{s_1{s}_2})\frac{{\left[X_{s_1{s}_2}\right]}_{s_2}}{{\left[X_{s_1}\right]}_{s_2}}\frac{{\left[X_{s_2}\right]}_1}{{\left[X_1\right]}_1})\left[X_1\right],\\ & =& f_{s_1{s}_2}\left[X_{s_1{s}_2}\right]+\frac{f_{s_1}-f_{s_1{s}_2}}{y_{-{\alpha }_2}}\left[X_{s_1}\right]+\frac{f_{s_2}-f_{s_1{s}_2}}{y_{-s_2{\alpha }_1}}\left[X_{s_2}\right]\\ & & +(\frac{f_1-f_{s_1{s}_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{f_{s_1}-f_{s_1{s}_2}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{f_{s_2}-f_{s_1{s}_2}}{y_{-s_2{\alpha }_1}{y}_{-{\alpha }_2}})\left[X_1\right],\\ & =& f_{s_1{s}_2}\left[X_{s_1{s}_2}\right]+\frac{f_{s_1}-f_{s_1{s}_2}}{y_{-{\alpha }_2}}\left[X_{s_1}\right]+\frac{f_{s_2}-f_{s_1{s}_2}}{y_{-s_2{\alpha }_{\mathrm{a}}}}\left[X_{s_2}\right]+(\frac{f_1-f_{s_1}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{f_{s_2}-f_{s_1{s}_2}}{y_{-s_2{\alpha }_1}{y}_{-{\alpha }_2}})\left[X_1\right],\\ & =& f_{s_1{s}_2}\left[X_{s_1{s}_2}\right]+\frac{f_{s_1}-f_{s_1{s}_2}}{y_{-{\alpha }_2}}\left[X_{s_1}\right]+\frac{f_{s_2}-f_{s_1{s}_2}}{y_{-s_2{\alpha }_1}}\left[X_{s_2}\right]\\ & & +\frac{1}{y_{-{\alpha }_2}}(\frac{(f_1-f_{s_1})}{y_{-{\alpha }_1}}-\frac{(f_{s_2}-f_{s_1{s}_2})}{y_{-s_2{\alpha }_1}})\left[X_1\right],\\ f\left[X_{s_1{s}_2{s}_1}\right]& =& f_{s_1{s}_2{s}_1}\left[X_{s_1{s}_2{s}_1}\right]\\ & & +(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2}}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}\left[X_{s_1{s}_2}\right]+(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_2{s}_1}}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}\left[X_{s_1{s}_2}\right]\\ & & +\left(\begin{array}{c}(f_{s_1}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1}}{{\left[X_{s_1}\right]}_{s_1}}\\ -(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2}}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}\frac{{\left[X_{s_1{s}_2}\right]}_{s_1}}{{\left[X_{s_1}\right]}_{s_1}}\\ -(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_2{s}_1}}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}\frac{{\left[X_{s_2{s}_1}\right]}_{s_1}}{{\left[X_{s_1}\right]}_{s_1}}\end{array}\right)\left[X_{s_1}\right]\\ & & +\left(\begin{array}{c}(f_{s_2}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_2}}{{\left[X_{s_2}\right]}_{s_2}}\\ -(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_1{s}_2}}{{\left[X_{s_1{s}_2}\right]}_{s_1{s}_2}}\frac{{\left[X_{s_1{s}_2}\right]}_{s_2}}{{\left[X_{s_2}\right]}_{s_2}}\\ -(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})\frac{{\left[X_{s_1{s}_2{s}_1}\right]}_{s_2{s}_1}}{{\left[X_{s_2{s}_1}\right]}_{s_2{s}_1}}\frac{{\left[X_{s_2{s}_1}\right]}_{s_2}}{{\left[X_{s_2}\right]}_{s_2}}\end{array}\right)\left[X_{s_2}\right]\\ & & +\left(\text{STUFF}\right)\left[X_1\right]\\ & =& f_{s_1{s}_2{s}_1}\left[X_{s_1{s}_2{s}_1}\right]+\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}}\left[X_{s_1{s}_2}\right]+\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-s_1{s}_2{\alpha }_1}}\left[X_{s_2{s}_1}\right]\\ & & +(\frac{N(f_{s_1}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}-\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})}{y_{-s_1{s}_2{\alpha }_1}{y}_{-s_1{\alpha }_2}})\left[X_{s_1}\right]\\ & & +(\frac{(f_{s_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}-\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}-\frac{(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-s_1{s}_2{\alpha }_1}})\left[X_{s_2}\right]\\ & & +\left(\text{STUFF}\right)\left[X_1\right]\\ & =& f_{s_1{s}_2{s}_1}\left[X_{s_1{s}_2{s}_1}\right]+\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}}\left[X_{s_1{s}_2}\right]+\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-s_1{s}_2{\alpha }_1}}\left[X_{s_2{s}_1}\right]\\ & & +(\frac{N(f_{s_1}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}-\frac{(f_{s_1{s}_2}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})}{y_{-s_1{s}_2{\alpha }_1}{y}_{-s_1{\alpha }_2}})\left[X_{s_1}\right]\\ & & +(\frac{(f_{s_2}-f_{s_1{s}_2})}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}-\frac{(f_{s_2{s}_1}-f_{s_1{s}_2{s}_1})}{y_{-{\alpha }_1}{y}_{-s_1{s}_2{\alpha }_1}})\left[X_{s_2}\right]\\ & & +\left(\text{STUFF}\right)\left[X_1\right]\end{array}$$

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

$${\left[X_1\right]}^2=y_{R^{-}}\left[X_1\right],\left[X_1\right]\left[X_{s_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}}\left[X_1\right],\left[X_1\right]\left[X_{s_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}}\left[X_1\right],$$ $$\left[X_1\right]\left[X_{s_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_1\right],\left[X_1\right]\left[X_{s_2{s}_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}}\left[X_1\right],$$ $$\left[X_1\right]\left[X_{s_1{s}_2{s}_1}\right]=\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left[X_1\right],\left[X_1\right]\left[X_{s_2{s}_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left[X_1\right],$$

The Schubert products are

$${\left[X_{s_1}\right]}^2=\frac{y_{R^{-}}}{y_{-{\alpha }_1}}\left[X_{s_1}\right],\left[X_{s_1}\right]\left[X_{s_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_{s_1}\right],\left[X_{s_1}\right]\left[X_{s_1{s}_2{s}_1}\right]=\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right],$$

$$\begin{array}{ccc}\left[X_{s_1}\right]\left[X_{s_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_1\right],\\ \left[X_{s_1}\right]\left[X_{s_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\right)\left[X_1\right],\\ \left[X_{s_1}\right]\left[X_{s_2{s}_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(\frac{y_{-s_1{\alpha }_2}-y_{-s_2{\alpha }_1}}{y_{-{\alpha }_1}}\right)\left[X_1\right],\end{array}$$

The next Schubert products are

$${\left[X_{s_2}\right]}^2=\frac{y_{R^{-}}}{y_{-{\alpha }_2}}\left[X_{s_2}\right],\left[X_{s_2}\right]\left[X_{s_2{s}_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}}\left[X_{s_2}\right],\left[X_{s_2}\right]\left[X_{s_2{s}_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left[X_{s_2}\right],$$

$$\begin{array}{ccc}\left[X_{s_2}\right]\left[X_{s_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left[X_{s_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}\right)\left[X_1\right],\\ \left[X_{s_2}\right]\left[X_{s_1{s}_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left[X_{s_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}^2}(\frac{N}{y_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_2{\alpha }_1}})\left[X_1\right],\end{array}$$

The next Schubert products are

$$\begin{array}{ccc}{\left[X_{s_1{s}_2}\right]}^2& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left[X_{s_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}\right)\left[X_{s_1}\right],\\ \left[X_{s_1{s}_2}\right]\left[X_{s_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left[X_{s_2}\right]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}{y}_{-s_2{\alpha }_1}}(\left(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}\right)\left(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\right)-1)\left[X_1\right],\\ \left[X_{s_1{s}_2}\right]\left[X_{s_1{s}_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left[X_{s_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}^2}(\frac{N}{y_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_2{\alpha }_1}})\left[X_{s_1}\right]\\ \left[X_{s_1{s}_2}\right]\left[X_{s_2{s}_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\left[X_{s_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2}^2}(\frac{1}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}})\left[X_{s_1}\right]\\ & & +\frac{y_{R^{-}}}{y_{-s_2{\alpha }_1}^2{y}_{-{\alpha }_2}}(\frac{1}{y_{-{\alpha }_1}}-\frac{1}{y_{-s_2{s}_1{\alpha }_2}})\left[X_{s_2}\right]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_2}^2}(\frac{1}{y_{-{\alpha }_1}^2{y}_{-s_2{\alpha }_1}}-\frac{1}{y_{-s_2{\alpha }_1}^2{y}_{-{\alpha }_1}}-\frac{1}{y_{-{\alpha }_1}^2{y}_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_2{\alpha }_1}^2{y}_{-s_2{s}_1{\alpha }_2}})\left[X_1\right],\end{array}$$

The next Schubert products are

$$\begin{array}{ccc}{\left[X_{s_2{s}_1}\right]}^2& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_2{s}_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\right)\left[X_{s_2}\right],\\ \left[X_{s_2{s}_1}\right]\left[X_{s_1{s}_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\left[X_{s_2{s}_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}^2}(\frac{1}{y_{-{\alpha }_2}}-\frac{1}{y_{-s_1{s}_2{\alpha }_1}})\left[X_{s_1}\right]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_1}^2}(\frac{1}{y_{-s_2{\alpha }_1}{y}_{-{\alpha }_2}}-\frac{1}{y_{-s_1{s}_2{\alpha }_1}{y}_{-s_1{\alpha }_2}})\left[X_{s_2}\right]+\left(\text{STUFF}\right)\left[X_{s_1}\right],\\ \left[X_{s_2{s}_1}\right]\left[X_{s_2{s}_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_2{s}_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}^2}(\frac{1}{y_{-s_2{\alpha }_1}}-\frac{1}{y_{-s_1{\alpha }_2}})\left[X_{s_2}\right],\end{array}$$

The next Schubert products are

$$\begin{array}{ccc}{\left[X_{s_1{s}_2{s}_1}\right]}^2& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\left[X_{s_1{s}_2{s}_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1^2}}(\frac{1}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}-\frac{1}{y_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}})\left[X_{s_1{s}_2}\right]\\ & \multicolumn{2}{c}{+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}(\frac{N^2}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}^2}-\frac{N}{y_{-s_1{\alpha }_2}^2{y}_{-s_1{s}_2{\alpha }_1}}-\frac{1}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}+\frac{1}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}})\left[X_1\right],}\\ \left[X_{s_1{s}_2{s}_1}\right]\left[X_{s_2{s}_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\left[X_{s_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\left[X_{s_2{s}_1}\right]\\ & \multicolumn{2}{c}{+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}(\frac{N}{y_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}^2}-\frac{1}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}-\frac{1}{y_{-s_1{\alpha }_2}^2{y}_{-s_1{s}_2{\alpha }_1}})\left[X_{s_1}\right]}\\ & \multicolumn{2}{c}{+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}(\frac{1}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}^2}-\frac{1}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}-\frac{1}{y_{-s_2{\alpha }_1}^2{y}_{-s_2{s}_1{\alpha }_2}})\left[X_{s_2}\right]}\\ {\left[X_{s_2{s}_1{s}_2}\right]}^2& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\left[X_{s_2{s}_1{s}_2}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2^2}}(\frac{1}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}-\frac{1}{y_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}})\left[X_{s_2{s}_1}\right]\\ & \multicolumn{2}{c}{+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}(\frac{1}{y_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}^2}-\frac{1}{y_{-s_2{\alpha }_1}^2{y}_{-s_2{s}_1{\alpha }_2}}-\frac{1}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}+\frac{1}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}})\left[X_1\right],}\end{array}$$

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