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: 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 $s_i^2=1,$ $s_1{\alpha }_1=-{\alpha }_1,$ $s_2{\alpha }_2=-{\alpha }_2,$

$$\begin{array}{ccc}\begin{array}{c}\begin{array}{cccccc}{s}_1{\alpha }_1& =& -{\alpha }_1,& s_1{\alpha }_2& =& j{\alpha }_1+{\alpha }_2,\\ s_2{\alpha }_1& =& {\alpha }_1+{\alpha }_2,& s_2{\alpha }_2& =& -{\alpha }_2,\end{array}\\ \text{with}\\ j=\{\begin{array}{cc}1,& \text{in Type}\hspace{0.17em}{A}_2,\\ 2,& \text{in Type}\hspace{0.17em}{B}_2,\\ 3,& \text{in Type}\hspace{0.17em}{G}_2,\end{array}\end{array}& \text{and}& \begin{array}{c}\begin{array}{cc}& b_1\\ b_{s_1}& & b_{s_2}\\ b_{s_1{s}_2}& & b_{s_2{s}_1}\\ b_{s_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2}\\ b_{s_1{s}_2{s}_1{s}_2}& & b_{s_2{s}_1{s}_2{s}_1}\\ b_{s_1{s}_2{s}_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2{s}_1{s}_2}\\ ⋮& & ⋮\end{array}\\ b_w\hspace{0.17em}\text{basis}\end{array}\end{array}$$ $$\begin{array}{cc}\begin{array}{c}\begin{array}{cc}& y_{-{\alpha }_1}\\ y_{{\alpha }_1}& & y_{-s_2{\alpha }_1}\\ y_{s_2{\alpha }_1}& & y_{-s_1{s}_2{\alpha }_1}\\ y_{s_1{s}_2{\alpha }_1}& & y_{-s_2{s}_1{s}_2{\alpha }_1}\\ y_{s_2{s}_1{s}_2{\alpha }_1}& & y_{-s_1{s}_2{s}_1{s}_2{\alpha }_1}\\ y_{s_1{s}_2{s}_1{s}_2{\alpha }_1}& & y_{-s_2{s}_1{s}_2{s}_1{s}_2{\alpha }_1}\\ ⋮& & ⋮\end{array}\\ x_{-{\alpha }_1}\end{array}& \begin{array}{c}\begin{array}{cc}& y_{-{\alpha }_2}\\ y_{-s_1{\alpha }_2}& & y_{{\alpha }_2}\\ y_{-s_2{s}_1{\alpha }_2}& & y_{s_1{\alpha }_2}\\ y_{-s_1{s}_2{s}_1{\alpha }_2}& & y_{s_2{s}_1{\alpha }_2}\\ y_{-s_2{s}_1{s}_2{s}_1{\alpha }_2}& & y_{s_1{s}_2{s}_1{\alpha }_2}\\ y_{-s_1{s}_2{s}_1{s}_2{s}_1{\alpha }_2}& & y_{s_2{s}_1{s}_2{s}_1{\alpha }_2}\\ ⋮& & ⋮\end{array}\\ x_{-{\alpha }_2}\end{array}\end{array}$$

Let

$$\begin{array}{cccc}{y}_{R^{-}}& =& \prod _{\alpha \in R^{+}}{y}_{-\alpha },& \text{(7.1)}\\ {\Delta }_{121}& =& y_{R^{-}}(\frac{1}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-{\alpha }_2}}+\frac{1}{y_{-s_2{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-{\alpha }_2}})\\ & =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}+p(y_{{\alpha }_1},y_{-{\alpha }_1})y_{-{\alpha }_2})\\ {\Delta }_{212}& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}(\frac{y_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-{\alpha }_2}}+p(y_{{\alpha }_2},y_{-{\alpha }_2})y_{-{\alpha }_1}),\text{and}& \text{(7.2)}\\ N=& =& 1+(1-p(y_{-{\alpha }_2},y_{-j{\alpha }_1})y_{-{\alpha }_2})\left(\sum _{k=1}^{j-1}(1-p(y_{-{\alpha }_1},y_{-k{\alpha }_1})y_{-k{\alpha }_1})\right)\text{.}& \text{(7.3)}\end{array}$$

We note that, for ordinary cohomology $H_T$ and K-theory $K_T,$

$$N=\{\begin{array}{cc}1+(j-1),& \text{in}\hspace{0.17em}{H}_T,\\ 1+e^{-{\alpha }_2}(e^{-{\alpha }_1}+\dots +e^{-(j-1){\alpha }_1}),& \text{in}\hspace{0.17em}{K}_T,\end{array}\text{and}{\Delta }_{121}=\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\text{.}$$

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

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

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

$$\begin{array}{c}{\left[X_1\right]}^2=y_{R^{-}}\left[X_1\right],\left[X_1\right]\left[X_{s_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}}\left[X_1\right],\left[X_1\right]\left[X_{s_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}}\left[X_1\right],\\ \left[X_1\right]\left[X_{s_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_1\right],\left[X_1\right]\left[X_{s_2{s}_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}}\left[X_1\right],\\ \left[X_1\right]\left[X_{s_1{s}_2{s}_1}\right]=\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left[X_1\right],\left[X_1\right]\left[X_{s_2{s}_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left[X_1\right],\\ {\left[X_{s_1}\right]}^2=\frac{y_{R^{-}}}{y_{-{\alpha }_1}}\left[X_{s_1}\right],\left[X_{s_1}\right]\left[X_{s_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_{s_1}\right],\left[X_{s_1}\right]\left[X_{s_1{s}_2{s}_1}\right]=\frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right],\end{array}$$ $$\begin{array}{ccc}\left[X_{s_1}\right]\left[X_{s_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}}\left[X_1\right],\\ \left[X_{s_1}\right]\left[X_{s_2{s}_1}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left(\frac{y_{-s_1{\alpha }_2}-y_{-{\alpha }_2}}{y_{-{\alpha }_1}}\right)\left[X_1\right],\\ \left[X_{s_1}\right]\left[X_{s_2{s}_1{s}_2}\right]& =& \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}\left[X_{s_1}\right]+\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(\frac{y_{-s_1{\alpha }_2}-y_{-s_2{\alpha }_1}}{y_{-{\alpha }_1}}\right)\left[X_1\right],\end{array}$$ $${\left[X_{s_2}\right]}^2=\frac{y_{R^{-}}}{y_{-{\alpha }_2}}\left[X_{s_2}\right],\left[X_{s_2}\right]\left[X_{s_2{s}_1}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}}\left[X_{s_2}\right],\left[X_{s_2}\right]\left[X_{s_2{s}_1{s}_2}\right]=\frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}{y}_{-s_2{\alpha }_1}}\left[X_{s_2}\right],$$ $$\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}{y}_{-s_1{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(\frac{N{y}_{-s_2{\alpha }_1}-y_{-{\alpha }_1}}{y_{-s_1{\alpha }_2}}\right)\left[X_1\right],\\ {\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}{y}_{-s_1{\alpha }_2}{y}_{-s_2{\alpha }_1}}\left(\frac{N{y}_{-s_2{\alpha }_1}-y_{-s_1{\alpha }_2}}{y_{-{\alpha }_2}}\right)\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 }_1}}\left[X_{s_1{s}_2}\right]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_2}{y}_{-s_1{\alpha }_2}{y}_{-s_2{s}_1{\alpha }_2}}\left(\frac{y_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}-y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}}{y_{-{\alpha }_2}}\right)\left[X_{s_1}\right]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\left(\frac{y_{-s_2{s}_1{\alpha }_2}-y_{-{\alpha }_1}}{y_{-s_2{\alpha }_1}}\right)\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],\\ {\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{N}{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]\\ & & +\frac{y_{R^{-}}}{y_{-{\alpha }_1}^2}(\frac{N}{y_{-{\alpha }_2}^2{y}_{-s_1{\alpha }_2}}-\frac{N}{y_{-{\alpha }_2}{y}_{-{\alpha }_1}^2}-\frac{1}{y_{-{\alpha }_2}^2{y}_{-s_2{\alpha }_1}}+\frac{1}{y_{-s_1{\alpha }_2}^2{y}_{-s_1{s}_2{\alpha }_1}})\left[X_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],\\ {\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]\\ & & +\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_2{s}_1}\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]\\ & & +\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]\\ & & +\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]\\ & & +\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]\text{.}\end{array}$$ $$\begin{array}{cc}\begin{array}{c}\begin{array}{cc}& \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\\ ⋮& & ⋮\end{array}\\ \left[X_{s_1{s}_2}\right]=\left[Z_{12}\right]\end{array}& \begin{array}{c}\begin{array}{cc}& \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\\ ⋮& & ⋮\end{array}\\ \left[X_{s_2{s}_1}\right]=\left[Z_{21}\right]\end{array}\\ \begin{array}{c}\begin{array}{cc}& \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{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}& & \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}\\ \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-s_1{\alpha }_2}{y}_{-s_1{s}_2{\alpha }_1}}& & 0\\ 0& & 0\\ ⋮& & ⋮\end{array}\\ \left[X_{s_1{s}_2{s}_1}\right]\end{array}& \begin{array}{c}\begin{array}{cc}& \frac{y_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}}\\ \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_2{\alpha }_1}}\\ \frac{N{y}_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}& & \frac{N{y}_{R^{-}}}{y_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}}\\ 0& & \frac{N{y}_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ 0& & 0\\ ⋮& & ⋮\end{array}\\ \left[X_{s_2{s}_1{s}_2}\right]\end{array}\\ \begin{array}{c}\begin{array}{cc}& {\Delta }_{121}\\ {\Delta }_{121}& & 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\\ ⋮& & ⋮\end{array}\\ \left[Z_{121}\right]\end{array}& \begin{array}{c}\begin{array}{cc}& {\Delta }_{212}\\ y_{R^{-}}{y}_{-{\alpha }_1}{y}_{-{\alpha }_2}{y}_{-s_1{\alpha }_2}& & {\Delta }_{212}\\ \frac{y_{R^{-}}}{y_{-{\alpha }_2}{y}_{-s_2{\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 }_1}{y}_{-s_2{\alpha }_1}{y}_{-s_2{s}_1{\alpha }_2}}\\ 0& & 0\\ ⋮& & ⋮\end{array}\\ \left[Z_{212}\right]\end{array}\end{array}$$ $$\text{Figure 2: Schubert and Bott-Samelson cycles for rank 2 and length}\hspace{0.17em}\le \hspace{0.17em}3\text{.}$$

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.

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