Schubert Products $G_2$

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 $G_2$

Here we will use

$W_0=⟨s_1,s_2\mid s_i^2=1,s_1{s}_2{s}_1{s}_2{s}_1{s}_2=s_2{s}_1{s}_2{s}_1{s}_2{s}_1⟩$

with

$\begin{array}{cc}{s}_1{\alpha }_1=-{\alpha }_1,& s_1{\alpha }_2=3{\alpha }_1+{\alpha }_2,\\ s_2{\alpha }_1={\alpha }_1+{\alpha }_2,& s_2{\alpha }_2=-{\alpha }_2,\end{array}$

The same in moment graph pictures is

$\begin{array}{ccc}& \begin{array}{ccc}& b_1& \\ b_{s_1}& & b_{s_2}\\ b_{s_1{s}_2}& & b_{s_2{s}_1}\\ b_{s_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2}\\ b_{s_1{s}_2{s}_1{s}_2}& & b_{s_2{s}_1{s}_2{s}_1}\\ b_{s_1{s}_2{s}_1{s}_2{s}_1}& & b_{s_2{s}_1{s}_2{s}_1{s}_2}\\ & b_{w_0}& \\ \\ & b_w\text{basis}& \end{array}& \\ \begin{array}{ccc}& D_{-{\alpha }_1}& \\ D_{{\alpha }_1}& & D_{-({\alpha }_1+{\alpha }_2)}\\ D_{{\alpha }_1+{\alpha }_2}& & D_{-(2{\alpha }_1+{\alpha }_2)}\\ D_{2{\alpha }_1+{\alpha }_2}& & D_{-(2{\alpha }_1+{\alpha }_2)}\\ D_{2{\alpha }_1+{\alpha }_2}& & D_{-({\alpha }_1+{\alpha }_2)}\\ D_{{\alpha }_1+{\alpha }_2}& & D_{-{\alpha }_1}\\ & D_{{\alpha }_1}& \\ \\ & 1\otimes D_{-{\alpha }_1}& \end{array}& & \begin{array}{ccc}& D_{-{\alpha }_2}& \\ D_{-(3{\alpha }_1+{\alpha }_2)}& & D_{{\alpha }_2}\\ D_{-(3{\alpha }_1+2{\alpha }_2)}& & D_{3{\alpha }_1+{\alpha }_2}\\ D_{-(3{\alpha }_1+2{\alpha }_2)}& & D_{3{\alpha }_1+2{\alpha }_2}\\ D_{-(3{\alpha }_1+{\alpha }_2)}& & D_{3{\alpha }_1+2{\alpha }_2}\\ D_{-{\alpha }_2}& & D_{3{\alpha }_1+{\alpha }_2}\\ & D_{{\alpha }_2}& \\ \\ & 1\otimes D_{-{\alpha }_2}& \end{array}\end{array}$

or, alternatively,

$\begin{array}{cc}\begin{array}{ccc}& D_{-{\alpha }_1}& \\ D_{{\alpha }_1}& & D_{-s_2{\alpha }_1}\\ D_{s_2{\alpha }_1}& & D_{-s_1{s}_2{\alpha }_1}\\ D_{s_1{s}_2{\alpha }_1}& & D_{-s_2{s}_1{s}_2{\alpha }_1}\\ D_{s_2{s}_1{s}_2{\alpha }_1}& & D_{-s_1{s}_2{s}_1{s}_2{\alpha }_1}\\ D_{s_1{s}_2{s}_1{s}_2{\alpha }_1}& & D_{-s_2{s}_1{s}_2{s}_1{s}_2{\alpha }_1}\\ & D_{{\alpha }_1}& \\ \\ & 1\otimes D_{-{\alpha }_1}& \end{array}& \begin{array}{ccc}& D_{-{\alpha }_2}& \\ D_{-s_1{\alpha }_2}& & D_{{\alpha }_2}\\ D_{-s_2{s}_1{\alpha }_2}& & D_{s_1{\alpha }_2}\\ D_{-s_1{s}_2{s}_1{\alpha }_2}& & D_{s_2{s}_1{\alpha }_2}\\ D_{-s_2{s}_1{s}_2{s}_1{\alpha }_2}& & D_{s_1{s}_2{s}_1{\alpha }_2}\\ D_{-s_1{s}_2{s}_1{s}_2{s}_1{\alpha }_2}& & D_{s_2{s}_1{s}_2{s}_1{\alpha }_2}\\ & D_{{\alpha }_2}& \\ \\ & 1\otimes D_{-{\alpha }_2}& \end{array}\end{array}$

Let

$D_{R^{-}}=D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-({\alpha }_2+{\alpha }_1)}{D}_{-({\alpha }_2+2{\alpha }_1)}{D}_{-({\alpha }_2+3{\alpha }_1)}{D}_{-(2{\alpha }_2+3{\alpha }_1)}$

$\begin{array}{ccc}& \begin{array}{ccc}& D_{R^{-}}& \\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{\text{pt}}& \end{array}& \\ \begin{array}{ccc}& {\Delta }_1& \\ {\Delta }_1& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_1& \end{array}& & \begin{array}{ccc}& {\Delta }_2& \\ 0& & D_2\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_2& \end{array}\\ \begin{array}{ccc}& {\Delta }_{12}& \\ {\Delta }_{12}& & {\Gamma }_{12}\\ {\Gamma }_{12}& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{12}& \end{array}& & \begin{array}{ccc}& {\Delta }_{21}& \\ {\Gamma }_{21}& & {\Delta }_{21}\\ 0& & {\Gamma }_{21}\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{21}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{121}& \\ {\Delta }_{121}& & {\Gamma }_{121}\\ {\Gamma }_{121}& & K_{121}\\ K_{121}& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{121}& \end{array}& & \begin{array}{ccc}& {\Delta }_{212}& \\ {\Gamma }_{212}& & {\Delta }_{212}\\ K_{212}& & {\Gamma }_{212}\\ 0& & K_{212}\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{212}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{1212}& \\ {\Delta }_{1212}& & {\Gamma }_{1212}\\ {\Gamma }_{1212}& & K_{1212}\\ K_{1212}& & L_{1212}\\ L_{1212}& & 0\\ 0& & 0\\ & 0& \\ \\ & Z_{1212}& \end{array}& & \begin{array}{ccc}& {\Delta }_{2121}& \\ {\Gamma }_{2121}& & {\Delta }_{2121}\\ K_{2121}& & {\Gamma }_{2121}\\ L_{2121}& & K_{2121}\\ 0& & L_{2121}\\ 0& & 0\\ & 0& \\ \\ & Z_{2121}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{12121}& \\ {\Delta }_{12121}& & {\Gamma }_{12121}\\ {\Gamma }_{12121}& & K_{12121}\\ K_{12121}& & L_{12121}\\ L_{12121}& & M_{12121}\\ M_{12121}& & 0\\ & 0& \\ \\ & Z_{12121}& \end{array}& & \begin{array}{ccc}& {\Delta }_{21212}& \\ {\Gamma }_{21212}& & {\Delta }_{21212}\\ K_{21212}& & {\Gamma }_{21212}\\ L_{21212}& & K_{21212}\\ M_{21212}& & L_{21212}\\ 0& & M_{21212}\\ & 0& \\ \\ & Z_{21212}& \end{array}\end{array}$

Where

${\displaystyle \begin{array}{ccc}{\Delta }_1& =& D_{R^{-}}\frac{1}{D_{-{\alpha }_1}}\\ {\Delta }_{12}& =& D_{R^{-}}\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}},\text{and}{\Gamma }_{12}={\Delta }_2\frac{1}{D_{-s_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}}\\ {\Delta }_{121}& =& {\Delta }_{21}\frac{1}{D_{-{\alpha }_1}}+{\Gamma }_{21}\frac{1}{D_{-s_1{\alpha }_1}}=D_{R^{-}}(\frac{1}{D_{-{\alpha }_1}^2{D}_{-{\alpha }_2}}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{{\alpha }_1}}),\\ & =& \frac{D_{R^{-}}}{D_{-{\alpha }_1}}(\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}+\frac{1}{D_{-s_1{\alpha }_1}{D}_{-s_1{\alpha }_2}}),\\ {\Gamma }_{121}& =& {\Delta }_{21}\frac{1}{D_{-s_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}},\\ K_{121}& =& {\Gamma }_{21}\frac{1}{D_{-s_1{s}_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}},\\ {\Delta }_{1212}& =& {\Delta }_{212}\frac{1}{D_{-{\alpha }_1}}+{\Gamma }_{212}\frac{1}{D_{-s_1{\alpha }_1}}\\ & =& D_{R^{-}}(\frac{1}{D_{-{\alpha }_2}^2{D}_{-{\alpha }_1}^2}+\frac{1}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}{D}_{-{\alpha }_1}{D}_{{\alpha }_2}}+\frac{1}{D_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}{D}_{-{\alpha }_1}{D}_{{\alpha }_1}})\\ & =& \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}(\frac{1}{D_{-{\alpha }_2}{D}_{-{\alpha }_1}}+\frac{1}{D_{-s_2{\alpha }_1}{D}_{-s_2{\alpha }_2}}+\frac{1}{D_{-s_1{\alpha }_2}{D}_{-s_1{\alpha }_1}}),\\ {\Gamma }_{1212}& =& {\Delta }_{212}\frac{1}{D_{-s_2{\alpha }_1}}+K_{212}\frac{1}{D_{-s_2{s}_1{\alpha }_1}}\\ & =& D_{R^{-}}(\frac{1}{D_{-{\alpha }_2}^2{D}_{-{\alpha }_1}{D}_{-s_2{\alpha }_1}}+\frac{1}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}^2{D}_{{\alpha }_2}}+\frac{1}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}{D}_{-s_2{s}_1{\alpha }_2}{D}_{s_2{\alpha }_1}})\\ & =& \frac{D_{R^{-}}}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}}(\frac{1}{D_{-{\alpha }_2}{D}_{-{\alpha }_1}}+\frac{1}{D_{-s_2{\alpha }_1}{D}_{-s_2{\alpha }_2}}+\frac{1}{D_{-s_2{s}_1{\alpha }_2}{D}_{-s_2{s}_1{\alpha }_1}}),\\ K_{1212}& =& {\Gamma }_{212}\frac{1}{D_{-s_1{s}_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_2}{D}_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}},\\ L_{1212}& =& K_{212}\frac{1}{D_{-s_2{s}_1{s}_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}{D}_{-s_2{s}_1{\alpha }_2}{D}_{-s_2{s}_1{s}_2{\alpha }_1}},\\ {\Delta }_{12121}& =& {\Delta }_{2121}\frac{1}{D_{-{\alpha }_1}}+{\Gamma }_{2121}\frac{1}{D_{-s_1{\alpha }_1}}\\ & =& D_{R^{-}}(\frac{1}{D_{-{\alpha }_1}^2{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}{D}_{{\alpha }_1}}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}^2{D}_{{\alpha }_1}^2}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}{D}_{s_1{\alpha }_2}{D}_{{\alpha }_1}})\\ & =& \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{\alpha }_1}}(\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}+\frac{1}{D_{-s_1{\alpha }_2}{D}_{-s_1{\alpha }_1}}+\frac{1}{D_{-s_1{s}_2{\alpha }_1}{D}_{-s_1{s}_2{\alpha }_2}}),\\ {\Gamma }_{12121}& =& {\Delta }_{2121}\frac{1}{D_{-s_2{\alpha }_1}}+K_{2121}\frac{1}{D_{-s_2{s}_1{\alpha }_1}}\\ & =& D_{R^{-}}\left(\begin{array}{c}\frac{1}{D_{-{\alpha }_1}^2{D}_{-{\alpha }_2}^2{D}_{-s_2{\alpha }_1}}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-{\alpha }_2}{D}_{{\alpha }_1}{D}_{-s_2{\alpha }_1}}\\ +\frac{1}{D_{-{\alpha }_1}{D}_{-s_2{\alpha }_1}^2{D}_{-{\alpha }_2}{D}_{{\alpha }_2}}+\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}{D}_{-s_2{s}_1{\alpha }_2}{D}_{s_2{\alpha }_1}}\end{array}\right)\\ & =& \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}}(\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}+\frac{1}{D_{-s_1{\alpha }_2}{D}_{-s_1{\alpha }_1}}+\frac{1}{D_{-s_2{\alpha }_1}{D}_{-s_2{\alpha }_2}}+\frac{1}{D_{-s_2{s}_1{\alpha }_2}{D}_{-s_2{s}_1{\alpha }_1}}),\\ K_{12121}& =& {\displaystyle {\Gamma }_{2121}\frac{1}{D_{-s_1{s}_2{\alpha }_1}}+L_{2121}\frac{1}{D_{-s_1{s}_2{s}_1{\alpha }_1}}}\\ & =& D_{R^{-}}\left(\begin{array}{c}{\displaystyle \frac{1}{D_{-{\alpha }_1}^2{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}^2{D}_{{\alpha }_1}{D}_{-s_1{s}_2{\alpha }_1}}}\\ {\displaystyle +\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}{D}_{s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}}+\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}{D}_{-s_1{s}_2{\alpha }_1}{D}_{-s_1{s}_2{s}_1{\alpha }_2}{D}_{s_1{s}_2{\alpha }_1}}}\end{array}\right),\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}}(\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}+\frac{1}{D_{-s_1{\alpha }_2}{D}_{-s_1{\alpha }_1}}+\frac{1}{D_{-s_1{s}_2{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}}+\frac{1}{D_{-s_1{s}_2{s}_1{\alpha }_2}{D}_{-s_1{s}_2{s}_1{\alpha }_1}})}\\ L_{12121}& =& {\displaystyle K_{2121}\frac{1}{D_{-s_2{s}_1{s}_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}{D}_{-s_2{s}_1{\alpha }_2}{D}_{-s_2{s}_1{s}_2{\alpha }_1}},}\\ M_{12121}& =& L_{2121}\frac{1}{D_{-s_1{s}_2{s}_1{s}_2{\alpha }_1}}=D_{R^{-}}\frac{1}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}{D}_{-s_1{s}_2{\alpha }_1}{D}_{-s_1{s}_2{s}_1{\alpha }_2}{D}_{-s_1{s}_2{s}_1{s}_2{\alpha }_1}}\end{array}}$

and the corresponding expressions for reduced words beginning with 2 are obtained by switching 1s and 2s in the above expressions. these formulas are pleasant enough that one can almost guess what happens for all rank 2 cases: Types $I_2\left(m\right)$ and Type $A_1^{\left(1\right)}$.

This formulation allows for efficient computations of products. For example

$\begin{array}{ccc}{Z}_{12}{Z}_{21}& =& {\displaystyle \frac{{\Delta }_{12}{\Gamma }_{21}}{{\Delta }_1}{Z}_1+\frac{{\Delta }_{21}{\Gamma }_{12}}{{\Delta }_2}{Z}_2+\frac{{\Delta }_{12}{\Delta }_{21}-{\Delta }_{12}{\Gamma }_{21}-{\Delta }_{21}{\Gamma }_{12}}{D_{R^{-}}}{Z}_{\text{pt}}}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}}{Z}_1+\frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}}{Z}_2}\\ & & {\displaystyle +\frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}{D}_{-s_2{\alpha }_1}}\frac{(D_{-s_2{\alpha }_1}{D}_{-s_1{\alpha }_2}-D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}-D_{-{\alpha }_2}{D}_{-s_2{\alpha }_1})}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}}{Z}_{\text{pt}}}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}}{Z}_1+\frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_2{\alpha }_1}}{Z}_2}\\ & & {\displaystyle +\frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}{D}_{-s_2{\alpha }_1}}(\left(\frac{D_{-s_2{\alpha }_1}-D_{-{\alpha }_1}}{D_{-{\alpha }_2}}\right)\left(\frac{D_{-s_1{\alpha }_2}-D_{-{\alpha }_2}}{D_{-{\alpha }_1}}\right)-1)Z_{\text{pt}},}\end{array}$

simultaneously generalizing 3 formulas for equivariant cohomology and K-theory given in [GR],

$\left[s_1{s}_2\right]\left[s_2{s}_1\right]=\{\begin{array}{ccc}\{-\left[1\right]\}+\left[s_1\right]+\left[s_2\right],& & \text{in Type}{A}_2,\\ (\left\{a_{11}\right\}+y_{21})\left[1\right]-{\alpha }_{11}\left[s_1\right]-{\alpha }_{21}\left[s_2\right],& & \text{in Type}{B}_2,\\ {\alpha }_{21}{\alpha }_{32}(y_{11}+y_{21}+{\alpha }_{31})\left[1\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right],& & \text{in Type}{G}_2.\end{array}$

Another example is

$\begin{array}{ccc}{Z}_{21}^2& =& {\displaystyle {\Gamma }_{21}{Z}_{21}+\frac{{\Delta }_{21}^2-{\Gamma }_{21}{\Delta }_{21}}{{\Delta }_2}{Z}_2}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-s_1{\alpha }_2}}{Z}_{21}+\frac{D_{R^{-}}}{D_{-{\alpha }_1}{D}_{-{\alpha }_2}{D}_{-s_1{\alpha }_2}}\left(\frac{D_{-s_1{\alpha }_2}-D_{-{\alpha }_2}}{D_{-{\alpha }_1}}\right)Z_2,}\end{array}$

simultaneously generalizing 3 formulas for equivariant cohomology and K-theory given in [GR],

${\left[s_1{s}_2\right]}^2=\{\begin{array}{ccc}-{\alpha }_{01}\left[s_1{s}_2\right]+y_{01}\left[s_2\right],& & \text{in Type}{A}_2,\\ {\alpha }_{01}{\alpha }_{11}\left[s_1{s}_2\right]-{\alpha }_{11}(y_{01}+y_{11})\left[s_2\right],& & \text{in Type}{B}_2,\\ {\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1{s}_2\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}(y_{01}+y_{11}+y_{21})\left[s_2\right],& & \text{in Type}{G}_2.\end{array}$

Switching 1s and 2s in the product $Z_{21}^2$ generalizes 3 more formulas from [GR]:

${\left[s_2{s}_1\right]}^2=\{\begin{array}{ccc}-{\alpha }_{10}\left[s_1{s}_2\right]+y_{10}\left[s_1\right],& & \text{in Type}{A}_2,\\ {\alpha }_{10}{\alpha }_{21}\left[s_2{s}_1\right]-{\alpha }_{21}{y}_{10}\left[s_1\right],& & \text{in Type}{B}_2,\\ {\alpha }_{10}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2{s}_1\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}{y}_{10}\left[s_1\right],& & \text{in Type}{G}_2.\end{array}$

Products

${\displaystyle Z_{\text{pt}}^2=D_{R^{-}}{Z}_{\text{pt}},Z_{\text{pt}}{Z}_1=\frac{D_{R^{-}}}{D_{-10}}{Z}_{\text{pt}},Z_{\text{pt}}{Z}_2=\frac{D_{R^{-}}}{D_{0-1}}{Z}_{\text{pt}},}$

${\displaystyle Z_{\text{pt}}{Z}_{12}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_{\text{pt}},Z_{\text{pt}}{Z}_{21}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_{\text{pt}},}$

${\displaystyle Z_1^2=\frac{D_{R^{-}}}{D_{-10}}{Z}_1,Z_1{Z}_2=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_{\text{pt}},Z_2^2=\frac{D_{R^{-}}}{D_{0-1}}{Z}_{\text{pt}},}$

${\displaystyle Z_{\text{pt}}{Z}_{12}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_{\text{pt}},Z_{\text{pt}}{Z}_{21}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_{\text{pt}}}$

${\displaystyle Z_1{Z}_{12}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_1,Z_1{Z}_{21}=\frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}}{Z}_1+(\frac{D_{R^{-}}}{D_{-10}^2{D}_{0-1}}-\frac{D_{R^{-}}}{D_{-10}^2{D}_{-j-1}})Z_{\text{pt}}}$

and this second formula is rewritten as

${\displaystyle Z_1{Z}_{21}=\frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}}{Z}_1+\frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}{D}_{0-1}}\left(\frac{D_{-j-1}-D_{0-1}}{D_{-10}}\right)Z_{\text{pt}}}$

since

${\displaystyle \frac{D_{R^{-}}}{D_{-10}^2{D}_{0-1}}-\frac{D_{R^{-}}}{D_{-10}^2{D}_{-j-1}}=\frac{D_{R^{-}}}{D_{-10}^2}(\frac{1}{D_{0-1}}-\frac{1}{D_{-j-1}})=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}{D}_{-j-1}}\left(\frac{D_{-j-1}-D_{0-1}}{D_{-10}}\right).}$

Next

${\displaystyle Z_2{Z}_{12}=\frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}}{Z}_2+(\frac{D_{R^{-}}}{D_{0-1}^2{D}_{-10}}-\frac{D_{R^{-}}}{D_{0-1}^2{D}_{-1-1}})Z_{\text{pt}},Z_2{Z}_{21}=\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}}{Z}_2}$

and the first of these can be rewritten as

${\displaystyle Z_2{Z}_{12}=\frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}}{Z}_2+\frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}{D}_{-10}}\left(\frac{D_{-1-1}-D_{-10}}{D_{0-1}}\right)Z_{\text{pt}}.}$

Then

$\begin{array}{ccc}{Z}_{12}^2& =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}}{Z}_{12}+\frac{D_{R^{-}}}{D_{0-1}^2{D}_{-10}}{Z}_1-\frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}{D}_{0-1}}{Z}_1}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}}{Z}_{12}+\frac{D_{R^{-}}}{D_{0-1}^2}(\frac{1}{D_{-10}}-\frac{1}{D_{-1-1}})Z_1}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-1-1}}{Z}_{12}+\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}}\left(\frac{D_{-1-1}-D_{-10}}{D_{0-1}}\right)Z_1,}\end{array}$

and

$\begin{array}{ccc}{Z}_{21}^2& =& {\displaystyle \frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}}{Z}_{21}+\frac{D_{R^{-}}}{D_{-10}^2{D}_{0-1}}{Z}_2-\frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}{D}_{-10}}{Z}_2}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}}{Z}_{21}+\frac{D_{R^{-}}}{D_{-10}^2}(\frac{1}{D_{0-1}}-\frac{1}{D_{-j-1}})Z_2}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{-10}{D}_{-j-1}}{Z}_{21}+\frac{D_{R^{-}}}{D_{-10}{D}_{0-1}{D}_{-j-1}}\left(\frac{D_{-j-1}-D_{0-1}}{D_{-10}}\right)Z_2.}\end{array}$

Then

$\begin{array}{ccc}{Z}_{12}{Z}_{21}& =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-j-1}}{Z}_1+\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}}{Z}_2}\\ & & {\displaystyle +(\frac{D_{R^{-}}}{D_{0-1}^2{D}_{-10}^2}-\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}^2{D}_{-j-1}}-\frac{D_{R^{-}}}{D_{0-1}^2{D}_{-10}{D}_{-1-1}})Z_{\text{pt}}}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-j-1}}{Z}_1+\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}}{Z}_2}\\ & & {\displaystyle +\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}}(\frac{1}{D_{0-1}{D}_{-10}}-\frac{1}{D_{-10}{D}_{-j-1}}-\frac{1}{D_{0-1}{D}_{-1-1}})Z_{\text{pt}}}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-j-1}}{Z}_1+\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}}{Z}_2}\\ & & {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-10}}\left(\frac{D_{-j-1}{D}_{-1-1}-D_{0-1}{D}_{-1-1}-D_{-10}{D}_{-j-1}}{D_{0-1}{D}_{-10}{D}_{-1-1}{D}_{-j-1}}\right)Z_{\text{pt}}}\\ & =& {\displaystyle \frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-j-1}}{Z}_1+\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}}{Z}_2}\\ & & {\displaystyle +\frac{D_{R^{-}}}{D_{0-1}{D}_{-10}{D}_{-1-1}{D}_{-j-1}}\left(\frac{D_{-j-1}{D}_{-1-1}-D_{0-1}{D}_{-1-1}-D_{-10}{D}_{-j-1}}{D_{0-1}{D}_{-10}}\right)Z_{\text{pt}}}\end{array}$

From [Ku, Prop. ???], the singular Schubert varieties in rank 2 groups are

$\begin{array}{ccc}\text{Type}& \text{Singular}& \text{Locus}\\ \\ B_2& X_{s_1{s}_2{s}_1}& X_{s_1}\\ \\ G_2& X_{s_1{s}_2{s}_1}& X_{s_1}\\ G_2& X_{s_1{s}_2{s}_1{s}_2}& X_{s_1{s}_2}\\ G_2& X_{s_2{s}_1{s}_2{s}_1}& X_{s_2{s}_1}\\ G_2& X_{s_1{s}_2{s}_1{s}_2{s}_1}& X_{s_1{s}_2{s}_1}\\ G_2& X_{s_2{s}_1{s}_2{s}_1{s}_2}& X_{s_2}\end{array}$

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