## Schubert Products ${G}_{2}$

Last update: 2 September 2012

## Examples

### Type ${G}_{2}$

Here we will use

${W}_{0}=⟨{s}_{1},\phantom{\rule{0.2em}{0ex}}{s}_{2}\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}{s}_{i}^{2}=1,\phantom{\rule{0.2em}{0ex}}{s}_{1}{s}_{2}{s}_{1}{s}_{2}{s}_{1}{s}_{2}={s}_{2}{s}_{1}{s}_{2}{s}_{1}{s}_{2}{s}_{1}⟩$

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

$\begin{array}{ccc}& \begin{array}{ccc}& {b}_{1}& \\ {b}_{{s}_{1}}& & {b}_{{s}_{2}}\\ {b}_{{s}_{1}{s}_{2}}& & {b}_{{s}_{2}{s}_{1}}\\ {b}_{{s}_{1}{s}_{2}{s}_{1}}& & {b}_{{s}_{2}{s}_{1}{s}_{2}}\\ {b}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}}& & {b}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}}\\ {b}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}{s}_{1}}& & {b}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}{s}_{2}}\\ & {b}_{{w}_{0}}& \\ \\ & {b}_{w}\phantom{\rule{0.2em}{0ex}}\text{basis}& \end{array}& \\ \begin{array}{ccc}& {D}_{-{\alpha }_{1}}& \\ {D}_{{\alpha }_{1}}& & {D}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\\ {D}_{{\alpha }_{1}+{\alpha }_{2}}& & {D}_{-\left(2{\alpha }_{1}+{\alpha }_{2}\right)}\\ {D}_{2{\alpha }_{1}+{\alpha }_{2}}& & {D}_{-\left(2{\alpha }_{1}+{\alpha }_{2}\right)}\\ {D}_{2{\alpha }_{1}+{\alpha }_{2}}& & {D}_{-\left({\alpha }_{1}+{\alpha }_{2}\right)}\\ {D}_{{\alpha }_{1}+{\alpha }_{2}}& & {D}_{-{\alpha }_{1}}\\ & {D}_{{\alpha }_{1}}& \\ \\ & 1\otimes {D}_{-{\alpha }_{1}}& \end{array}& & \begin{array}{ccc}& {D}_{-{\alpha }_{2}}& \\ {D}_{-\left(3{\alpha }_{1}+{\alpha }_{2}\right)}& & {D}_{{\alpha }_{2}}\\ {D}_{-\left(3{\alpha }_{1}+2{\alpha }_{2}\right)}& & {D}_{3{\alpha }_{1}+{\alpha }_{2}}\\ {D}_{-\left(3{\alpha }_{1}+2{\alpha }_{2}\right)}& & {D}_{3{\alpha }_{1}+2{\alpha }_{2}}\\ {D}_{-\left(3{\alpha }_{1}+{\alpha }_{2}\right)}& & {D}_{3{\alpha }_{1}+2{\alpha }_{2}}\\ {D}_{-{\alpha }_{2}}& & {D}_{3{\alpha }_{1}+{\alpha }_{2}}\\ & {D}_{{\alpha }_{2}}& \\ \\ & 1\otimes {D}_{-{\alpha }_{2}}& \end{array}\end{array}$

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

${D}_{{R}^{-}}={D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-\left({\alpha }_{2}+{\alpha }_{1}\right)}{D}_{-\left({\alpha }_{2}+2{\alpha }_{1}\right)}{D}_{-\left({\alpha }_{2}+3{\alpha }_{1}\right)}{D}_{-\left(2{\alpha }_{2}+3{\alpha }_{1}\right)}$

$\begin{array}{ccc}& \begin{array}{ccc}& {D}_{{R}^{-}}& \\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{\text{pt}}& \end{array}& \\ \begin{array}{ccc}& {\Delta }_{1}& \\ {\Delta }_{1}& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{1}& \end{array}& & \begin{array}{ccc}& {\Delta }_{2}& \\ 0& & {D}_{2}\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{2}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{12}& \\ {\Delta }_{12}& & {\Gamma }_{12}\\ {\Gamma }_{12}& & 0\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{12}& \end{array}& & \begin{array}{ccc}& {\Delta }_{21}& \\ {\Gamma }_{21}& & {\Delta }_{21}\\ 0& & {\Gamma }_{21}\\ 0& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{21}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{121}& \\ {\Delta }_{121}& & {\Gamma }_{121}\\ {\Gamma }_{121}& & {K}_{121}\\ {K}_{121}& & 0\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{121}& \end{array}& & \begin{array}{ccc}& {\Delta }_{212}& \\ {\Gamma }_{212}& & {\Delta }_{212}\\ {K}_{212}& & {\Gamma }_{212}\\ 0& & {K}_{212}\\ 0& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{212}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{1212}& \\ {\Delta }_{1212}& & {\Gamma }_{1212}\\ {\Gamma }_{1212}& & {K}_{1212}\\ {K}_{1212}& & {L}_{1212}\\ {L}_{1212}& & 0\\ 0& & 0\\ & 0& \\ \\ & {Z}_{1212}& \end{array}& & \begin{array}{ccc}& {\Delta }_{2121}& \\ {\Gamma }_{2121}& & {\Delta }_{2121}\\ {K}_{2121}& & {\Gamma }_{2121}\\ {L}_{2121}& & {K}_{2121}\\ 0& & {L}_{2121}\\ 0& & 0\\ & 0& \\ \\ & {Z}_{2121}& \end{array}\\ \begin{array}{ccc}& {\Delta }_{12121}& \\ {\Delta }_{12121}& & {\Gamma }_{12121}\\ {\Gamma }_{12121}& & {K}_{12121}\\ {K}_{12121}& & {L}_{12121}\\ {L}_{12121}& & {M}_{12121}\\ {M}_{12121}& & 0\\ & 0& \\ \\ & {Z}_{12121}& \end{array}& & \begin{array}{ccc}& {\Delta }_{21212}& \\ {\Gamma }_{21212}& & {\Delta }_{21212}\\ {K}_{21212}& & {\Gamma }_{21212}\\ {L}_{21212}& & {K}_{21212}\\ {M}_{21212}& & {L}_{21212}\\ 0& & {M}_{21212}\\ & 0& \\ \\ & {Z}_{21212}& \end{array}\end{array}$

Where

$\begin{array}{ccc}{\Delta }_{1}& =& {D}_{{R}^{-}}\frac{1}{{D}_{-{\alpha }_{1}}}\\ {\Delta }_{12}& =& {D}_{{R}^{-}}\frac{1}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\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}^{-}}\left(\frac{1}{{D}_{-{\alpha }_{1}}^{2}{D}_{-{\alpha }_{2}}}+\frac{1}{{D}_{-{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}{D}_{{\alpha }_{1}}}\right),\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}}\left(\frac{1}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}}+\frac{1}{{D}_{-{s}_{1}{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}}\right),\\ {\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}^{-}}\left(\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}}}\right)\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}}\left(\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}}}\right),\\ {\Gamma }_{1212}& =& {\Delta }_{212}\frac{1}{{D}_{-{s}_{2}{\alpha }_{1}}}+{K}_{212}\frac{1}{{D}_{-{s}_{2}{s}_{1}{\alpha }_{1}}}\\ & =& {D}_{{R}^{-}}\left(\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}}}\right)\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}\left(\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}}}\right),\\ {K}_{1212}& =& {\Gamma }_{212}\frac{1}{{D}_{-{s}_{1}{s}_{2}{\alpha }_{1}}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}{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}}}\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}{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}^{-}}\left(\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}}}\right)\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{1}}}\left(\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}}}\right),\\ {\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}}}\\ \phantom{\rule{1em}{0ex}}+\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}\phantom{\rule{1em}{0ex}}\right)\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}\left(\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}}}\right),\\ {K}_{12121}& =& {\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}\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}}}\\ +\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),\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}{D}_{-{s}_{1}{s}_{2}{\alpha }_{1}}}\left(\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}}}\right)\\ {L}_{12121}& =& {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

$\begin{array}{ccc}{Z}_{12}{Z}_{21}& =& \frac{{\Delta }_{12}{\Gamma }_{21}}{{\Delta }_{1}}{Z}_{1}+\frac{{\Delta }_{21}{\Gamma }_{12}}{{\Delta }_{2}}{Z}_{2}+\frac{{\Delta }_{12}{\Delta }_{21}-{\Delta }_{12}{\Gamma }_{21}-{\Delta }_{21}{\Gamma }_{12}}{{D}_{{R}^{-}}}{Z}_{\text{pt}}\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{2}}}{Z}_{1}+\frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}{Z}_{2}\\ & & +\frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}\frac{\left({D}_{-{s}_{2}{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}-{D}_{-{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}-{D}_{-{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}\right)}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}}{Z}_{\text{pt}}\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{2}}}{Z}_{1}+\frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}{Z}_{2}\\ & & +\frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{2}}{D}_{-{s}_{2}{\alpha }_{1}}}\left(\left(\frac{{D}_{-{s}_{2}{\alpha }_{1}}-{D}_{-{\alpha }_{1}}}{{D}_{-{\alpha }_{2}}}\right)\left(\frac{{D}_{-{s}_{1}{\alpha }_{2}}-{D}_{-{\alpha }_{2}}}{{D}_{-{\alpha }_{1}}}\right)-1\right){Z}_{\text{pt}},\end{array}$

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

$\left[{s}_{1}{s}_{2}\right]\left[{s}_{2}{s}_{1}\right]=\left\{\begin{array}{ccc}\left\{-\left[1\right]\right\}+\left[{s}_{1}\right]+\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{A}_{2},\\ \left(\left\{{a}_{11}\right\}+{y}_{21}\right)\left[1\right]-{\alpha }_{11}\left[{s}_{1}\right]-{\alpha }_{21}\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{B}_{2},\\ {\alpha }_{21}{\alpha }_{32}\left({y}_{11}+{y}_{21}+{\alpha }_{31}\right)\left[1\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[{s}_{1}\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{G}_{2}.\end{array}$

Another example is

$\begin{array}{ccc}{Z}_{21}^{2}& =& {\Gamma }_{21}{Z}_{21}+\frac{{\Delta }_{21}^{2}-{\Gamma }_{21}{\Delta }_{21}}{{\Delta }_{2}}{Z}_{2}\\ & =& \frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{s}_{1}{\alpha }_{2}}}{Z}_{21}+\frac{{D}_{{R}^{-}}}{{D}_{-{\alpha }_{1}}{D}_{-{\alpha }_{2}}{D}_{-{s}_{1}{\alpha }_{2}}}\left(\frac{{D}_{-{s}_{1}{\alpha }_{2}}-{D}_{-{\alpha }_{2}}}{{D}_{-{\alpha }_{1}}}\right){Z}_{2},\end{array}$

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

${\left[{s}_{1}{s}_{2}\right]}^{2}=\left\{\begin{array}{ccc}-{\alpha }_{01}\left[{s}_{1}{s}_{2}\right]+{y}_{01}\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{A}_{2},\\ {\alpha }_{01}{\alpha }_{11}\left[{s}_{1}{s}_{2}\right]-{\alpha }_{11}\left({y}_{01}+{y}_{11}\right)\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{B}_{2},\\ {\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[{s}_{1}{s}_{2}\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left({y}_{01}+{y}_{11}+{y}_{21}\right)\left[{s}_{2}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{G}_{2}.\end{array}$

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

${\left[{s}_{2}{s}_{1}\right]}^{2}=\left\{\begin{array}{ccc}-{\alpha }_{10}\left[{s}_{1}{s}_{2}\right]+{y}_{10}\left[{s}_{1}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{A}_{2},\\ {\alpha }_{10}{\alpha }_{21}\left[{s}_{2}{s}_{1}\right]-{\alpha }_{21}{y}_{10}\left[{s}_{1}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{B}_{2},\\ {\alpha }_{10}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[{s}_{2}{s}_{1}\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}{y}_{10}\left[{s}_{1}\right],& & \text{in Type}\phantom{\rule{0.2em}{0ex}}{G}_{2}.\end{array}$

## Products

${Z}_{\text{pt}}^{2}={D}_{{R}^{-}}{Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{\text{pt}}{Z}_{1}=\frac{{D}_{{R}^{-}}}{{D}_{-10}}{Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{\text{pt}}{Z}_{2}=\frac{{D}_{{R}^{-}}}{{D}_{0-1}}{Z}_{\text{pt}},$

${Z}_{\text{pt}}{Z}_{12}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{\text{pt}}{Z}_{21}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{\text{pt}},$

${Z}_{1}^{2}=\frac{{D}_{{R}^{-}}}{{D}_{-10}}{Z}_{1},\phantom{\rule{1em}{0ex}}{Z}_{1}{Z}_{2}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{2}^{2}=\frac{{D}_{{R}^{-}}}{{D}_{0-1}}{Z}_{\text{pt}},$

${Z}_{\text{pt}}{Z}_{12}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{\text{pt}}{Z}_{21}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{\text{pt}}$

${Z}_{1}{Z}_{12}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{1},\phantom{\rule{1em}{0ex}}{Z}_{1}{Z}_{21}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{-j-1}}{Z}_{1}+\left(\frac{{D}_{{R}^{-}}}{{D}_{-10}^{2}{D}_{0-1}}-\frac{{D}_{{R}^{-}}}{{D}_{-10}^{2}{D}_{-j-1}}\right){Z}_{\text{pt}}$

and this second formula is rewritten as

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

since

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

Next

${Z}_{2}{Z}_{12}=\frac{{D}_{{R}^{-}}}{{D}_{0-1}{D}_{-1-1}}{Z}_{2}+\left(\frac{{D}_{{R}^{-}}}{{D}_{0-1}^{2}{D}_{-10}}-\frac{{D}_{{R}^{-}}}{{D}_{0-1}^{2}{D}_{-1-1}}\right){Z}_{\text{pt}},\phantom{\rule{1em}{0ex}}{Z}_{2}{Z}_{21}=\frac{{D}_{{R}^{-}}}{{D}_{-10}{D}_{0-1}}{Z}_{2}$

and the first of these can be rewritten as

${Z}_{2}{Z}_{12}=\frac{{D}_{{R}^{-}}}{{D}_{0-1}{D}_{-1-1}}{Z}_{2}+\frac{{D}_{{R}^{-}}}{{D}_{0-1}{D}_{-1-1}{D}_{-10}}\left(\frac{{D}_{-1-1}-{D}_{-10}}{{D}_{0-1}}\right){Z}_{\text{pt}}.$

Then

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

and

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

Then

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

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

$\begin{array}{ccc}\text{Type}& \text{Singular}& \text{Locus}\\ \phantom{\rule{0ex}{0.7ex}}\\ {B}_{2}& {X}_{{s}_{1}{s}_{2}{s}_{1}}& {X}_{{s}_{1}}\\ \phantom{\rule{0ex}{0.7ex}}\\ {G}_{2}& {X}_{{s}_{1}{s}_{2}{s}_{1}}& {X}_{{s}_{1}}\\ {G}_{2}& {X}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}}& {X}_{{s}_{1}{s}_{2}}\\ {G}_{2}& {X}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}}& {X}_{{s}_{2}{s}_{1}}\\ {G}_{2}& {X}_{{s}_{1}{s}_{2}{s}_{1}{s}_{2}{s}_{1}}& {X}_{{s}_{1}{s}_{2}{s}_{1}}\\ {G}_{2}& {X}_{{s}_{2}{s}_{1}{s}_{2}{s}_{1}{s}_{2}}& {X}_{{s}_{2}}\end{array}$