Rank two and a positivity conjecture

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

Last update: 25 February 2013

Rank two and a positivity conjecture

In this section we will give explicit formulas for the rank two root systems. The data supports the following positivity conjecture which generalizes the theorems of Brion [Bri2002, formula before Theorem 1] and Graham [Gra9908172, Corollary 4.1].

Conjecture 5.9. For $\beta \in R^{+}$ let $y_{\beta }=e^{-\beta }$ and ${\alpha }_{\beta }=e^{-\beta }-1$ and let $d\left(w\right)=\ell \left(w_0\right)-\ell \left(w\right)$ for $w\in W\text{.}$ Let $c_{wv}^z$ be the structure constants of $K_T(G/B)$ with respect to the basis $\{\left[{𝒪}_{X_w}\right]\hspace{0.17em}\mid \hspace{0.17em}w\in W\}$ as defined in (0.1). Then

$$c_{wv}^z={(-1)}^{d\left(w\right)+d\left(v\right)-d\left(z\right)}f(\alpha ,y),\text{where}\hspace{0.17em}f(\alpha ,y)\in {\mathbb{Z}}_{\ge 0}[{\alpha }_{\beta },y_{\beta }\hspace{0.17em}\mid \hspace{0.17em}\beta \in R^{+}],$$

that is, $f(\alpha ,y)$ is a polynomial in the variables ${\alpha }_{\beta }$ and $y_{\beta },$ $\beta \in R^{+},$ which has non-negative integral coefficients.

In the following, for brevity, use the following notations:

$$\begin{array}{c}\text{in}\hspace{0.17em}{K}_T(G/B),\left[w\right]=\left[{𝒪}_{X_w}\right],{\alpha }_{rs}=e^{-(r{\alpha }_1+s{\alpha }_2)}-1,\text{and}{y}_{rs}=e^{-(r{\alpha }_1+s{\alpha }_2)},\\ \text{in}\hspace{0.17em}K(G/B),\left[w\right]=\left[{𝒪}_{X_w}\right],{\alpha }_{rs}=0,\text{and}{y}_{rs}=1,\\ \text{in}\hspace{0.17em}{H}_T^{*}(G/B),\left[w\right]=\left[X_w\right],{\alpha }_{rs}=r{\alpha }_1+s{\alpha }_2,,\text{and}{y}_{rs}=1,\\ \text{in}\hspace{0.17em}{H}^{*}(G/B),\left[w\right]=\left[X_w\right],{\alpha }_{rs}=0,,\text{and}{y}_{rs}=1,\end{array}$$

and in $H_T^{*}(G/B)$ and in $H^{*}(G/B)$ the terms in $\left\{\right\}$ brackets do not appear.

Type $A_2\text{.}$ For the root system $R$ of type $A_2$

$$\begin{array}{c}{\alpha }_1=-{\omega }_1+2{\omega }_2,{\lambda }_1=\rho ,{\lambda }_{s_1}={\omega }_2=\frac{1}{3}{\alpha }_1+\frac{2}{3}{\alpha }_2,{\lambda }_{s_2{s}_1}=s_2{\omega }_2=\frac{1}{3}{\alpha }_1-\frac{1}{3}{\alpha }_2,\\ {\alpha }_2=2{\omega }_1-{\omega }_2,{\lambda }_{w_0}=0,{\lambda }_{s_2}={\omega }_1=\frac{2}{3}{\alpha }_1+\frac{1}{3}{\alpha }_2,{\lambda }_{s_1{s}_2}=s_1{\omega }_1=-\frac{1}{3}{\alpha }_1+\frac{1}{3}{\alpha }_2\text{.}\end{array}$$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$$\begin{array}{c}\left[s_1{s}_2{s}_1\right]=1,\left[1\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2\right],\\ \left[s_2{s}_1\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_2},\left[s_1{s}_2\right]=1-e^{-{\omega }_2}{X}^{-{\omega }_1}\\ \left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2{s}_1\right],\left[s_2\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1{s}_2\right],\end{array}$$

and

$$\begin{array}{c}\left[s_1{s}_2{s}_1\right]=1,\left[s_1{s}_2\right]=1-e^{-{\omega }_2}{X}^{-{\omega }_1},\left[s_2{s}_1\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_2},\\ \left[s_1\right]=1-e^{-{\omega }_2}{X}^{-s_1{\omega }_1}-e^{-{\omega }_2}{X}^{-{\omega }_1}+e^{-2{\omega }_2}{X}^{-{\omega }_2},\\ \left[s_2\right]=1-e^{-{\omega }_1}{X}^{-s_2{\omega }_2}-e^{-{\omega }_1}{X}^{-{\omega }_2}+e^{-2{\omega }_1}{X}^{-{\omega }_1},\\ \left[1\right]=1-e^{-{\omega }_2}{X}^{-s_1{\omega }_1}-e^{-{\omega }_1}{X}^{-s_2{\omega }_2}+e^{-2{\omega }_1}{X}^{-{\omega }_1}+e^{-2{\omega }_2}{X}^{-{\omega }_2}-e^{-\rho }{X}^{-\rho }\text{.}\end{array}$$

The multiplication of the Schubert classes is given by

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

Type $B_2\text{.}$ For the root system $R$ of type $B_2$

$$\begin{array}{c}{\alpha }_1=2{\omega }_1-{\omega }_2,{\lambda }_1=\rho =2{\alpha }_1+\frac{3}{2}{\alpha }_2,{\lambda }_{s_1}={\omega }_2={\alpha }_1+{\alpha }_2,\\ {\alpha }_2=-2{\omega }_1+2{\omega }_2,{\lambda }_{w_0}=0,{\lambda }_{s_2}={\omega }_1={\alpha }_1+\frac{1}{2}{\alpha }_2,\\ {\lambda }_{s_2{s}_1}=s_2{\omega }_2={\alpha }_1,{\lambda }_{s_1{s}_2{s}_1}=s_1{s}_2{\omega }_2=-{\alpha }_1,\\ {\lambda }_{s_1{s}_2}=s_1{\omega }_1=\frac{1}{2}{\alpha }_2,{\lambda }_{s_2{s}_1{s}_2}=s_2{s}_1{\omega }_1=-\frac{1}{2}{\alpha }_2\text{.}\end{array}$$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$$\begin{array}{c}\left[s_1{s}_2{s}_1{s}_2\right]=1,\left[1\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2\right],\\ \left[s_1{s}_2{s}_1\right]=1-e^{-{\omega }_2}{X}^{-{\omega }_2},\left[s_2{s}_1{s}_2\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_1},\\ \left[s_2{s}_1\right]=(1-e^{-{\omega }_1}{X}^{-s_1{\omega }_1})\left[s_2{s}_1{s}_2\right],\left[s_2{s}_1\right]=(1-e^{s_2{s}_1{\omega }_1}{X}^{-{\omega }_1})\left[s_2{s}_1{s}_2\right],\\ \left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2{s}_1\right],\left[s_2\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1{s}_2\right],\end{array}$$

and

$$\begin{array}{c}\left[s_1{s}_2{s}_1{s}_2\right]=1,\left[s_1{s}_2{s}_1\right]=1-e^{-{\omega }_2}{X}^{-{\omega }_2},\left[s_2{s}_1{s}_2\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_1},\\ \left[s_1{s}_2\right]=(1-e^{-{\omega }_2})-e^{-{\omega }_2}{X}^{-{\omega }_2}-e^{-{\omega }_2}{X}^{-s_2{\omega }_2}+(e^{-\rho }+e^{-s_1\rho })X^{-{\omega }_1},\\ \left[s_2{s}_1\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_1}-e^{-{\omega }_1}{X}^{-s_1{\omega }_1}+e^{-2{\omega }_1}{X}^{-{\omega }_2},\\ \left[s_1\right]=(1-e^{-{\omega }_2})+(e^{-\rho }+e^{-s_1\rho })X^{-s_1{\omega }_1}+(e^{-\rho }+e^{-s_1\rho })X^{-{\omega }_1}\\ -e^{-{\omega }_2}{X}^{-s_1{s}_2{\omega }_2}-e^{-{\omega }_2}{X}^{-s_2{\omega }_2}-(e^{-2{\omega }_2}+e^{-{\omega }_2})X^{-{\omega }_2},\\ \left[s_2\right]=(1+e^{-2{\omega }_1})+e^{-2{\omega }_1}{X}^{-s_2{\omega }_2}+e^{-2{\omega }_1}{X}^{-{\omega }_2}\\ -e^{-{\omega }_1}{X}^{-s_2{s}_1{\omega }_1}-e^{-{\omega }_1}{X}^{-s_1{\omega }_1}-(e^{-3{\omega }_1}+e^{-{\omega }_1})X^{-{\omega }_1},\\ \left[1\right]=(1+e^{-2{\omega }_1})-e^{-{\omega }_1}{X}^{-s_2{s}_1{\omega }_1}+(e^{-\rho }+e^{-s_1\rho })X^{-s_1{\omega }_1}-(e^{-3{\omega }_1}+e^{-{\omega }_1})X^{-{\omega }_1}\\ -e^{-{\omega }_2}{X}^{-s_1{s}_2{\omega }_2}+e^{-2{\omega }_1}{X}^{-s_2{\omega }_2}-(e^{-2{\omega }_2}+e^{-{\omega }_2})X^{-{\omega }_2}+e^{-\rho }{X}^{-\rho }\text{.}\end{array}$$

The multiplication of the Schubert classes is given by

$$\begin{array}{c}{\left[1\right]}^2={\alpha }_{10}{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[1\right]\left[s_1\right]=-{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[1\right]\left[s_2\right]-{\alpha }_{10}{\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[1\right]\left[s_1{s}_2\right]={\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[1\right]\left[s_2{s}_1\right]={\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[1\right]\left[s_1{s}_2{s}_1\right]=-{\alpha }_{11}(1+y_{11})\left[1\right],\\ \left[1\right]\left[s_2{s}_1{s}_2\right]=-{\alpha }_{21}\left[1\right],\\ \\ {\left[s_1\right]}^2=-{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}\left[s_1\right],\\ \left[s_1\right]\left[s_2\right]={\alpha }_{11}{\alpha }_{21}\left[1\right],\\ \left[s_1\right]\left[s_1{s}_2\right]=-{\alpha }_{11}(y_{01}+y_{11})\left[1\right]+{\alpha }_{01}{\alpha }_{11}\left[s_1\right],\\ \left[s_1\right]\left[s_2{s}_1\right]={\alpha }_{11}{\alpha }_{21}\left[s_1\right],\\ \left[s_1\right]\left[s_1{s}_2{s}_1\right]=-{\alpha }_{11}(1+y_{11})\left[s_1\right],\\ \left[s_1\right]\left[s_2{s}_1{s}_2\right]=y_{11}\left[1\right]-{\alpha }_{11}\left[s_1\right],\\ \\ {\left[s_2\right]}^2=-{\alpha }_{10}{\alpha }_{11}{\alpha }_{21}\left[s_2\right],\\ \left[s_2\right]\left[s_1{s}_2\right]={\alpha }_{11}{\alpha }_{21}\left[s_2\right],\\ \left[s_2\right]\left[s_2{s}_1\right]=-{\alpha }_{21}{y}_{10}\left[1\right]+{\alpha }_{10}{\alpha }_{21}\left[s_2\right],\\ \left[s_2\right]\left[s_1{s}_2{s}_1\right]=y_{21}\left[1\right]-{\alpha }_{21}\left[s_2\right],\\ \left[s_2\right]\left[s_2{s}_1{s}_2\right]=-{\alpha }_{21}\left[s_2\right],\\ \\ {\left[s_1{s}_2\right]}^2=-{\alpha }_{11}(y_{01}+y_{11})\left[s_2\right]+{\alpha }_{01}{\alpha }_{11}\left[s_1{s}_2\right],\\ \left[s_1{s}_2\right]\left[s_2{s}_1\right]=(\left\{{\alpha }_{11}\right\}+y_{21})\left[1\right]-{\alpha }_{11}\left[s_1\right]-{\alpha }_{21}\left[s_2\right],\\ \left[s_1{s}_2\right]\left[s_1{s}_2{s}_1\right]=\{-(y_{01}+y_{11})\left[1\right]\}+y_{01}\left[s_1\right]+(y_{11}+y_{12})\left[s_2\right]-{\alpha }_{01}\left[s_1{s}_2\right],\\ \left[s_1{s}_2\right]\left[s_2{s}_1{s}_2\right]=y_{11}\left[s_2\right]-{\alpha }_{11}\left[s_1{s}_2\right],\\ \\ {\left[s_2{s}_1\right]}^2=-{\alpha }_{21}{y}_{10}\left[s_1\right]+{\alpha }_{10}{\alpha }_{21}\left[s_2{s}_1\right],\\ \left[s_2{s}_1\right]\left[s_1{s}_2{s}_1\right]=y_{21}\left[s_1\right]-{\alpha }_{21}\left[s_2{s}_1\right],\\ \left[s_2{s}_1\right]\left[s_2{s}_1{s}_2\right]=\{-y_{10}\left[1\right]\}+y_{10}\left[s_1\right]+y_{10}\left[s_2\right]-{\alpha }_{10}\left[s_2{s}_1\right],\\ \\ {\left[s_1{s}_2{s}_1\right]}^2=\{-y_{11}\left[s_1\right]\}+(y_{01}+y_{11})\left[s_2{s}_1\right]-{\alpha }_{01}\left[s_1{s}_2{s}_1\right],\\ \left[s_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2\right]=\{\left[1\right]-\left[s_1\right]-\left[s_2\right]\}+\left[s_1{s}_2\right]+\left[s_2{s}_1\right],\\ \\ {\left[s_2{s}_1{s}_2\right]}^2=y_{10}\left[s_1{s}_2\right]-{\alpha }_{10}\left[s_2{s}_1{s}_2\right]\text{.}\end{array}$$

Type $G_2\text{.}$ For the root system $R$ of type $G_2$

$$\begin{array}{c}{\lambda }_1=\rho =5\alpha +3{\alpha }_2,{\lambda }_{s_1{s}_2{s}_1}=s_1{s}_2{\omega }_2={\alpha }_2,\\ {\lambda }_{s_1}={\omega }_2=3{\alpha }_1+2{\alpha }_2,{\lambda }_{s_2{s}_1{s}_2{s}_1}=s_2{s}_1{s}_2{\omega }_2=-{\alpha }_2,\\ {\lambda }_{s_2}={\omega }_1=2{\alpha }_1+{\alpha }_2,{\lambda }_{s_1{s}_2{s}_1{s}_2}=s_1{s}_2{s}_1{\omega }_1=-{\alpha }_1,\\ {\lambda }_{s_2{s}_1}=s_2{\omega }_2=3{\alpha }_1+{\alpha }_2,{\lambda }_{s_1{s}_2{s}_1{s}_2{s}_1}=s_1{s}_2{s}_1{s}_2{\omega }_2=-3{\alpha }_1-{\alpha }_2,\\ {\lambda }_{s_1{s}_2}=s_1{\omega }_1={\alpha }_1+{\alpha }_2,{\lambda }_{s_2{s}_1{s}_2{s}_1{s}_2}=s_2{s}_1{s}_2{s}_1{\omega }_1=-{\alpha }_1-{\alpha }_2,\\ {\lambda }_{s_2{s}_1{s}_2}=s_2{s}_1{\omega }_1={\alpha }_1,{\lambda }_{w_0}=0\text{.}\end{array}$$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$$\begin{array}{c}\left[s_1{s}_2{s}_1{s}_2{s}_1{s}_2\right]=1,\left[1\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2\right],\\ \left[s_1{s}_2{s}_1{s}_2{s}_1\right]=1-e^{-{\omega }_2}{X}^{-{\omega }_2},\left[s_2{s}_1{s}_2{s}_1{s}_2\right]=1-e^{-{\omega }_1}{X}^{-{\omega }_1},\\ \left[s_2{s}_1{s}_2{s}_1\right]=(1-e^{-{\omega }_1}{X}^{-s_1{\omega }_1})\left[s_2{s}_1{s}_2{s}_1{s}_2\right],\\ \left[s_1{s}_2{s}_1{s}_2\right]=(1-e^{-s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_2{s}_1{s}_2{s}_1{s}_2\right],\\ \left[s_1{s}_2{s}_1\right]=\text{see below,}\left[s_2{s}_1{s}_2\right]=\frac{1-e^{s_2{s}_2{\omega }_1}{X}^{-{\omega }_1}}{1+X^{-{\omega }_1}}\left[s_1{s}_2{s}_1{s}_2\right],\\ \left[s_2{s}_1\right]=(1-e^{-{\omega }_1}{X}^{-s_1{s}_2{s}_1{\omega }_1})\left[s_2{s}_1{s}_2\right],\left[s_1{s}_2\right]=(1-e^{s_2{s}_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1{s}_2\right],\\ \left[s_1\right]=(1-e^{s_2{\omega }_2}{X}^{-{\omega }_2})\left[s_2{s}_1\right],\left[s_2\right]=(1-e^{s_1{\omega }_1}{X}^{-{\omega }_1})\left[s_1{s}_2\right],\\ \left[s_1{s}_2{s}_1\right]=\frac{(1-e^{-{\alpha }_2}{X}^{-{\omega }_2})\left[s_2{s}_1{s}_2{s}_1\right]+e^{-{\alpha }_2}(1+e^{{\omega }_1}{X}^{-{\omega }_2})\left[s_2{s}_1\right]}{1+e^{-{\alpha }_2}},\end{array}$$

and

$$\begin{array}{ccc}\left[w_0\right]& =& 1,\\ \left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& 1-y_{21}{X}^{-{\omega }_1},\\ \left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& 1-y_{32}{X}^{-{\omega }_2},\\ \left[s_2{s}_1{s}_2{s}_1\right]& =& 1-y_{21}{X}^{-{\omega }_1}-y_{21}{X}^{-s_1{\omega }_1}+y_{42}{X}^{-{\omega }_2},\\ \left[s_1{s}_2{s}_1{s}_2\right]& =& (1-y_{32})+(y_{22}+y_{42}+y_{43}+y_{53})X^{-{\omega }_1}-y_{32}{X}^{-s_1{\omega }_1}\\ & & -y_{32}{X}^{-s_2{s}_1{\omega }_1}-y_{32}{X}^{-{\omega }_2}-y_{32}{X}^{-s_2{\omega }_2},\\ \left[s_2{s}_1{s}_2\right]& =& (1-y_{21}+y_{42})+(y_{42}-y_{21}-y_{52}-y_{53}-y_{63})X^{-{\omega }_1}\\ & & +(y_{42}-y_{21})X^{-s_1{\omega }_1}+(y_{42}-y_{21})X^{-s_2{s}_1{\omega }_1}+y_{42}{X}^{-{\omega }_2}+y_{42}{X}^{-s_2{\omega }_2},\\ \left[s_1{s}_2{s}_1\right]& =& (1-2y_{32})+(y_{22}+y_{42}+y_{43}+y_{53})X^{-{\omega }_1}\\ & & +(y_{22}+y_{42}+y_{43}+y_{53})X^{-s_1{\omega }_1}-y_{32}{X}^{-s_2{s}_1{\omega }_1}-y_{32}{X}^{-s_1{s}_2{s}_1{\omega }_1}\\ & & -(y_{32}+y_{43}+y_{53})X^{-{\omega }_2}-y_{32}{X}^{-s_2{\omega }_2}-y_{32}{X}^{-s_1{s}_2{\omega }_2},\\ \left[s_2{s}_1\right]& =& (1-y_{21}+2y_{42})+(y_{42}-y_{21}-y_{52}-y_{53}-y_{63})X^{-{\omega }_1}\\ & & +(y_{42}-y_{21}-y_{32}-y_{53}-y_{63})X^{-s_1{\omega }_1}+(y_{42}-y_{21})X^{-s_2{s}_1{\omega }_1}\\ & & +(y_{42}-y_{21})X^{-s_1{s}_2{s}_1{\omega }_1}+(y_{42}+y_{63})X^{-{\omega }_2}+y_{42}{X}^{-s_2{\omega }_2}+y_{42}{X}^{-s_1{s}_2{\omega }_2},\\ \left[s_1{s}_2\right]& =& 1-y_{11}-y_{21}-y_{32}-y_{43}-y_{53}+(y_{22}+y_{32})(1+y_{10}+y_{20})X^{-{\omega }_1}\\ & & +(y_{22}+y_{32}+y_{42})X^{-s_1{\omega }_1}+(y_{22}+y_{32}+y_{42})X^{-s_2{s}_1{\omega }_1}\\ & & -(y_{32}+y_{43}+y_{53})X^{-{\omega }_2}-(y_{32}+y_{43}+y_{53})X^{-s_2{\omega }_2}-y_{32}{X}^{-s_1{s}_2{\omega }_2}\\ & & -y_{32}{X}^{-s_2{s}_1{s}_2{\omega }_2},\\ \left[s_2\right]& =& (1+y_{31}+y_{32}+2y_{42}+y_{63})-(y_{21}+y_{52}+y_{53}+y_{84})X^{-{\omega }_1}\\ & & -(y_{21}+y_{52}+y_{53})X^{-s_1{\omega }_1}-(y_{21}+y_{52}+y_{53})X^{-s_2{s}_1{\omega }_1}-y_{21}{X}^{-s_1{s}_2{s}_1{\omega }_1}\\ & & -y_{21}{X}^{-s_2{s}_1{s}_2{s}_1{\omega }_1}+(y_{42}+y_{63})X^{-{\omega }_2}+(y_{42}+y_{63})X^{-s_2{\omega }_2}\\ & & +y_{42}{X}^{-s_1{s}_2{\omega }_2}+y_{42}{X}^{-s_2{s}_1{s}_2{\omega }_2},\\ \left[s_1\right]& =& 1-(y_{11}+y_{21}+y_{32}+2y_{43}+2y_{53})+(y_{22}+y_{54})(1+y_{10}+y_{20})X^{-{\omega }_1}\\ & & +(y_{22}+y_{54})(1+y_{10}+y_{20})X^{-s_1{\omega }_1}+(y_{22}+y_{32}+y_{42})X^{-s_2{s}_1{\omega }_1}\\ & & +(y_{22}+y_{32}+y_{42})X^{-s_1{s}_2{s}_1{\omega }_1}-(y_{32}+y_{43}+y_{53}+y_{64})X^{-{\omega }_2}\\ & & -(y_{32}+y_{43}+y_{53})X^{-s_2{\omega }_2}-(y_{32}+y_{43}+y_{53})X^{-s_1{s}_2{\omega }_2}-y_{32}{X}^{-s_2{s}_1{s}_2{\omega }_2}\\ & & -y_{32}{X}^{-s_1{s}_2{s}_1{s}_2{\omega }_2},\\ \left[1\right]& =& (1+y_{31}+y_{42}+y_{63}-y_{53}-y_{43})-y_{21}{(1+y_{32})}^2{X}^{-{\omega }_1}\\ & & +y_{22}(1+y_{10}+y_{20})(1+y_{21}+y_{31})X^{-s_1{\omega }_1}-(y_{21}+y_{52}+y_{53})X^{-s_2{s}_1{\omega }_1}\\ & & +y_{22}{X}^{-s_1{s}_2{s}_1{\omega }_1}-y_{21}{X}^{-s_2{s}_1{s}_2{s}_1{\omega }_1}-y_{32}(1+y_{11})(1+y_{21})X^{-{\omega }_2}\\ & & +(y_{42}+y_{63})X^{-s_2{\omega }_2}-(y_{32}+y_{43}+y_{53})X^{-s_1{s}_2{\omega }_2}+y_{42}{X}^{-s_2{s}_1{s}_2{\omega }_2}\\ & & -y_{32}{X}^{-s_1{s}_2{s}_1{\omega }_2{\omega }_2}+y_{53}{X}^{-\rho }\text{.}\end{array}$$

The multiplication of the Schubert classes is given by

$$\begin{array}{ccc}{\left[1\right]}^2& =& {\alpha }_{10}{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_1\right]& =& -{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_2\right]& =& -{\alpha }_{10}{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_1{s}_2\right]& =& {\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_2{s}_1\right]& =& {\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_1{s}_2{s}_1\right]& =& -{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}(1+y_{11}+y_{21})\left[1\right],\\ \left[1\right]\left[s_2{s}_1{s}_2\right]& =& -{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right],\\ \left[1\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& {\alpha }_{31}{\alpha }_{32}(1+y_{21})\left[1\right],\\ \left[1\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& {\alpha }_{31}{\alpha }_{32}(1+y_{21})\left[1\right],\\ \left[1\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& -{\alpha }_{32}(1+y_{32})\left[1\right],\\ \left[1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& -{\alpha }_{21}(1+y_{21})\left[1\right],\\ \\ {\left[s_1\right]}^2& =& -{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_1\right]\\ \left[s_1\right]\left[s_2\right]& =& {\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[1\right]\\ \left[s_1\right]\left[s_1{s}_2\right]& =& -{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}(y_{01}+y_{11}+y_{21})\left[1\right]+{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1\right]\\ \left[s_1\right]\left[s_2{s}_1\right]& =& {\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_1\right]\\ \left[s_1\right]\left[s_1{s}_2{s}_1\right]& =& -{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}(1+y_{11}+y_{21})\left[s_1\right]\\ \left[s_1\right]\left[s_2{s}_1{s}_2\right]& =& {\alpha }_{21}{\alpha }_{32}(y_{11}+y_{21})\left[1\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1\right]\\ \left[s_1\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& -{\alpha }_{32}(y_{22}+y_{32})\left[1\right]+{\alpha }_{11}{\alpha }_{32}(1+y_{11})\left[s_1\right]\\ \left[s_1\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& {\alpha }_{21}{\alpha }_{32}(1+y_{21})\left[s_1\right]\\ \left[s_1\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& -{\alpha }_{32}(1+y_{32})\left[s_1\right]\\ \left[s_1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& y_{32}\left[1\right]-{\alpha }_{32}\left[s_1\right]\\ \\ {\left[s_2\right]}^2& =& -{\alpha }_{10}{\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right]\\ \left[s_2\right]\left[s_1{s}_2\right]& =& {\alpha }_{11}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right]\\ \left[s_2\right]\left[s_2{s}_1\right]& =& -{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}{y}_{10}\left[1\right]+{\alpha }_{10}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right]\\ \left[s_2\right]\left[s_1{s}_2{s}_1\right]& =& {\alpha }_{21}{\alpha }_{32}(y_{21}+y_{31})\left[1\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right]\\ \left[s_2\right]\left[s_2{s}_1{s}_2\right]& =& -{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2\right]\\ \left[s_2\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& {\alpha }_{21}{\alpha }_{32}(1+y_{21})\left[s_2\right]\\ \left[s_2\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& -{\alpha }_{21}(y_{31}+y_{52})\left[1\right]+{\alpha }_{21}{\alpha }_{31}(1+y_{21})\left[s_2\right]\\ \left[s_2\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& y_{63}\left[1\right]-{\alpha }_{21}(1+y_{21}+y_{42})\left[s_2\right]\\ \left[s_2\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& -{\alpha }_{21}(1+y_{21})\left[s_2\right]\\ \\ {\left[s_1{s}_2\right]}^2& =& -{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}(y_{01}+y_{11}+y_{21})\left[s_2\right]+{\alpha }_{01}{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_2{s}_1\right]& =& {\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]\\ \left[s_1{s}_2\right]\left[s_1{s}_2{s}_1\right]& =& -{\alpha }_{32}(y_{32}+y_{42}\{+{\alpha }_{11}(y_{01}+2y_{11}+y_{21})\})\left[1\right]\\ & & +{\alpha }_{11}{\alpha }_{32}(y_{01}+y_{11})\left[s_1\right]+({\alpha }_{31}{\alpha }_{32}{y}_{11}+{\alpha }_{11}{\alpha }_{32}(y_{01}+y_{11}+y_{21}))\left[s_2\right]\\ & & -{\alpha }_{01}{\alpha }_{11}{\alpha }_{32}\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_2{s}_1{s}_2\right]& =& {\alpha }_{21}{\alpha }_{32}(y_{11}+y_{21})\left[s_2\right]-{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& -{\alpha }_{32}(y_{22}+y_{32})\left[s_2\right]+{\alpha }_{11}{\alpha }_{32}(1+y_{11})\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& \left(y_{63}\{+{\alpha }_{32}(y_{11}+y_{21})\}\right)\left[1\right]-{\alpha }_{32}{y}_{11}\left[s_1\right]\\ & & -({\alpha }_{32}(y_{11}+y_{21})+{\alpha }_{31}{y}_{32})\left[s_2\right]+{\alpha }_{11}{\alpha }_{32}\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& \{-(y_{33}+y_{43}+y_{53})\left[1\right]\}+y_{33}\left[s_1\right]+(y_{33}+y_{43}+y_{53})\left[s_2\right]\\ & & -{\alpha }_{11}(1+y_{11}+y_{22})\left[s_1{s}_2\right]\\ \left[s_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& y_{32}\left[s_2\right]-{\alpha }_{32}\left[s_1{s}_2\right]\\ \\ {\left[s_2{s}_1\right]}^2& =& -{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}{y}_{10}\left[s_1\right]{\alpha }_{10}{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_1{s}_2{s}_1\right]& =& {\alpha }_{21}{\alpha }_{31}(y_{21}+y_{31})\left[s_1\right]-{\alpha }_{21}{\alpha }_{31}{\alpha }_{32}\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_2{s}_1{s}_2\right]& =& -{\alpha }_{21}(y_{51}+y_{52}\{+{\alpha }_{31}{y}_{10}\})\left[1\right]+{\alpha }_{21}({\alpha }_{10}{y}_{31}+{\alpha }_{32}{y}_{10})\left[s_1\right]\\ & & +{\alpha }_{21}{\alpha }_{31}(y_{10}+y_{21})\left[s_2\right]-{\alpha }_{10}{\alpha }_{21}{\alpha }_{31}\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& \left(y_{62}\{+{\alpha }_{31}(y_{21}+y_{31})\}\right)\left[1\right]-({\alpha }_{31}{y}_{21}+{\alpha }_{10}(y_{31}+y_{41}))\left[s_1\right]\\ & & -({\alpha }_{31}{y}_{21}+{\alpha }_{32}{y}_{31})\left[s_2\right]+{\alpha }_{21}{\alpha }_{31}\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& -{\alpha }_{21}(y_{31}+y_{52})\left[s_1\right]+{\alpha }_{21}{\alpha }_{31}(1+y_{21})\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& y_{63}\left[s_1\right]-{\alpha }_{21}(1+y_{21}+y_{42})\left[s_2{s}_1\right]\\ \left[s_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& \{-y_{31}\left[1\right]\}+y_{31}\left[s_1\right]+y_{31}\left[s_2\right]-{\alpha }_{31}\left[s_2{s}_1\right]\\ \\ {\left[s_1{s}_2{s}_1\right]}^2& =& -{\alpha }_{32}(y_{32}+y_{42}\{+{\alpha }_{11}(y_{11}+y_{21})\})\left[s_1\right]\\ & & +({\alpha }_{11}{\alpha }_{32}(y_{01}+y_{11}+y_{21})+{\alpha }_{31}{\alpha }_{32}{y}_{11})\left[s_2{s}_1\right]-{\alpha }_{01}{\alpha }_{11}{\alpha }_{32}\left[s_1{s}_2{s}_1\right]\\ \left[s_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2\right]& =& \left(1\{+{\alpha }_{11}(y_{11}+y_{22}+y_{33}+y_{31}+y_{42})+{\alpha }_{31}(y_{21}+y_{32})+{\alpha }_{32}{y}_{21}\}\right)\left[1\right]\\ & & -({\alpha }_{11}(y_{21}+{\alpha }_{32})+{\alpha }_{10}(y_{31}+y_{41}+y_{32}+y_{42}))\left[s_1\right]\\ & & -({\alpha }_{31}(y_{21}+y_{32})+{\alpha }_{11}(y_{21}+y_{32}+y_{31}+{\alpha }_{42}))\left[s_2\right]\\ & & +{\alpha }_{11}+{\alpha }_{32}\left[s_1{s}_2\right]+{\alpha }_{21}{\alpha }_{31}\left[s_2{s}_1\right]\\ \left[s_1{s}_2{s}_1\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& \{-(y_{33}+2y_{43}+y_{53}+{\alpha }_{11}(y_{01}+y_{11})+{\alpha }_{21}(y_{11}+y_{21}))\left[1\right]\}\\ & & +(y_{33}+y_{43}\{+{\alpha }_{11}(y_{01}+y_{11})+{\alpha }_{21}(y_{11}+y_{21})\})\left[s_1\right]\\ & & \left((y_{33}+y_{43}+y_{53})\{+{\alpha }_{11}(y_{01}+y_{11})+{\alpha }_{21}(y_{11}+y_{21})\}\right)\left[s_2\right]\\ & & -{\alpha }_{11}(y_{01}+y_{11}+y_{22})\left[s_1{s}_2\right]-({\alpha }_{11}(y_{01}+y_{11})+{\alpha }_{21}(y_{11}+y_{21}))\left[s_2{s}_1\right]\\ & & +{\alpha }_{01}{\alpha }_{11}\left[s_1{s}_2{s}_1\right]\\ \left[s_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& \left(y_{62}\{+{\alpha }_{32}{y}_{21}\}\right)\left[s_1\right]-({\alpha }_{31}{y}_{32}+{\alpha }_{32}(y_{11}+y_{21}))\left[s_2{s}_1\right]\\ & & +{\alpha }_{11}{\alpha }_{32}\left[s_1{s}_2{s}_1\right]\\ \left[s_1{s}_2{s}_1\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& \{-(y_{43}+y_{53})\left[s_1\right]\}+(y_{33}+y_{43}+y_{53})\left[s_2{s}_1\right]\\ & & -{\alpha }_{11}(1+y_{11}+y_{22})\left[s_1{s}_2{s}_1\right]\\ \left[s_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& \{(y_{11}+y_{21})\left[1\right]-(y_{11}+y_{21})\left[s_1\right]-(y_{11}+y_{21})\left[s_2\right]\}\\ & & +y_{11}\left[s_1{s}_2\right]+(y_{11}+y_{21})\left[s_2{s}_1\right]-{\alpha }_{11}\left[s_1{s}_2{s}_1\right]\\ \\ {\left[s_2{s}_1{s}_2\right]}^2& =& -{\alpha }_{21}(y_{21}+y_{42})\left[s_2\right]+({\alpha }_{11}{\alpha }_{21}{y}_{31}+{\alpha }_{21}{\alpha }_{31}{y}_{10})\left[s_1{s}_2\right]-{\alpha }_{10}{\alpha }_{21}{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \left[s_2{s}_1{s}_2\right]\left[s_1{s}_2{s}_1{s}_2\right]& =& y_{53}\left[s_2\right]-({\alpha }_{21}{y}_{31}+{\alpha }_{11}{\alpha }_{21}{\alpha }_{32}{y}_{21})\left[s_1{s}_2\right]+{\alpha }_{21}{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \left[s_2{s}_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& \{-(y_{51}+y_{52}+{\alpha }_{31}{y}_{10})\left[1\right]\}+\left(y_{41}\{+{\alpha }_{31}{y}_{10}\}\right)\left[s_1\right]\\ & & +(y_{42}+y_{52}\{+{\alpha }_{31}{y}_{10}\})\left[s_2\right]-({\alpha }_{11}{y}_{31}+{\alpha }_{31}{y}_{10})\left[s_1{s}_2\right]\\ & & -{\alpha }_{31}{y}_{10}\left[s_2{s}_1\right]+{\alpha }_{10}{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \left[s_2{s}_1{s}_2\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& \{(y_{31}+y_{32}+y_{42})\left[1\right]-(y_{31}+y_{32})\left[s_1\right]-(y_{31}+y_{32}+y_{42})\left[s_2\right]\}\\ & & +(y_{31}+y_{32})\left[s_1{s}_2\right]+y_{31}\left[s_2{s}_1\right]-{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \left[s_2{s}_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& y_{31}\left[s_1{s}_2\right]-{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \\ {\left[s_1{s}_2{s}_1{s}_2\right]}^2& =& \{-y_{42}\left[s_2\right]\}+(y_{31}+y_{42}\{+{\alpha }_{01}{y}_{21}+{\alpha }_{31}{y}_{11}\})\left[s_1{s}_2\right]\\ & & -({\alpha }_{01}(y_{11}+y_{21})+{\alpha }_{31}(y_{01}+y_{11}))\left[s_2{s}_1{s}_2\right]+{\alpha }_{01}{\alpha }_{11}\left[s_1{s}_2{s}_1{s}_2\right]\\ \left[s_1{s}_2{s}_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1\right]& =& \{(y_{21}+y_{31}+y_{32}+y_{42}+{\alpha }_{11})\left[1\right]-(y_{21}+y_{31}+y_{32}+{\alpha }_{11})\left[s_1\right]-(y_{21}+y_{31}+y_{32}+y_{42}+{\alpha }_{11})\left[s_2\right]\}\\ & & +(y_{31}+y_{42}\{+{\alpha }_{11}\})\left[s_1{s}_2\right]+(y_{21}+y_{31}\{+{\alpha }_{11}\})\left[s_2{s}_1\right]-{\alpha }_{11}\left[s_1{s}_2{s}_1\right]-{\alpha }_{31}\left[s_2{s}_1{s}_2\right]\\ \left[s_1{s}_2{s}_1{s}_2\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& \{-(y_{01}+y_{11}+y_{21}+y_{22}+y_{32})\left[1\right]+(y_{01}+y_{11}+y_{21}+y_{22})\left[s_1\right]\\ & & +(y_{01}+y_{11}+y_{21}+y_{22}+y_{32})\left[s_2\right]-(y_{01}+y_{11}+y_{21}+y_{22})\left[s_1{s}_2\right]\\ & & -(y_{01}+y_{11}+y_{21})\left[s_2{s}_1\right]\}+y_{01}\left[s_1{s}_2{s}_1\right]+(y_{01}+y_{11}+y_{21})\left[s_2{s}_1{s}_2\right]-{\alpha }_{01}\left[s_1{s}_2{s}_1{s}_2\right]\\ \left[s_1{s}_2{s}_1{s}_2\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& \{-y_{21}\left[s_1{s}_2\right]\}+(y_{11}+y_{21})\left[s_2{s}_1{s}_2\right]-{\alpha }_{11}\left[s_1{s}_2{s}_1{s}_2\right]\\ \\ {\left[s_2{s}_1{s}_2{s}_1\right]}^2& =& \{-y_{52}\left[s_1\right]+(y_{42}+y_{52})\left[s_2{s}_1\right]\}-({\alpha }_{11}{y}_{31}+{\alpha }_{31}{y}_{10})\left[s_1{s}_2{s}_1\right]\\ & & +{\alpha }_{10}{\alpha }_{31}\left[s_2{s}_1{s}_2{s}_1\right]\\ \left[s_2{s}_1{s}_2{s}_1\right]\left[s_1{s}_2{s}_1{s}_2{s}_1\right]& =& \{y_{42}\left[s_1\right]-(y_{31}+y_{41})\left[s_2{s}_1\right]\}+(y_{31}+y_{32})\left[s_1{s}_2{s}_1\right]\\ & & -{\alpha }_{31}\left[s_2{s}_1{s}_2{s}_1\right]\\ \left[s_2{s}_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& \{-y_{10}\left[1\right]+y_{10}\left[s_1\right]+y_{10}\left[s_2\right]-y_{10}\left[s_1{s}_2\right]-y_{10}\left[s_2{s}_1\right]\}\\ & & +y_{10}\left[s_1{s}_2{s}_1\right]+y_{10}\left[s_2{s}_1{s}_2\right]-{\alpha }_{10}\left[s_2{s}_1{s}_2{s}_1\right]\\ \\ {\left[s_1{s}_2{s}_1{s}_2{s}_1\right]}^2& =& \{-y_{32}\left[s_1\right]+(y_{22}+y_{32})\left[s_2{s}_1\right]-(y_{11}+y_{21}+y_{22})\left[s_1{s}_2{s}_1\right]\}\\ & & +(y_{01}+y_{11}+y_{21})\left[s_2{s}_1{s}_2{s}_1\right]-{\alpha }_{01}\left[s_1{s}_2{s}_1{s}_2{s}_1\right]\\ \left[s_1{s}_2{s}_1{s}_2{s}_1\right]\left[s_2{s}_1{s}_2{s}_1{s}_2\right]& =& \{\left[1\right]-\left[s_1\right]-\left[s_2\right]+\left[s_1{s}_2\right]+\left[s_2{s}_1\right]-\left[s_1{s}_2{s}_1\right]-\left[s_2{s}_1{s}_2\right]\}\\ & & +\left[s_1{s}_2{s}_1{s}_2\right]+\left[s_2{s}_1{s}_2{s}_1\right]\\ {\left[s_2{s}_1{s}_2{s}_1{s}_2\right]}^2& =& y_{10}\left[s_1{s}_2{s}_1{s}_2\right]-{\alpha }_{10}\left[s_2{s}_1{s}_2{s}_1{s}_2\right]\end{array}$$

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.

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