-2.5)(9 -7.5)(1 -1)(-8 -12.6)[$\vdots$]{}(9 -12.6)[$\vdots$]{}(1 -12.6)[$\vdots$]{}(0 0)[${Y_1(-1)}^{(1,0)}$]{} (0 -0.7)[$\cdot {Y_2(-2)}^{(1,0)}$]{} (-4 -3)[${Y_1(-1)}^{(1,-1)}{Y_1(0)}^{(0,-1)}$]{} (-4 -3.7)[$\cdot {Y_2(-2)}^{(1,0)}{Y_2(-1)}^{(0,1)}$]{}

(4 -3)[${Y_1(-1)}^{(1,3)}$]{} (4 -3.7)[$\cdot {Y_2(-2)}^{(1,-1)}{Y_2(-1)}^{(0,-1)}$]{} (-9 -6)[${Y_1(-1)}^{(1,-2)}{Y_1(0)}^{(0,-2)}$]{} (-9 -6.7)[$\cdot {Y_2(-2)}^{(1,0)}{Y_2(-1)}^{(0,2)}$]{}

(-2.5 -6)[${Y_1(-1)}^{(1,-1)}{Y_1(0)}^{(0,2)}$]{} (-2.5 -6.7)[$\cdot {Y_2(-2)}^{(1,0)}{Y_2(0)}^{(0,-1)}$]{}

(3.5 -6)[${Y_1(-1)}^{(1,2)}{Y_1(0)}^{(0,-1)}$]{} (3.5 -6.7)[$\cdot {Y_2(-2)}^{(1,-1)}$]{}

(9 -6)[${Y_1(-1)}^{(1,6)}$]{} (9 -6.7)[$\cdot {Y_2(-2)}^{(1,-2)}{Y_2(-1)}^{(0,-2)}$]{} (-9 -9)[${Y_1(-1)}^{(1,-3)}{Y_1(0)}^{(0,-3)}$]{} (-9 -9.7)[$\cdot {Y_2(-2)}^{(1,0)}{Y_2(-1)}^{(0,3)}$]{}

(-7.5 -11)[${Y_1(-1)}^{(1,-2)}{Y_1(0)}^{(0,1)}$]{} (-7.5 -11.7)[$\cdot {Y_2(-2)}^{(1,0)}{Y_2(-1)}^{(0,1)}
                       {Y_2(0)}^{(0,-1)}$]{}

(-2.5 -9)[${Y_1(-1)}^{(1,-1)}{Y_1(0)}^{(0,1)}$]{} (-2.5 -9.7)[$\cdot {Y_1(1)}^{(0,-1)}{Y_2(-2)}^{(1,0)}$]{}

(1 -11)[${Y_1(-1)}^{(1,2)}{Y_1(0)}^{(0,2)}{Y_2(-2)}^{(1,-1)}$]{} (1 -11.7)[$\cdot {Y_2(-1)}^{(0,-1)}{Y_2(0)}^{(0,-1)}$]{}

(3.8 -9)[${Y_1(-1)}^{(1,1)}{Y_1(0)}^{(0,-2)}$]{} (3.8 -9.7)[$\cdot {Y_2(-2)}^{(1,-1)}{Y_2(-1)}^{(0,1)}$]{}

(9 -11)[${Y_1(-1)}^{(1,5)}{Y_1(0)}^{(0,-1)}$]{} (9 -11.7)[$\cdot {Y_2(-2)}^{(1,-2)}{Y_2(-1)}^{(0,-1)}$]{}

(10.5 -9)[${Y_1(-1)}^{(1,9)}$]{} (10.5 -9.7)[$\cdot {Y_2(-2)}^{(1,-3)}{Y_2(-1)}^{(0,-3)}$]{} (0 0) (-1.8 -1.5)[$1$]{} (1.8 -1.5)[$2$]{} (0 -2.7) (-5.7 -2.1)[$1$]{} (-2.2 -2.1)[$2$]{} (2.2 -2.1)[$1$]{} (6.1 -2.1)[$2$]{} (0 -6.2) (-9 -1.7)[$1$]{} (-6 -1.7)[$2$]{} (-2.9 -1.7)[$1$]{} (0.2 -1.7)[$2$]{} (3.1 -1.7)[$1$]{} (5.6 -1.7)[$2$]{} (7.5 -1.7)[$1$]{} (9.8 -1.7)[$2$]{} (0 -13.3)

We now introduce new expressions for elements of $\M(\infty)$. First, we introduce the following notation.

For $u\in\Z_{\ge 0}$, $v\in\Z$, and $m\in\Z$, we use the notation $$\begin{aligned}
\label{order}
{X_j(m)}^{(u,v)}&=
\begin{cases}
{Y_j(m)}^{(u,v)}{Y_{j-1}(m+1)}^{(-u,-v)} &\text{for $j=1,2$},\\
{Y_1(m+1)}^{(2u,2v)}{Y_2(m+1)}^{(-u,-v)} &\text{for $j=3$},
\end{cases}\\
{X_0(m)}^{(u,v)}
&={Y_1(m+1)}^{(u,v)}{Y_1(m\!+\!2)}^{(-u,-v)},\\
{X_{\bar{j}}(m)}^{(u,v)}&=
\begin{cases}
{Y_{j-1}(m+(4-j))}^{(u,v)}{Y_j(m+(4-j))}^{(-u,-v)} &\text{for $j=1,2$},\\
{Y_2(m+1)}^{(u,v)}{Y_1(m+2)}^{(-2u,-2v)} &\text{for $j=3$}.
\end{cases}\end{aligned}$$ Here, we set $Y_0(k)^{(u,v)}=1$.

Using the above notation, we may write $$\begin{aligned}
A_1(m)&= X_1(m)^{(0,1)}X_2(m)^{(0,-1)}\\
      &= X_3(m-1)^{(0,1)}X_0(m-1)^{(0,-1)}\\
      &= X_0(m-1)^{(0,1)}X_{\bar3}(m-1)^{(0,-1)}\\
      &= X_{\bar2}(m-2)^{(0,1)}X_{\bar1}(m-2)^{(0,-1)},\\
A_2(m) &= X_2(m)^{(0,1)}X_3(m)^{(0,-1)}\\
       &= X_{\bar3}(m-1)^{(0,1)}X_{\bar2}(m-1)^{(0,-1)}.\end{aligned}$$ This is very useful when computing Kashiwara action on monomials written in terms of $X_j(m)^{(u,v)}$ or 