terms $X_j(m)^{(u,v)}$ and $X_{\bar
j}(m)^{(u,v)}$. The product form  was chosen because it allows us to relate monomials of the set $\M(\infty)$ to tableaux in $\T(\infty)$ directly.

Now, we translate the Kashiwara actions , into a form suitable for the new monomial expression of $\M(\infty)$.

\[anaysiskashi2\] The Kashiwara operator action on $\Me$ may be rewritten as given below for elements $$~\label{equ:jin8}
\begin{aligned}
      M=& X_1(-1)^{(2,-b^{-1}_2-b^{-1}_3-b^{-1}_0-
                      b^{-1}_{\bar3}-b^{-1}_{\bar2}-b^{-1}_{\bar1})}
          X_2(-1)^{(0,b_2^{-1})}
          X_3(-1)^{(0,b_3^{-1})}\\
        & \cdot
          X_0(-1)^{(0,b_0^{-1})}
          X_{\bar3}(-1)^{(0,b_{\bar3}^{-1})}
          X_{\bar2}(-1)^{(0,b_{\bar2}^{-1})}
          X_{\bar1}(-1)^{(0,b_{\bar1}^{-1})}\\
        & \cdot
          X_2(-2)^{(1,-b_3^{-2})}
          X_3(-2)^{(0,b_3^{-2})}
\end{aligned}$$ of $\M(\infty)$. Elements of the above form constitutes $\M(\infty)$ and this set is closed under Kashiwara operator actions.

Kashiwara actions $\tilde f_1$ and $\tilde
e_1$:

-   Consider the following ordered sequence of some components of $M$. $${X_{\bar1}(-1)}^{(0,b_{\bar1}^{-1})}
          {X_{\bar2}(-1)}^{(0,b_{\bar2}^{-1})}
          {X_{\bar3}(-1)}^{(0,b_{\bar3}^{-1})}
          {X_0(-1)}^{(0,b_0^{-1})}
          {X_3(-1)}^{(0,b_3^{-1})}
          {X_2(-1)}^{(0,b_2^{-1})}.$$

-   Under each of the components $${X_{\bar1}(-1)}^{(0,b_{\bar1}^{-1})},
          {X_0(-1)}^{(0,b_0^{-1})},
          {X_2(-1)}^{(0,b_2^{-1})},$$ given in the above sequence, write $b_j^{-1}$-many $1$’s and under ${X_{\bar3}(-1)}^{(0,b_{\bar3}^{-1})}$, write $(2b_{\bar 3}^{-1})$-many $1$’s. Also, under each of the components $${X_{\bar2}(-1)}^{(0,b_{\bar2}^{-1})},{X_0(-1)}^{(0,b_0^{-1})},$$ write $b_j^{-1}$-many $0$’s and under ${X_3(-1)}^{(0,b_3^{-1})}$, write $(2b_3^{-1})$-many $0$’s.

-   From this sequence of $1$’s and $0$’s, successively cancel out each $(0,1)$-pair to obtain a sequence of $1$’s followed by $0$’s reading from left to right. This remaining $1$ and $0$ sequence is called the .

-   Depending on the 