monomials\].

The monomial $M_{(p_1,p_2;r;\infty)}$ with $\operatorname{\tilde{wt}}(M_{(p_1,p_2;r;\infty)})=\sum_{i\in I} (p_i,0)\La_i$, also satisfies the condition for a monomial $M$ mentioned in the conjecture.

It should be clear from the proof of Proposition \[cor4\], that in developing any theory for ${\M(p_1,p_2;r;\infty)}$ the actual values of integer $r$ or $(p_1,p_2)$ will not be very important. Arguments made for any set of such values can easily be adapted to applied to other set of such values. Hence, we shall concentrate on the theory for $\M(\infty)$ only.

Now, we will show that $\M(\infty)$ is a new realization of $\hwc(\infty)$ by giving a crystal isomorphism. Recall from Theorem \[HL\] that the set $\T(\infty)$ gives a realization of the crystal $\hwc(\infty)$.

Here is one of our two main realization theorems.

\[mainreal5\] There exists a $\uq(G_2)$-crystal isomorphism $$\B(\infty) \overset{\sim} \longrightarrow \T(\infty)
\overset {\sim} \longrightarrow \M(\infty)$$ which maps $T_\infty$ to $M_{\infty}$.

We define a canonical map $\Theta:\T(\infty)\rightarrow\M(\infty)$ by setting, for each tableau $T\in\T(\infty)$ with second row consists of $b^2_3$-many $3$-boxes and just one $2$-box, and with first row consists of $b^1_j$-many $j$-boxes, for each $j\succ 1$, and $(b^2_3+2)$-many $1$-boxes, $\Theta(T)=M$, where $$\begin{aligned}
      M=& X_1(-1)^{(2,-b^1_2-b^1_3-b^1_0-
                      b^1_{\bar3}-b^1_{\bar2}-b^1_{\bar1})}
          X_2(-1)^{(0,b_2^1)}
          X_3(-1)^{(0,b_3^1)}\\
        & \cdot
          X_0(-1)^{(0,b_0^1)}
          X_{\bar3}(-1)^{(0,b_{\bar3}^1)}
          X_{\bar2}(-1)^{(0,b_{\bar2}^1)}
          X_{\bar1}(-1)^{(0,b_{\bar1}^1)}\\
        & \cdot
          X_2(-2)^{(1,-b_3^2)}
          X_3(-2)^{(0,b_3^2)}\in\M(\infty).
\end{aligned}$$ It is obvious 