\htext(2.5 0.5){$3$}
\end{texdraw}}\in\T(\infty).$$

Note that Remark \[wkwmd3\] and Remark \[final2\] provide evidence supporting the conjecture introduced in Section \[Nakajima monomials\], for the $G_2$ case.

Cliff’s description of $\B(\infty)$ {#Cliff's description}
===================================

Let us recall the abstract crystal $\B_i=\{b_i(k)\vert k\in{\bf Z}\}$ introduced in [@MR1240605] for each $i\in I$. It has the following maps defining the crystal structure. $$\begin{aligned}
{3}
\operatorname{wt}b_i(k)      &=k\alpha_i,\\
\vphi_i(b_i(k)) &=k,         & \veps_i(b_i(k))&=-k,\\
\vphi_i(b_j(k)) &=-\infty,   & \veps_i(b_j(k))&=-\infty,
& &\text{ for }i\ne j,\\
\fit(b_i(k))    &=b_i(k-1),  \quad & \eit(b_i(k))&=b_i(k+1),\\
\fit(b_j(k))    &=0,         & \eit(b_j(k))&=0, & &\text{ for
}i\ne j.\end{aligned}$$ From now on, we will denote the element $b_i(0)$ by $b_i$. We next cite the tensor product rule on crystals.

 \[propq\] Let $\B^k(1\le k\le n)$ be crystals with $b^k\in \B^k$. We set $$\label{kashi}
a_k=\veps_i(b^k)-\sum_{1\le v<k} \langle h_i,\operatorname{wt}(b^v)\rangle.$$ Then we have

1.  $\eit(b^1\otimes\cdots\otimes
          b^n)=b^1\otimes\cdots\otimes b^{k-1}\otimes
          \eit b^k\otimes b^{k+1}\otimes\cdots\otimes b^n$\
    if $a_k>a_v$ for $1\le v<k$ and $a_k\ge a_v$ for $k<v\le n$,

2.  $\fit(b^1\otimes\cdots\otimes
          b^n)=b^1\otimes\cdots\otimes b^{k-1}\otimes
          \fit b^k\otimes b^{k+1}\otimes\cdots\otimes b^n$\
    if $a_k\ge a_v$ for $1\le v<k$ and $a_k>a_v$ for $k<v\le n$.

Kashiwara has shown [@MR1240605] the existence of an injective strict crystal morphism $$\Psi:\B(\infty)\to\B(\infty)\otimes
     \B_{i_k}\otimes\B_{i_{k-1}}\otimes\cdots\otimes\B_{i_1}$$ which sends the highest weight element $u_\infty$ to $u_\infty\otimes
b_{i_k}\otimes\cdots\otimes b_{i_1}$, for any sequence $S={i_1,i_2,\cdots,i_k}$ of numbers in the index set 