the references cited in the introduction or books on quantum groups [@MR1881971; @MR1359532] for the basic concepts on quantum groups and crystal bases.

Let us first fix the basic notation.

-   $I=\{1,2\}$ : index set for $G_2$-type.

-   $A=(a_{ij})_{i,j\in I}$ : Cartan matrix of type $G_2$.

-   $\ali$, $\La_i$ $(i\in I)$ : simple root, fundamental weight.

-   $\Pi =\{\ali\vert i\in I\}$ : the set of simple root.

-   $P =\op_{i\in I} \Z\La_i$ : weight lattice.

-   $\uq(G_2)$ : quantum group for $G_2$.

-   $\uq^-(G_2)$ : subalgebra of $\uq(G_2)$ generated by $f_i$ ($i\in I$).

-   $\fit, \eit$ : Kashiwara operators.

-   $\hwc(\infty)$ : crystal base of $\uq^-(G_2)$.

Throughout this paper, a $U_q(G_2)$-crystal will refer to a (abstract) crystal associated with the Cartan datum $(A,\Pi,P)$.

Nakajima monomials {#Nakajima monomials}
------------------

We now recall the set of monomials and its crystal structure discovered by Nakajima [@MR1988990] and also recall their extension introduced in [@lee]. Both of these sets were defined for all symmetrizable Kac-Moody algebras, but we shall restrict ourselves to the $G_2$ case in this paper.

Let $\Me$ be a certain set of formal monomials in the variables $Y_i(m)$ ($i\in I$, $m\in\Z$). More explicitly, $$\Me =
\left\{\prod_{(i,m)\in I\times\Z}
        {Y_i(m)}^{y_i(m)}
        \Big\vert
\begin{aligned}
 & 