to be true [@lee].

*If a monomial $M\in\Me$ of $\operatorname{\tilde{wt}}(M)=\sum_i(p_i,0)\La_i$, where each $p_i$ is a positive integer, satisfies $\eit(M)=0$ for all $i\in I$, then the connected component of $\Me$ containing $M$ is isomorphic to $\B(\infty)$ as a $\uq(\g)$-crystal.*

In [@lee; @lee2], result for $\B(\infty)$ of type $\an$ and $\anone$ was given as evidence supporting this conjecture. In the next section, we will give a concrete listing of elements containing a weight zero vector $M\in\Me$ mentioned in the above theorem, for the case of $G_2$. We shall show this set to be a crystal and give an isomorphism between the crystal and crystal $\B(\infty)$. This is another result supporting the above conjecture.

Young tableaux {#Young tableaux}
--------------

In this section, we recall a Young tableaux description for the crystal $\B(\infty)$ over type $G_2$ introduced in [@HL].

For the $G_2$-type, we shall take the Young tableau realization of highest weight crystal $\B(\la)$ given in [@KM] as the definition of semi-standard tableaux. Since the work is a rather well known result, we refer readers to the original papers and shall not repeat the complicated definition here. The alphabet to be used inside the boxes constituting the Young tableaux will be denoted by $J$, 