is a marginally large tableau, then we define $\eit T$ to be $T'$.

    -   If $T'$ is large but not marginally large, then we define $\eit T$ to be the large tableau obtained by removing the column containing the changed box. It will be of $i$ rows and have a $k$-box at the $k$-th row for $1\leq k \leq i$.

5.  If there is no $1$ in the $i$-signature of $T$, we define $\eit T=0$.

Let $T$ be a tableau in $\T(\infty)$ with the second row consisting of $b^2_3$-many $3$s, one $2$ and the first row consisting of $b^1_j$-many $j$s ($1\prec j\preceq \bar 1$), $(b^2_3+2)$-many $1$s. We define the maps $\operatorname{wt}: \T(\infty)
\rightarrow P$, $\vphi_i, \veps_i : \T(\infty) \rightarrow \Z$ by setting $$\begin{aligned}
\operatorname{wt}(T)      &=(-b_2^1-b_3^1-2b_0^1-3b_{\bar 3}^1-3b_{\bar 2}^1
              -4b_{\bar 1}^1)\alpha_1 \label{same}\\
            &\quad+(-b_3^1-b_0^1-b_{\bar 3}^1-2b_{\bar 2}^1
              -2b_{\bar 1}^1-b_3^2)\alpha_2,\notag\\
\veps_i(T)  &=\text{the number of $1$s
             in the $i$-signature of $T$},\label{same2}\\
\vphi_i(T)  &=\veps_i(T)+\langle h_i,\operatorname{wt}(T)\rangle.\label{same3}\end{aligned}$$

[@HL]\[HL\] The operator given by equations  to , with Kashiwara operators define a crystal structure on $\Tinf$. And the crystal $\T(\infty)$ is isomorphic to $\B(\infty)$ as a $U_q(G_2)$-crystal.

Monomial description of $\B(\infty)$ {#Extended monomial description}
====================================

We give a new realization of the crystal $\B(\infty)$, for $G_2$-type, in terms of extended monomials.

