and it will be equipped with an ordering $\prec$, as given in [@KM].

$$J=\{1\prec{2}\prec{3}\prec{0}\prec\bar{3}\prec\bar{2}\prec\bar{1}\}.$$

1.  A semi-standard tableau $T$ of shape $\la\in P^+$, equivalently, an element of an irreducible highest weight crystal $\B(\la)$ for the $G_2$ type, is *large* if it consists of $2$ non-empty rows, and if the number of $1$-boxes in the first row is strictly greater than the number of all boxes in the second row and the second row contains at least one $2$-box.

2.  A large tableau $T$ is *marginally* large, if the number of $1$-boxes in the first row of $T$ is greater than the number of all boxes in the second row by exactly one and the second row of $T$ contain one $2$-box.

In Figure \[tbl:1\], we give examples of semi-standard tableaux. The one on the left is large, the one on the middle is marginally large, and the one on the right is not large.

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We denote by $\T(\infty)$ the set of all marginally large tableaux. The marginally large tableau whose $i$-th row consists only of $i$-boxes ($i\in I$) is denoted by $T_{\infty}$.

The set $\Tinf$, consists of all tableaux of the following form. 