Ultra-Discretization of $D_6^{(1)}$- Geometric Crystal at the spin node

Let $\mathfrak{g}$ be an affine Lie algebra with index set $I = {0, 1, 2, \cdots , n}$\mathrm{.}Itisconjecturedin\text{cite}KNOthatforeachDynkinnode$k \in I \setminus {0}$theaffineLiealgebra$\mathfrak g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of a coherent family of perfect crystals for the Langland dual ${\mathfrak{g}}^L$. In this paper we show that at the spin node $k=6$, the family of perfect crystals given in \cite{KMN2} form a coherent family and show that its limit $B^{6,\mathrm{\infty }}$ is isomorphic to the ultra-discretization of the positive geometric crystal we constructed in \cite{MP} for the affine Lie algebra $D_6^{(1)}$ which proves the conjecture in this case.